1lP7nIG

IN MEMORY OF ... 

Professor Emeritus Theodore S. 

Papatheodorou, a prominent 

faculty member of the Department 

of Computer Engineering & 

Informatics at the University of 

Patras at Patras-Greece, died on 

December 10, 2012. Theo, as we all 

called him, was an internationally 

renowned educator and 

researcher. He was a leading and 

recognized expert in Numerical 

Analysis. 

He was born in Patras-Greece in 

1945 and graduated with honors Theodore S. Papatheodorou 

from the University of Athens with a 

1945 - 2012 

B.S. degree in Mathematics in 1968. 

He entered the graduate program at 

Purdue University in 1969, and 

subsequently earned his M.S. in Applied Mathematics and in Civil 

Engineering, and Ph.D. in Science in 1971, 1975 and 1973, 

respectively under the guidance of Professor Robert Lynch. He 

joined Clarkson University in 1976 and served there until 1984 

when he was invited to join the newly formed Greek Ministry of 

Research & Development as scientific advisor. With his efforts 

the Computer Technology Institute was established and he 

served as its first Director until 1990. His numerous 

accomplishments over the years contributed immensely to 

building the reputation of the Institute as a leader in 

Computing research. Meanwhile he joined the faculty of the 

Department of Computer Engineering & Informatics at the 

University of Patras where he established the High Performance 

Information Systems Lab (HPCLab) and also served as Dean, 

Head, member of the University Senate, Director of graduate 

studies, etc. 

His research contributions covered various aspects of 

Computational Science and Computer Engineering. Seminal 

contributions include research and development work on system 

and application software for parallel and distributed 

computing, scientific computing, web and multimedia 

applications, 3D virtual reconstruction of monuments and other 

applications for culture and education, while, at the early years, 

his contribution in the development of a general methodology 

for generating arbitrary high order finite difference methods 

has influenced the invention of the so called HODIE methods 

and contributed significant fast methods that became part of 

ACM algorithms and ELLPACK libraries. He published over 150 

papers, including a book and he supervised and guided the 

doctoral work of fifteen students.

Theo was member of many international scientific committees 

and was awarded several prizes and awards. 

Theo taught a wide range of courses for his entire academic 

career. He was a gifted teacher with selfless service and with 

keen interest for his students. He influenced with his academic 

presence the lives of a great number of students and 

colleagues. 

Professor Emeritus Theodore S. Papatheodorou will be 

remembered as a gentleman, a scholar and an unassuming 

researcher. On a personal note, being his advisee, collaborator 

and close friend for over thirty years, he will always be 

remembered as “MY TEACHER” and beloved friend. 

Yiannis G. Saridakis 

Technical University of Crete.

NumAn2014 Book of Abstracts iv 

numan2014.amcl.tuc.gr — Conference on Numerical Analysis, Chania, Greece, Sept 2-5, 2014

Contents 

Invited speakers 

Houstis N. E. 

Remembering Theo Papatheodorou . . . . . . . . . . . . . . . . . . . . . . . . . 1 

Bai Z-Z. 

Scalable and Fast Iteration Methods for Complex Linear Systems . . . . . . . . 2 

Fokas S. A. 

The interplay of the concrete and general: from PDEs to medical imaging . . . 3 

Iserles A. 

Fast computation of there semiclassical Schrödinger equation . . . . . . . . . . 4 

Noutsos D. 

Perron-Frobenius Theory Some Extensions and Applications . . . . . . . . . . 5 

Vrahatis N. M. 

Sign Methods for Imprecise Problems . . . . . . . . . . . . . . . . . . . . . . . 6 

Contributed speakers 

Abhulimen E.C. 

A new class of second derivative methods for numerical integration of stiff initial 

value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 

Agaoglou M., Rothos V.M. and Susanto H. 

Homoclinic chaos in a pair of parametrically-driven coupled SQUIDs . . . . . . 8 

Alefragis P., Spyrou A. and Likothanassis S. 

Application of a hybrid parallel Monte Carlo PDE Solver on rectangular multidomains 

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

Antoniadou I. K. and Voyatzis G. 

Continuation and stability deduction of resonant periodic orbits in three dimensional 

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 

Antonopoulos C. and Bellas N. 

SOpenCL: An Infrastructure for Transparently Integrating FPGAs in Heterogeneous, 

Accelerator-Based Systems . . . . . . . . . . . . . . . . . . . . . . . . 11 

Antonopoulos C., Maroudas M. and Vavalis M. 

Software Platforms for Multi-Domain Multi-Physics Simulations . . . . . . . . 12 

Antonopoulos C. D. and Dougalis A. V. 

Error estimates for the standard Galerkin-Finite Element method for the shallow 

water equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 

Antonopoulos G. C., Srivastava S., Pinto S. S., Baptista S. M. 

Do Brain Networks Evolve by Maximizing Flow of Information? . . . . . . . . 14 

Antonopoulou D. 

Finite elements for a class of nonlinear stochastic pdes from phase transition 

problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 

v

NumAn2014 Book of Abstracts vi 

Antunes R. S. P. 

Numerical Solution of the Magnetic Laplacian Eigenvalue Problem using Radial 

Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 

Árciga A. M. P., Ariza H. F. J. and Sánchez O. J. 

Stochastic Riez-Fractional Partial Differential Equation with White Noise on 

the Half-Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 

Ashton A. 

Functional Analytic Framework of the Fokas Method for Elliptic Boundary Value 

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

Athanasakis E. I., Papadopoulou P. E. and Saridakis G. Y. 

Discontinuous Hermite Collocation and Runge-Kutta schemes for multi-domain 

linear and non-linear brain tumor invasion models . . . . . . . . . . . . . . . . 19 

Athanasakis E. I., Vilanakis D. N., Mathioudakis N. E., Papadopoulou P. E. and Saridakis 

G. Y. 

Solving discontinuous collocation equations for a class of brain tumor models on 

GPUs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 

Atisattapong W. and Maruphanton P. 

Obviating the Bin Width Effect of the 1/t Algorithm for Multidimensional Numerical 

Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 

Bacigaluppi P. and Ricchiuto M. 

A 1D stabilized finite element model for non-hydrostatic wave breaking and run-up 22 

Barrera D., Ibáñez J. M., Roldán M. A., Roldán B. J. and Yáñez R. 

Parameter determination in MOSFETs transitors based on Discrete Orthogonal 

Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

Bebiano N. 

Spectral inclusion regions for matrix pencils . . . . . . . . . . . . . . . . . . . . 24 

Belhah Z., Kouibia A., Pasadas M. 

Variational iterative method for solving a nonlinear partial differential equation. 

Application to the two-dimensional Bratu’s problem . . . . . . . . . . . . . . . 25 

Bellas N. and Antonopoulos C. 

Significance-Based Computing for Reliability and Power Optimization . . . . . 26 

Bellavia S., Governi L., Papini A. and Puggelli L. 

Quadratic Penalty Methods for Shape from Shading . . . . . . . . . . . . . . . 27 

Benmir M., Bessonov N., Boujena S. and Volpert V. 

Multi-scale hybrid model of cell differentiation propagation as traveling waves . 28 

Benzi M., Duff S. I. and Guo X-P. 

Preconditioned derivative-free globally convergent Newton-GMRES methods for 

large sparse nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

Berisha F., Sadiku M. and Berisha N. 

Using an Euler type transform for accelerating convergence of series . . . . . . 30 

Bobolakis D.E., Delis A.I. and Mathioudakis E.N. 

Efficient Solution of the Two-Dimensional Shallow-Water Equations using GPUs 31 

Bountis T., Antonopoulos C. and Skokos H. 

Complex Statistics and Diffusion in Nonlinear Disordered Particle Chains . . . 32 

Bratsos G. A. 

A modified predictor-corrector method for the generalized BurgersHuxley equation 33 

Bueno I. M., Curlett K. and Furtado S. 

Structured Strong Linearizations obtained from Fiedler Pencils with Repetition 34 

Burde I. G., Nasibullayev Sh. I. and Zhalij A. 

Unified Semi-Analytical, Semi-Numerical Approach to Stability Analysis of Nonparallel 

Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 


NumAn2014 Book of Abstracts vii 

Bu Y-M. and Carpentieri B. 

A recursive multilevel approximate inverse-based preconditioner for solving general 

linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 

Carpio J, Prieto L. J. 

A local anisotropic adaptive algorithm to solve time-dependent dominated convection 

problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 

Charalampaki E. N. and Mathioudakis N. E. 

CPU-GPU computations for MultiGrid techniques coupled with Fourth-Order 

Compact Discretizations for Isotropic and Anisotropic Poisson problems . . . . 38 

Chaturantabut S. 

Nonlinear Model Reduction with Localized Basis for Two-Phase Miscible Flow 

in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 

Chatzipantelidis P. 

On positivity preservation for finite element based methods for the heat equation 40 

Chollom P. J. and Kumleng M. G. 

Block Hybrid Numerical Integrators for the Solution of Stiff Equations . . . . . 41 

Dang D.-M., Christara C. and Jackson K. 

Efficient GPU pricing of interest rate derivatives: PDE formulation and ADI 

methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 

Christodoulidi H., Cirto L., Bountis T. and Tsallis C. 

Dynamical and statistical behavior of the Fermi-Pasta-Ulam model with longrange 

interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 

Crooks K. 

Two numerical implementations of the Fokas method for elliptic equations in a 

polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 

Crowdy G. D. and Luca E. 

Solving Wiener-Hopf problems without kernel factorisation . . . . . . . . . . . 45 

De Bonis M. C. and Occorsio D. 

Numerical evaluation of hypersingular integrals on the semiaxis . . . . . . . . . 46 

Demetriou C. I. 

A Characterization Theorem for the Discrete Best L 1 Monotonic Approximation 

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 

Dieci L., Elia C. and Lopez L. 

Numerical techniques for sliding motion in Filippov discontinuous systems . . . 48 

El-Gindy T.M., Salim M.S. and Ahmed A.I. 

A new filled function method applied to unconstrained global optimization . . 49 

Fernández L., Fortes A. M. and Rodríguez M. L. 

Multiresolution analysis for 3D scattered data sets . . . . . . . . . . . . . . . . 51 

Fevgas A., Tsompanopoulou P. and Bozanis P. 

Exploring the Performance of Out-of-Core Linear Algebra Algorithms in Flash 

based Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 

Filelis-Papadopoulos K. C. and Gravvanis A. G. 

A comparative study on the effect of the ordering schemes for solving sparse 

linear systems, based on factored approximate sparse inverse matrix methods . 53 

Flouri E., Dougalis V. and Synolakis C. 

Tsunami hazard and inundation for the northern coast of Crete . . . . . . . . . 54 

Fokas S. A. and Kalimeris K. 

Eigenvalues and eigenfunctions for the Laplace Operator . . . . . . . . . . . . . 55 

Fortes A. M., González P., Palomares A. and Pasadas M. 

Filling holes with geometric constraints . . . . . . . . . . . . . . . . . . . . . . 56 

Fortes A. M., González P., Palomares A. and Pasadas M. 

Matrix-free resolution of PDEs using the Powell-Sabin FE . . . . . . . . . . . . 57 


NumAn2014 Book of Abstracts viii 

Gaitani M., Kazolea M. and Delis A. 

Numerical Solution for Sparse Linear Systems that occur from the discretization 

of Boussinesq-type equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

Georgieva I., Hofreither C. and Uluchev R. 

Approximations Using Radon Projection Data in the Unit Disc . . . . . . . . . 59 

González-Pinto S., Hernández-Abreu D. 

Splitting methods based on Approximate Matrix Factorization and Radau-IIA 

formulas for the time integration of advection diffusion reaction PDEs . . . . . 60 

Grylonakis G. E.N., Filelis-Papadopoulos K. C. and Gravvanis A. G. 

On the numerical modelling and solution of multi-asset Black-Scholes equation 

based on Generic Approximate Sparse Inverse Preconditioning . . . . . . . . . 61 

Gu C. and Zhang K. 

The Error Analysis of the Indirect Pade Method for Matrix Exponential . . . . 62 

Guedouar R., Bouzabia A. and Zarrad B. 

Optimization of pre-recontruction restoration filtering for filtered back projection 

reconstruction (FBP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

Hadjidimos A. and Tzoumas M. 

On the Solution of the Linear Complementarity Problem by the Generalized 

Accelerated Overrelaxation Iterative Method . . . . . . . . . . . . . . . . . . . 64 

Hadjimichael Y. and Ketcheson I. D. 

Strong-stability-preserving additive linear multistep methods . . . . . . . . . . 65 

Hadjinicolaou M. 

Fokas method and Kelvin transformation applied to potential problems in non 

convex unbounded domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 

Hananel A., Pasadas M. and Rodríguez L. M. 

Construction and approximation of surfaces by smoothing meshless methods . 67 

Hashemzadeh P. and Fokas S A. 

The definitive estimation of the neuronal current via EEG and MEG using real 

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 

Hashemzadeh P. and Fokas S A. 

Numerical Solution of The Unified Transform For Linear Elliptic PDEs in Polygonal 

Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 

Hassoun Y. and Othman H. 

Symmetric Key Cryptography Algorithms Based on Numerical Methods . . . . 70 

Hitzazis I. 

The Fokas Method and Initial-Boundary Value Problems for Multidimensional 

Integrable PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 

Hong X.-L., Meng L.-S. and Zheng B. 

Some new perturbation bounds of generalized polar decomposition . . . . . . . 73 

Huang Yu.-M. and Zhang X.-Y. 

On block preconditioners for PDE-constrained optimization problems . . . . . 74 

Kalosakas G. 

Modeling drug release kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 

Kanellopoulos G. and van der Weele k. 

Granular Transport Dynamics: Numerics and Analysis . . . . . . . . . . . . . . 76 

Kastis A. G., Gaitanis A. and Fokas S A. 

Quantitative evaluation of SRT for PET imaging: Comparison with FBP and 

OSEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 

Kazolea M., Delis I. A. and Synolakis E. C. 

A wave breaking mechanism for an unstructured finite volume scheme . . . . . 78 

Khoshkhoo R. and Jahangirian A. 

Numerical Simulation of Flow Separation Control using Dielectric Barrier Discharge 

plasma actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 


NumAn2014 Book of Abstracts ix 

Kim M., Jung H.-K. and Park S. 

An effective approach on finite-difference-time-domain method for quasi-static 

electromagnetic field analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 

Kim S. and Zhang H. 

Domain decomposition method with complete radiation boundary conditions for 

the Helmholtz equation in waveguides . . . . . . . . . . . . . . . . . . . . . . . 81 

Kincaid R. D., Chen J-Y. and Li Yu-C. 

Generalizations and Modifications of Iterative Methods for Solving Large Sparse 

Indefinite Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 

Kontogiorgos P., Sarri E., Vrahatis N. M. and Papavassilopoulos P. G. 

An energy market stackelberg game solved with particle swarm optimization . 83 

Korfiati A., Tsompanopoulou P. and Likothanassis S. 

Serial and Parallel Implementation of the Interface Relaxation Method GEO . 84 

Kouloukas T. 

A special class of integrable Lotka-Voltera systems and their Kahan discretization 85 

Kourounis D. 

Constraint handling for gradient-based optimization of compositional reservoir 

flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 

Lisitsa V. and Tcheverda V. 

Combining discontinuous Galerkin and Finite Differences methods for simulation 

of seismic wave propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 

Liu Z., Yamanea Y., Tsujib T. and Tanaka T. 

Decreasing Computational Load by Using Similarity for Lagrangian Approach 

to Gas-solid Two-phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 

Makhanov S. 

Curvilinear Grids for Five-Axis Machining . . . . . . . . . . . . . . . . . . . . . 89 

Manapova A. and Lubyshev F. 

Numerical Solution of Optimization Problems for Semilinear Elliptic Equations 

with Discontinuous Coefficients and Solutions . . . . . . . . . . . . . . . . . . . 90 

Mandikas G. V., Mathioudakis N. E., Kozyrakis V. G., Ekaterinaris A. J. and Kampanis 

A. N. 

A MultiGrid accelerated high-order pressure correction compact scheme for incompressible 

Navier-Stokes solvers . . . . . . . . . . . . . . . . . . . . . . . . . 91 

Muslu M. G. and Borluk H. 

A Fourier Collocation Method for the Nonlocal Nonlinear Wave Equation . . . 92 

Mylonas K. I., Rothos M. V., Kevrekidis G. P. and Frantzeskakis J. D. 

Perturbation Theory of Dark-Bright solitons in Bose-Einstein condensates . . . 93 

Nikas A. I. 

Efficient Unconstrained Optimization Multistart Solvers Using a Self-Clustering 

Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 

Noutsos D., Serra-Capizzano S. and Vassalos P. 

Essential spectral equivalence via multiple step preconditioning and applications 

to ill conditioned Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . 95 

Occorsio D. and Russo G. M. 

Nyström methods for two-dimensional Fredholm integral equations on unbounded 

domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 

Okulicka-D̷lużewska F., Smoktunowicz A. 

Numerical stability of block direct methods for solving symmetric saddle point 

problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 

Owolabi M. K. and Patidar C. K. 

Robust numerical simulation of reaction-diffusion models arising in Mathematical 

Ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 


NumAn2014 Book of Abstracts x 

Pelloni B. and Smith A. D. 

Unified Tranforms and classical spectral theory of operators . . . . . . . . . . . 99 

Petsounis K. 

MATLAB : Parallel and Distributed Computing using CPUs and GPUs . . . . 100 

Prusińska A. and Tretýakov A.A. 

Method for solving nonlinear singular problems . . . . . . . . . . . . . . . . . . 101 

Sablonniére P. a and Barrera D. 

Solving the Fredholm integral equation of the second kind by global spline quasiinterpolation 

of the kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 

Sacconi A. 

On the comparison between fitted and unfitted finite element methods for the 

approximation of void electromigration . . . . . . . . . . . . . . . . . . . . . . . 103 

Sattarzadeh S. and Jahangirian A. 

A Numerical Mesh-Less Method for Solving Unsteady Compressible Flows . . . 104 

Sattarzadeh S., Jahangirian A. and Ebrahimi M. 

A Numerical Adaptive Mesh-Less Method for Solution of Compressible Flows . 105 

Schioppa Jr. E., Verkerke W., Visser J. and Koffeman E. 

Solving CT reconstruction with a particle physics tool (RooFit) . . . . . . . . . 106 

Shmerling E. 

Ziggurat algorithm for sampling from bivariate distributions . . . . . . . . . . . 107 

Sifalakis G. A., Papadomanolaki G. M., Papadopoulou P. E. and Saridakis G. Y. 

Fokas transform method for classes of advection-diffusion IBVPs . . . . . . . . 108 

Sintunavarat W. 

Approximate algorithm for single valued nonexpansive and multi-valued strictly 

pseudo contractive mappings in Hilbert . . . . . . . . . . . . . . . . . . . . . . 109 

Spanakis C., Marias K. and Kampanis A. N. 

Application of an image registration method based on maximization of mutual 

information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 

Spivak A. 

Successive approximations for optimal control in some nonlinear systems with 

small parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 

Stylianopoulos N. 

Inverse moment problems with applications in shape reconstruction . . . . . . 112 

Szczepanik E., Tretýakov A. 

Method for solving degenerate sub-definite nonlinear equations . . . . . . . . . 113 

Stratis P.N., Karatzas G.P., Papadopoulou E.P. and Saridakis Y.G. 

Stochastic optimization for a problem of saltwater intrusion in coastal aquifers 

with heterogeneous hydraulic conductivity . . . . . . . . . . . . . . . . . . . . . 114 

Taha T. 

Numerical simulations for 1+2 dimensional coupled nonlinear Schrödinger type 

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 

Tsakiri K. and Marsellos A. 

A Numerical Model for the prediction of flooding in Water Rivers . . . . . . . 116 

Tsompanopoulou P. 

Interface Rexation Methods for the solution of Multi-Physics Problems . . . . . 117 

Ukpebor A. L. 

An Order 19-rational integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 

Valtchev S S., Alves J. S. C. and Martins F. M. N. 

A Meshfree Method with Fundamental Solutions for Inhomogeneous Elastic 

Wave Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 

Vavalis M. and Zimeris D. 

On the Numerical Solution of Power Flow Problems . . . . . . . . . . . . . . . 120 


NumAn2014 Book of Abstracts xi 

Venetis E. I., Kouris A., Nikoloutsakos N., Sobczyk A. and Gallopoulos E. 

Towards robust parallel solvers for tridiagonal systems for multiGPUs . . . . . 121 

Venetis E. I., Nikoloutsakos N., Gallopoulos E. and Ekaterinaris J. 

Local Stiffness Matrix Calculations for FSI Applications on multi-GPU Systems 122 

Wang Z.-Q. 

Chebyshev accelerated preconditioned MHSS iteration methods for a class of 

block two-by-two linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 

Yang X. 

The WR-HSS Methods for Non-Self-Adjoint Positive Definite Linear Differential 

Equations and Applications to the Unsteady Discrete Elliptic Problem . . . . . 124 

Zaitseva A. and Lisitsa V. 

Sensitivity of the Domain Decomposition Method to Perturbation of the Transmission 

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 

Zakynthinaki M. 

An improved model of heart rate kinetics . . . . . . . . . . . . . . . . . . . . . 126 

Zambelli A. 

Normalizations of the Proposal Density in Markov Chain Monte Carlo Algorithms127 

Zhang G.-F. and Zheng Z. 

A local preconditioned alternating direction iteration method for generalized 

saddle point problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 

Zhang Y. and Li Q. 

Katservich Algorithm Based on Spherical Detector for Cone-Beam CT and the 

Implementation on GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 

Zhao Z., Bai Z.-J. and Jin X.-Q. 

A Riemannian Newton Algorithm for Nonlinear Eigenvalue Problems . . . . . 130 

Zouraris E. G. 

Finite element approximations for a linear stochastic Cahn-Hilliard-Cook equation131 


NumAn2014 Book of Abstracts xii 


NumAn2014 Book of Abstracts 1 

Remembering Theo Papatheodorou 

Elias N. Houstis 

Department of Electrical and Computer Engineering, 

University of Thessaly, 

Volos, Greece 

enh@inf.uth.gr 

”On the day I die, don’t say he’s gone 

Death has nothing to do with going away 

The sun sets, and the moon sets but theyre not gone 

Death is a coming together 

The human seed goes down into the ground like a bucket, 

and comes up with some unimagined beauty 

Your mouth closes here, and immediately opens 

with a shout of joy there” 

Rumi 



Conference in Numerical Analysis 2014 (NumAn 2014) 

September 2-5, 2014 

Chania,Greece 

Scalable and Fast Iteration Methods 

for Complex Linear Systems 

Zhong-Zhi Bai a , 

a State Key Laboratory of Scientific/Engineering Computing 

Institute of Computational Mathematics and Scientific/Engineering Computing 

Academy of Mathematics and Systems Science 

Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, P.R. China 

bzz@lsec.cc.ac.cn 

Abstract 

Complex system of linear equations arises in many important applications. We further explore 

algebraic and convergence properties and present analytical and numerical comparisons among 

several available iteration methods such as C-to-R and PMHSS for solving such a class of linear 

systems. Theoretical analyses and computational results show that reformulating the complex 

linear system into an equivalent real form is a feasible and effective approach, for which we 

can construct, analyze and implement accurate, efficient and robust preconditioned iteration 

methods. 

Key words: complex symmetric linear system, real reformulation, PMHSS iteration, preconditioning, 

convergence theory, spectral properties. 





Chania,Greece 

The interplay of the concrete and general: 

from PDEs to medical imaging 

Athanassios S. Fokas 

Department of Applied Mathematics and Theoretical Physics 

University of Cambridge, Cambridge, CB3 0WA, UK 

T.Fokas@damtp.cam.ac.uk 

Abstract 

The need to solve a concrete problem of physical significance occasionally leads to the development 

of a new mathematical technique. It is often realised that this technique can actually 

be used for the solution of a plethora of other problems, and thus it becomes a mathematical 

method. In this lecture a review will be presented on how a problem posed by the late Julian 

Cole led to the development of the so called unified transform, which provides a novel and 

powerful treatment to boundary value problems for linear and integrable non-linear PDEs. Interesting 

connections with the Riemann hypothesis, as well as the development of several effective 

algorithms for Medical Imaging, will also be reviewed. 



Fast computation of the semiclassical Schrödinger equation 

Arieh Iserles 

Department of Applied Mathematics and Theoretical Physics, 

Centre for Mathematical Sciences, 

University of Cambridge, 

Cambridge CB3 OWA United Kingdom 

ai10@cam.ac.uk 

Abstract 

The computation of the semiclassical Schrödinger equation presents a number of difficult challenges 

because of the presence of high oscillation and the need to respect unitarity. Typical strategy 

involves a spectral method in space and Strang splitting in time, but it is of low accuracy and 

sensitive to high oscillation. In this talk we sketch an alternative strategy, based on high-order 

symmetric Zassenhaus splittings, combined with spectral collocation, which preserve unitarity 

and whose accuracy is immune to high oscillation. These splittings can be implemented with 

large time steps and allow for an exceedingly affordable computation of underlying exponentials. 

The talk will be illustrated by the computation of different quantum phenomena. 


NumAn2014 Book Conference of Abstracts in Numerical Analysis 2014 (NumAn 2014) 5 


Chania,Greece 

Perron-Frobenius Theory – Some Extensions and Applications 

Dimitrios Noutsos 

Department of Mathematics, University of Ioannina, 

Ioannina, Greece 

dnoutsos@uoi.gr 

Abstract 

The Perron-Frobenius theory on nonnegative matrices was introduced by Perron and Frobenius 

in the beginning of the 20th century. Since its construction the Perron-Frobenius theory has been 

developed and constituted a basic Linear Algebra tool to study and solve problems arising from 

applications in discretization of Differential and Integral Equations, Markov chains, Economics, 

Biosciences, etc. The class of M-matrices, which was introduced and studied in the meantime, 

appears in many of the aforementioned applications. Also, some classes of splittings (regular, 

weak regular, nonnegative, etc.) were proposed for the solution of large linear algebraic systems 

of equations by iterative methods. 

The Perron-Frobenius theory of nonnegative matrices was extended to matrices which have 

some negative entries, by D. Noutsos [Linear Algebra Appl., 412 (2006), no 2–3, 132–153]. 

Some properties which give information, when a matrix possesses a Perron-Frobenius eigenpair, 

were presented and proved. This class of matrices is associated to eventually nonnegative 

matrices, namely matrices whose powers become and remain nonnegative. The class of 

Perron-Frobenius splitting was proposed and studied for the solution of linear systems by classical 

iterative methods. Some properties of the type of Stein-Rosenberg theorem were extended 

to the class of Perron-Frobenius splittings by D. Noutsos [Linear Algebra Appl., 429 (2008), 

1983–1996]. 

Linear differential systems ẋ(t) = Ax(t), A ∈ R n,n , whose solutions become and remain 

nonnegative, were studied by D. Noutsos and M. J. Tsatsomeros [SIAM J. Matrix Anal. Appl., 

30 (2008), no 2, 700–712]. Initial conditions that result to nonnegative states are shown to form 

a convex cone that is related to the matrix exponential e tA and its eventual nonnegativity. 

Further extension of the Perron-Frobenius theory of nonnegative matrices to certain complex 

matrices was proposed and proved by D. Noutsos and R. S. Varga [Linear Algebra Appl., 437 

(2012), 1071–1088]. 

Recently, B. Iannazzo and D. Noutsos, in a forthcoming paper, have considered an extension 

of the Perron-Frobenius theory to matrices obtained by a suitable scaling (with positive and 

negative entries) applied to an M-matrix. This problem appears, for instance, in the study of 

Algebraic Riccati Equations arising in fluid queues, where one is interested in the invariant 

subspaces of the matrix 

[ ] 

A −B 

H = 

, 

−C −D 

obtained by changing the sign of the last m rows of an M-matrix. 

Key words: Nonnegative Matrices, Perron-Frobenius Theory, Eventually Nonnegative Matrices, M-Matrices, 

Riccati Equation. 




Chania, Greece 

Sign Methods for Imprecise Problems 

Michael N. Vrahatis 

Computational Intelligence Laboratory (CILab), 

Department of Mathematics, University of Patras, 

GR-26110 Patras, Greece 

vrahatis@math.upatras.gr 

Abstract 

Tackling problems with imprecise (not exactly known) information occur in different scientific 

fields including mathematics, physics, astronomy, meteorology, engineering, computer science, 

biomedical informatics, medicine and bioengineering among others. In many applications, precise 

function values are either impossible or time consuming to obtain. For example, when the 

function values depend on the results of numerical simulations, then it may be difficult or impossible 

to get very precise values. Or in other cases, it may be necessary to integrate numerically 

a system of differential equations in order to obtain a function value, so that the precision of the 

computed value is limited. Furthermore, in many problems the accurate values of the function 

are computationally expensive. 

Ideas from the topological degree theory and combinatorial topology (algebraic topology) 

have led to the introduction of iterative root-finding and fixed point methods as well as numerical 

optimization methods for tackling problems with impressions. We call these methods sign 

methods since the only computable information required is the algebraic sign of the function that 

is the smallest amount of information (one bit of information) necessary for the purpose needed, 

and not any additional information. In this contribution, some of these methods are reviewed 

and applications to computational mathematics and computational intelligence are presented. 

Key words: Sign Methods, Root-finding Methods, Fixed Point Methods, Numerical Optimization Methods, 

Imprecise Problems 



A new class of Second Derivative methods for numerical 

Integration of Stiff Initial Value Problems 

C. E. Abhulimen 

Department of Mathematics 

Ambrose Alli University 

Ekpoma, Nigeria 

cletusabhulimen@yahoo.co.uk 

Abstract 

In this paper, we construct a new class of four-step second derivative exponential fitting 

method of order six for the numerical integration of stiff initial-value problems of the type: 

y ′ = f(x, y); y(x 0 ) = y 0 

The implicit method which is based on the work of Cash [1], possess free parameters which allow 

it to be fitted automatically to exponential functions. For the purpose of effective implementation 

of the new proposed method, we adopt the mechanism in[1] by splitting the method into 

predictor and corrector schemes. The numerical analysis of the stability of the new method was 

discussed and some numerical experiments confirming theoretical expectations are provided. 

Finally, the numerical results show that the new method is A-stable and compete favorably with 

the existing methods in terms of efficiency and accuracy. 

Keywords: Second derivative four-step, exponentially fitted, A-stable, stiff initial value problems 

2010 MSC: 65L05, 65L2O 

References 

[1] J.R Cash (1981). ”On the exponential fitting of composite multiderivative linear multistep 

methods” SIAM J. Numer Annal 18(5), (1981), 808-821 



Homoclinic chaos in a pair of parametrically-driven coupled 

SQUIDs 

Makrina Agaoglou a , Vassilios M Rothos a and Hadi Susanto b 

a Department of Mechanical Engineering, Faculty of Engineering, Aristotle 

University of Thessaloniki, Thessaloniki 54124, Greece 

b Department of Mathematical Sciences, University of Essex, Wivenhoe Park, 

Colchester CO4 3SQ, United Kingdom 

rothos@auth.gr,hsusanto@essex.ac.uk 

Abstract 

An rf superconducting quantum interference device (SQUID) consists of a superconducting ring 

interrupted by a Josephson junction (JJ). When driven by an alternating magnetic field, the induced 

supercurrents around the ring are determined by the JJ through the celebrated Josephson 

relations. This system exhibits rich nonlinear behavior, including chaotic effects. We study 

the dynamics of a pair of parametrically-driven coupled SQUIDs arranged in series. We take 

advantage of the weak damping that characterizes these systems to perform a multiple-scales 

analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture 

allows us to expose the existence of homoclinic orbits in the dynamics of the integrable 

part of the slow equations of motion. Using high-dimensional Melnikov theory, we are able to 

obtain explicit parameter values for which these orbits persist in the full system, consisting of 

both Hamiltonian and non-Hamiltonian perturbations, to form so-called Silnikov orbits, indicating 

a loss of integrability and the existence of chaos. Extensive numerical analysis requiring 

algorithms of rapid numerical integration are required to follow the solutions for long times and 

verify the accuracy of the analytical results. 

Key words: superconducting quantum interference device, Josephson junction, Near-Integrable Hamiltonian 

Systems, Silnikov Chaos, Numerical Simulations 






Chania,Greece 

Continuation and stability deduction of resonant periodic orbits 

in three dimensional systems 

Kyriaki I. Antoniadou and George Voyatzis 

Section of Astrophysics, Astronomy and Mechanics, Department of Physics, 

Aristotle University of Thessaloniki, 

Thessaloniki, 54124, Greece 

kyant@auth.gr, voyatzis@auth.gr 

Abstract 

The general three body problem (GTBP) through the implementation of periodic orbits computed 

in a suitable rotating frame of reference can be used in order to describe the dynamics 

of planets locked in a mean motion resonance. The families of periodic orbits, either planar or 

spatial, derived by specific continuation processes can, also, constitute paths that can drive the 

planetary migration. 

In Hamiltonian systems, it is known that in phase the stable periodic orbits are surrounded 

by invariant tori, while in the neighbourhood of unstable periodic orbits chaotic regions exist. 

It has been shown numerically that in the vicinity of stable periodic orbits, where the motion 

is regular and bounded, exoplanetary systems can survive, whereas in case they are found near 

unstable ones, they will eventually destabilize and the planets may collide or even escape. The 

significance of periodic orbits is therefore taken for granted and the accuracy of their computation 

is apparently crucial. 

We herein depict examples of resonant periodic orbits, exploit analytic continuation and 

elaborate on matters of both horizontal and vertical stability. Particularly, we consider the spatial 

GTBP and study the dynamics of planetary systems consisting of a Star and two inclined 

Planets, which evolve into mean motion resonance. We attempt a comparative study between 

three methods used for the deduction of the stability of a periodic orbit: the computation of 

eigenvalues, stability indices and Fast Lyapunov Indicator. Finally, we construct maps of dynamical 

stability in the vicinity of periodic orbits, in order to identify the extent of stable regions 

in phase space. 

Key words: periodic orbits, horizontal and vertical stability, mean motion resonance 



SOpenCL: An Infrastructure for Transparently Integrating 

FPGAs in Heterogeneous, Accelerator-Based Systems 1 

Christos Antonopoulos and Nikolaos Bellas 

Department of Electrical and Computer Engineering, University of Thessaly, 

Volos, Greece 

cda, nbellas@uth.gr 

Abstract 

The use of heterogeneous parallel architectures appears as a promising approach in the HPC 

domain, due to both the absolute performance and the high performance/power ratio these architectures 

offer. Heterogeneous systems are typically organized as a number of computational 

accelerators, such as GPUs, DSPs etc., complementing one or more general purpose CPUs. 

Field Programmable Gate Arrays (FPGAs) are hardware devices that offer a sea of gates and 

memory islands which can be configured as digital circuits, thus implementing algorithms at 

the hardware level. FPGAs are an excellent accelerator choice, as they can often prove more 

power-efficient than conventional CPUs and even GPUs. 

Despite the favorable power/performance characteristics, the adoption of FPGAs in the HPC 

domain is rather limited. The implementation of algorithms at the hardware-level requires experience 

on hardware design and the use of specialized hardware-description languages (Verilog, 

VHDL, SystemC), thus remaining outside the realm of domain experts and software engineers. 

In this talk we present SOpenCL, a tool infrastructure that facilitates the wider use of FPGAs 

in reconfigurable systems. SOpenCL translates algorithmic descriptions at the software level 

to equivalent circuit descriptions in Verilog, which can then be directly implemented on an 

FPGA. We use OpenCL, a popular and industry supported parallel programming standard for 

heterogeneous systems, as the programming model of choice for the software-level algorithmic 

descriptions. This way, programs targeted at CPUs or GPUs can transparently and without any 

further development effort be executed on FPGA-based accelerator systems as well. 

Key words: Heterogeneous Systems, OpenCL, FPGAs, High-level synthesis. 

1 The present research work has been co-financed by the European Union (European Social Fund ESF) and Greek 

national funds through the Operational Program ”Education and Lifelong Learning” of the National Strategic Reference 

Framework (NSRF) - Research Funding Program: THALIS. Investing in knowledge society through the European 

Social Fund. 



Software Platforms for Multi-Domain Multi-Physics 

Simulations 1 

Christos Antonopoulos, Manolis Maroudas and Manolis Vavalis 


Volos, Greece 

{cda,kapamaroo,mav}@uth.gr 

Abstract 

Advances in hardware and software technologies in the 1980s led to the modern era of scientific 

modeling and simulation. This era seems to come to an end. The simulation needs in both industry 

and academia mismatch with the existing software platforms and practices, which to a great 

extent have remained unchanged for the past several decades. We foresee that this mismatch, 

together with the emerging ICT advances and the cultural changes in scientific approaches will 

lead to a new generation of modeling and simulation. 

This paper proposes approaches for designing, analyzing, implementing and evaluating new 

simulation frameworks particularly suited to multi-domain and multi-physics (MDMP) problems 

that have Partial Differential Equations (PDEs) in their foundations. We focus on introducing 

software platforms that facilitate the numerical solution of PDEs associated with MDMP 

mathematical models. 

In particular, we propose an enhanced meta-computing environment which is based on: (a) 

scripting languages (like python) and their practices, and (b) on the Service Oriented Architecture 

(SOA) paradigm and the associated web services technologies. 

The proposed environment has been designed and engineered having in mind a set of characteristics 

particularly suited for MDMP problems. More specifically, it: 

• Allows domain experts to focus on expressing the models, rather than delving into implementation 

details. 

• Fully utilizes the plethora of PDE solving modules available. 

• Allows the programmer to effectively select the most appropriate available software module 

for the particular component of the problem, as this is defined by its associated single 

physics model and its simple/single domain 

• Transparently benefits from recent algorithmic advances (e.g. domain decomposition) 

• Allows users to efficiently deploy and run the MDMP computations on loosely coupled 

distributed and heterogeneous compute engines. 

Although our design is generic, covering a wide range of problems, our proof of concept 

implementation is restricted to elliptic PDEs in two or three dimensions. Furthermore, it clearly 

shows that our tool can easily exploit state of the art numerical solvers like those available in 

FENICS and deal.II, domain decomposition methods with or without overlapping, Monte Carlo 

based hybrid solvers, rectangular or curvilinear domains and interfaces and beyond. 

Key words: Numerical Solution of PDEs, Multi-domain, Multi-physics, Problem Solving Environments. 


national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference 


Social Fund. 



Error Estimates for the Standard Galerkin-Finite Element 

Method for the Shallow Water Equations 

D. C. Antonopoulos and V. A. Dougalis 

Institute of Computational and Applied Mathematics, FORTH, 70013 Heraklion, 

Greece 

dougalis@iacm.forth.gr 

Abstract 

We consider a simple initial-boundary-value problem for the shallow water equations on a finite 

interval, and also the analogous problem for a symmetric variant of the system that we justify 

for small-amplitude solutions. Assuming smoothness of solutions we discretize these problems 

in space using the standard Galerkin-finite element method and prove L 2 -error estimates for 

the semidiscrete problem for quasiuniform and uniform meshes. In particular we show that 

the semidiscretization with piecewise linear, continuous functions on a uniform mesh posseses 

optimal-order O(h 2 ) L 2 -error estimates. We also examine time-stepping of the semidiscrete 

problems with three explicit Runge-Kutta methods (the Euler, improved Euler, and the Shu- 

Osher scheme), and prove L 2 -error estimates for the resulting full discretizations that are of 

optimal order in the temporal variable. We also discuss the cases of periodic and absorbing 

boundary conditions. 

Key words: Shallow water equations, fully discrete Galerkin methods, error estimates. 






Do Brain Networks Evolve by Maximizing Flow of Information? 

Chris G. Antonopoulos, Shambhavi Srivastava, Sandro S. Pinto, Murilo S. 

Baptista 

Department of Physics, University of Aberdeen, Institute for Complex Systems 

and Mathematical Biology (ICSMB), 

Aberdeen, AB24 3UE, UK 

chris.antonopoulos@abdn.ac.uk 

Abstract 

In this talk, I will first present unexpected structural and functional similarities we have, recently, 

been able to find in the C.elegans and human brain networks. Based on these findings, we then 

propose an appropriately constructed model for the evolution of such networks by adding and 

retaining new connections between neurons of the network that lead to a subsequent increase 

of the upper bound of Mutual Information Rate (MIR), a quantity related to the amount of 

information the brain network can process. This idea is reminiscent of the Hebbian rule of 

learning and synaptic plasticity. I will show the ability of our model for brain evolution to 

capture important properties, such as synchronization and upper bound of MIR patterns, of 

realistic already evolved brain networks. Finally, I will comment on some of the computational 

aspects arising in this study, like numerical integration methods, accuracy and computation time 

regarding the serial or parallel implementation of the model. 

Key words: 

Brain networks, Evolutionary process, Hindmarsh-Rose dynamics, Synchronization, Mutual Information 

Rate (MIR), Upper bound of MIR 





Chania,Greece 

Finite elements for a class of nonlinear stochastic pdes 

from phase transition problems 

Dimitra Antonopoulou a , 

a Department of Mathematics and Applied Mathematics, University of Crete, 

GR-714 09 Heraklion, Greece, and Institute of Applied and Computational 

Mathematics, FORTH, GR-711 10 Heraklion, Greece 

danton@tem.uoc.gr 

Abstract 

We construct Galerkin numerical schemes with possible discontinuities in time for a class of 

nonlinear evolutionary pdes with additive noise. These equations appear in phase transitions 

problems and may involve a positive parameter ε which stands as a measure for the inner interfacial 

regions width. Our goal is to establish existence of numerical solution and derive optimal 

error estimates even for the discontinuous Galerkin case in the presence of noise. 

Key words: Finite elements, nonlinear stochastic pdes, dG methods. 




Chania,Greece 

Numerical Solution of the Magnetic Laplacian Eigenvalue 

Problem using Radial Basis Functions 

Pedro R. S. Antunes a,b 

a Group of Mathematical Physics of the University of Lisbon, 

Lisbon, Portugal 

b Department of Mathematics, Lusophone University of Humanities and 

Technologies, 

Lisbon, Portugal 

pant@cii.fc.ul.pt 

Abstract 

We consider the numerical solution of the Magnetic Laplacian eigenvalue problem, 

{ 

(i∇ + F ) 2 u = λu in Ω, 

u = 0 on ∂Ω. 

(1) 

where u(x) is complex-valued and the vector potential is F (x) = β(−x 2 , x 1 ). The magnetic 

field is ∇ × F = (0, 0, 2β), where β ∈ R is constant. In this work we study the application 

of a numerical method based on radial basis functions. It is well known that the Kansa method 

allows for the numerical solution of boundary value problems using radial basis functions and 

the efficiency has been verified in a wide range of problems. It is a meshfree method, which 

can have high accuracy, provided an appropriate shape parameter is chosen. On the other hand, 

a disadvantage of the method is that the matrices involved tend to become progressively more 

ill-conditioned as the rank increases. In this work we propose a numerical algorithm based 

on the Generalized Singular Value Decomposition to circumvent the ill-conditioning. Several 

numerical simulations are presented to illustrate the good performance of the method. 

Key words: Magnetic Laplacian, eigenvalue problem, radial basis functions. 





Chania,Greece 

Stochastic Riez-Fractional Partial Differential Equation with 

White Noise on the Half-Line 

Árciga A. Martín P. a , Ariza H. Francisco J. a and Sánchez O. Jorge a 

a Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, 

Chilpancingo, Guerrero, México 

mparcigae@gmail.com,aarizahfj@gmail.com,jsanchezmate@gmail.com 

Abstract 

We consider a Bayesian numerical solution of a stochastic Riesz-fractional partial differential 

equation with white noise on the half-line. This equation is given by 

u t = Dx α u + N u + Ḃ(x, t), x, t > 0 (1) 

where Dx α is the Riesz-fractional derivative, N is a nonlinear operator and Ḃ(x, t) is the white 

noise. To construct the integral representation of solution we use the Fokas’ Method. 

Key words: Fractional derivative, Fokas’ Method, Brownian motion. 




Chania,Greece 

Functional Analytic Framework of the Fokas Method 

for Elliptic Boundary Value Problems 

Anthony Ashton a 

a DAMTP, University of Cambridge, 

United Kingdom 

acla2@damtp.cam.ac.uk 

Abstract 

We give an overview of the functional analytic framework for the Fokas approach to elliptic 

boundary value problems in convex planar domains. The global relation can be interpreted as 

an operator equation of the form Ax = y, where y depends on the known data of a given boundary 

value problem and x corresponds to the unknown boundary values. We study the functional 

analytic properties of the operator A : X → Y where X, Y are Banach spaces of complex 

analytic functions that are similar to the classical Paley-Wiener spaces. These results are important, 

not only from a theoretical perspective, but are essential for establishing convergence and 

stability results for the numerical implementation of this approach to boundary value problems. 

Finally, we give a brief account of some recent results that are applicable to elliptic boundary 

value problems in three dimensional polyhedra. 

Key words: Fokas Method, Boundary Value Problems, Functional Analysis, Operator Theory. 



Discontinuous Hermite Collocation and Runge-Kutta schemes 

for multi-domain linear and non-linear brain tumor invasion 

models 1 

I.E. Athanasakis ∗ , E.P. Papadopoulou and Y.G. Saridakis 

Applied Mathematics and Computers Laboratory (AMCL) 

Technical University of Crete 

Chania 73100, Greece 

∗ g.athanasakis@amcl.tuc.gr 

Abstract 

Growth simulation models of aggressive forms of malignant brain tumors have been well developed 

over the past years. In our recent works we have considered both novel analytical and 

numerical methods for the efficient treatment of brain tumor models that, apart from proliferation 

and diffusion, are being characterized by a discontinuous diffusion coefficient to incorporate 

the heterogeneity of the brain tissue. In this direction we have recently introduced a Discontinuous 

Hermite Collocation (DHC) finite element method, with appropriately discontinuous basis 

functions associated with the discontinuity nodes. The method was coupled with Diagonally 

Implicit (DI) Runge-Kutta schemes and studied for a three region linear model to reveal its high 

order approximation properties. In this work, we consider extending our results in the following 

directions: 

• Employment of both linear and non-linear multi-domain brain tumor models 

• Coupling of the DHC with both DI and Strong Stability Preserving (SSP) Runge-Kutta 

schemes. 

Their behavior is being examined and several experiments are included to demonstrate their 

performance. 




Social Fund. 





Chania,Greece 

Solving discontinuous collocation equations for a class of 

brain tumor models on GPUs 1 

I.E. Athanasakis, N.D. Vilanakis ∗ , E.N. Mathioudakis, 

E.P. Papadopoulou and Y.G. Saridakis 

Applied Mathematics and Computers Laboratory 


Chania, Crete, Greece 

∗ nivilanakis@amcl.tuc.gr 

Abstract 

Brain tumor models, that incorporate brain’s heterogeneity, have been well developed in the last 

decades. The core PDE, that models tumor’s cell diffusion and proliferation properties, is being 

characterized by a discontinuous diffusion coefficient, since tumor cells migrate with different 

rates in brain’s white and gray matter. In recent years, working towards the development of 

high order approximation methods, we have introduced and studied Discontinuous Hermite 

Collocation (DHC) methods coupled with traditional as well as high order semi implicit and 

strongly stable Runge-Kutta (RK) time discretization schemes. In this work the problem at 

hand is the efficient solution of the linear model tumor invasion problem in 1+2 dimensions. 

Tensor product formulated fourth order DHC method is used as spatial discretization to produce 

a system of ODEs, to be solved, in the sequel, by third order Diagonally-Implicit RK (DIRK) 

schemes. Therefore, in each time step a large linear system of order O(N 2 ), where N is the 

number of elements in each dimension, has to be solved demanding quite intense computational 

effort. Its efficient solution by incomplete factorization preconditioned BiCG stabilized iterative 

method (as the eigenvalue topology suggests) in GPU computational environments is presented 

and several numerical experiments are used to demonstrate its performance. 


national funds through the Operational Program ’Education and Lifelong Learning’ of the National Strategic Reference 


Social Fund. 




Chania,Greece 

Obviating the Bin Width Effect of the 1/t Algorithm 

for Multidimensional Numerical Integration 

Wanyok Atisattapong a and Pasin Maruphanton a 

a Department of Mathematics and Statistics, Faculty of Science and Technology, 

Thammasat University, Pathum Thani, Thailand 12120 

wanyok@mathstat.sci.tu.ac.th, oporkabbb@hotmail.com 

Abstract 

In this work we improve the accuracy and the convergence of the 1/t algorithm [1] for multidimensional 

numerical integration. The 1/t algorithm has been proposed as an improved version 

of the Wang-Landau algorithm [2] which belongs to the class of Monte Carlo methods. After 

the lower bound y min and the upper bound y max of the integral are determined by a domain 

sampling run [3], the integral can then be approximated by 

I = 

∫ b 

a 

y∑ 

max 

y(x)dx ≃ g(y).y, (1) 

y min 

where g(y) ≡ {x|x ∈ [a, b], y ≤ y(x) ≤ y + dy} and dy is the bin width of y. The distribution 

g(y) can be obtained from the 1/t algorithm. However, the errors of estimated integrals saturate 

because of the bin width effect. To obviate this effect, we introduce a new approximation 

method based on the simple sampling Monte Carlo method by using the average of y values 

in the subinterval [y, y + dy], which varies as the number of Monte carlo trials changes, instead 

of the fixed value of y. 

The non-convergence of the 1/t algorithm [4, 5] and the convergence of the new method are 

proved by theoretical analysis. A potential of the method is illustrated by the evaluation of one-, 

two- and multidimensional integrals up to six dimensions. The dynamic behavior of accuracy 

shows that the numerical estimates from our method converge to their exact values without 

either error saturation or the bin with effect in contrast with the conventional 1/t algorithm. 

Key words: Monte Carlo method, Numerical integration, the 1/t algorithm, Bin width effect 

References 

[1] R. E. Belardinelli, S. Manzi, and V. D. Pereyra (2008), Phys. Rev. E. 78, 067701. 

[2] Y. W. Li, Wüst, D. P. Landau, and H. Q. Lin (2007), Comput. Phys. Commum. 177, 524. 

[3] A. Tröster and C. Dellago (2005), Phys. Rev. E. 71, 066705. 

[4] C. Zhou and J. Su (2008), Phys. Rev. E. 78, 046705. 

[5] Y. Komura and Y. Okabe (2012), Phys. Rev. E. 85, 010102 (R). 



NumAn2014 Book of Abstracts September 2-5, 2014 22 

Chania,Greece 

A 1D stabilized finite element model for 

non-hydrostatic wave breaking and run-up 

Paola Bacigaluppi a and Mario Ricchiuto b 

a Department of Mathematics, Universität Zürich, 

Zürich, Switzerland 

b Inria Bordeaux Sud-Ouest, 

Talence cedex, France 

paola.bacigaluppi@gmail.com,Mario.Ricchiuto@inria.fr 

Abstract. 

A new methodology is presented to model the propagation, wave breaking and run-up of waves in coastal 

zones. Propagation is modelled by a form of the enhanced Boussinesq equations (Madsen and Sorensen, 

Coast.Eng. 1992), while the forming of a roller in breaking regions is captured by reverting to the shallow 

water equations and allowing waves to locally converge into discontinuities. The switch between the two 

models is defined by a wave breaking criterion that depends on several physical parameters, including 

the shape and celerity of the wave and the presence of dry areas. To discretize the system we propose 

a non-linear variant of the stabilized finite element method of (Ricchiuto and Filippini, J.Comput.Phys. 

2014). To guarantee monotone shock capturing, a technique based on a non-linear mass-lumping allows 

to provide local non-oscillatory approximations of discontinuities reverting from a third order scheme in 

smooth regions to a first order upwind scheme. The local character of the mass-lumping is guaranteed by 

the use of limiters, or of properly defined smoothness sensors. The presented scheme guarantees positivity 

preservation, well balancedness and the treatment of wet/dry fronts. The wave breaking is triggered 

by means of three different criteria, including a local implementation of the theoretical convective criterion 

of (Bjørkavåg and Kalisch, Phys.Letters A 2011), which have been thoroughly analysed and tested. 

The model obtained is validated on several benchmarks showing excellent agreement with the available 

experimental data. As an example the figure below shows the run-up of a periodic wave over a constant 

slope. In particular the solution between the black lines corresponds to the detected breaking area and is 

computed through the shallow water model. 

Key words: Wave propagation, wave breaking, shock-capturing, stabilized finite elements, SUPG scheme, 

Boussinesq equations, shallow water equations, wet/dry fronts, wave breaking model. 

Fig.: Snapshots of the first and last breaking instants for a periodic wave run up on a constant slope. 

Result obtained with the local variant of the convective criteria. 






Parameter determination in MOSFETs transitors based 

on Discrete Orthogonal Chebyshev polynomials 

D. Barrera a , M. J. Ibáñez a , A. M. Roldán b , J. B. Roldán b and R. Yáñez a 

a Department of Applied Mathematics, University of Granada, 

Granada, Spain 

b Department of Electronics, University of Granada, 


{dbarrera,mibanez,amroldan,jroldan,ryanez}@ugr.es 

Abstract 

Transistors, and in particular MOSFETs (Metal Oxide Semiconductor Field Effect Transistors), 

are the most used basic building blocks of integrated circuits (ICs). The complexity of current 

chips makes essential their accurate characterization to use them for circuit design purposes. 

For each generation of transistors the main electrical features have to be modeled in order to 

reproduce them as a function of the voltages differences applied between their terminals. The 

models (usually known as compact models) consist of a set of analytical equations and a set 

of parameters to include in those equations. A different set of parameters is used for each 

fabrication technology. These models are used in TCAD circuit simulation tools and also for 

hand-calculations used at the first stages of circuit design. 

The extraction of the parameters of new technologies is essential since the capacities of 

circuit designers are dependant on the accuracy of model parameters that in many cases are 

linked to important physical effects. 

Each parameter is obtained in a different way. However, few of them share some features in 

common, at least from the numerical viewpoint. In this respect, several parameters are obtained 

by means of extrapolation methods (for example threshold voltage calculation), linear regression 

(determination of the body factor), slope calculations (extraction of the DIBL parameter), etc. 

In all these procedures, the determination of portions of curves that can be approximated by a 

straight line is crucial. In this work we just deal with this issue trying to shed light by means of 

advanced numerical techniques. 

We have developed a method to determine the number of straight line portions contained 

in a curve in an automatic manner. The algorithm developed, based on discrete orthogonal 

polynomials, can be used for parameter extraction purposes. It consist on the isolation of straight 

line portions in experimental or simulated data and the determination of the slope of those curve 

sections to calculate one or more parameters of a compact model. 

Key words: Discrete orthogonal polynomials, straight line portion, MOSFET. 




Chania,Greece 

Spectral inclusion regions for matrix pencils 

Natalia Bebiano 

Department of Mathematics, University of Coimbra, 

Coimbra, Portugal 

bebiano@mat.uc.pt 

Abstract 

Consider the linear pencil A − λB, where A and B are n × n complex matrices and λ ∈ C. 

Our main purpose is to obtain spectral inclusion regions for the pencil based on certain fields 

of values. Namely, we propose efficient methods for the numerical approximation of the field 

of values of A − λB denoted by W (A, B). Our approach builds on the fact that the field of 

values can be reduced under compressions to the bidimensional case, in which case these sets 

can be exactly determined. The obtained results are illustrated by numerical examples. We point 

out that the given procedure to approximate W (A, B) compares well with those existing in the 

literature. 

Key words: Field of values, Numerical range, Linear pencil, Eigenvalue, Compression. 





Chania,Greece 

Variational iterative method for solving a nonlinear partial 

differential equation. Application to the two-dimensional 

Bratu’s problem 

Z. Belhah a , A. Kouibia b , and Miguel Pasadas b 

a LERMA–Engineering Mohammedia School,Rabat, Morocco. 

b Department of Applied Mathematics, University of Granada, 


z.belhaj@gmail.com, kouibia@ugr.es, mpasadas@ugr.es 

Abstract 

In this paper we present a variational approximation method for solving the two–dimensional 

Bratu’s problem. The existence and the uniqueness of this problem are shown. Moreover, we 

construct a sequence of bicubic splines approximating the problem solution and depending of 

the knots number of a sequence of partitions of the domain. Such sequence converge to the 

exact solution of the problem. Finally, we analyze some numerical examples in order to show 

the efficiency of our method. 

Key words: Bratu’s problem, PDE, variational method, bicubic splines. 



Significance-Based Computing for Reliability and Power 

Optimization 1 

N¯ ikolaos Bellas and Christos D. Antonopoulos 


Volos, Greece 

{nbellas, cda}@uth.gr 

Abstract 

Manufacturing process variability at low geometries and energy dissipation are the most challenging 

problems in the design of future computing systems. Currently, manufacturers go to 

great lengths to guarantee fault-free operation of their products by introducing redundancy in 

voltage margins, conservative layout rules, and extra protection circuitry. However, such design 

redundancy leading to significant energy overheads may not be really required, given that 

many modern workloads, such as multimedia, machine learning, visualization, etc. can tolerate 

a degree of imprecision in computations and data. 

In this talk, I will introduce an approach which seeks to exploit this observation and to relax 

reliability requirements for the hardware layer by allowing a controlled degree of imprecision to 

be introduced to computations and data. It proposes to research methods that allow the systemand 

application-software layers to synergistically characterize the significance of various parts 

of the program for the quality of the end result, and their tolerance to faults. Based on this 

information, extracted automatically or manually, the system software will steer computations 

and data to either low-power, yet unreliable or higher-power and reliable functional and storage 

components. In addition, the system will be able to aggressively reduce its power footprint by 

opportunistically powering hardware modules below nominal values. Significance-based computing 

lays the foundations for not only approaching the theoretical limits of energy reduction 

of CMOS technology, but also moving beyond those limits by accepting hardware faults in a 

controlled manner. 

Key words: Computational significance, Low-power design, Reliable Design. 




Social Fund. 



Quadratic Penalty Methods 

for Shape from Shading 

Stefania Bellavia, Lapo Governi, Alessandra Papini and Luca Puggelli 

Department of Industrial Engineering, University of Florence, 

Florence, Italy 

stefania.bellavia@unifi.it,lapo.governi@unifi.it, 

alessandra.papini@unifi.it,luca.puggelli@unifi.it 

Abstract 

“Shape from shading” (SFS) denotes the problem of reconstructing a 3D surface, starting from 

only one image showing a shaded representation of the surface itself. Minimization techniques 

are commonly used for solving the SFS problem, where the functional that must be optimized 

is a weighted combination of the brightness functional plus one or more regularization terms. 

A critical role in this context is played by the weights used in the functional, which markedly 

affect the possibility of obtaining a good reconstruction. However the choice of these weights 

in not trivial. 

In this work we present a quadratic penalty method where an a-priori choice of the weights 

is not needed. In this approach the SFS problem is formulated as a constrained minimization 

problem, where the objective function is given by a term accounting for the smoothness of the 

reconstructed surface, and the constraints consist of the image irradiance equation (representing 

how well the reconstructed surface reproduces the original image) and of an integrability term. 

Using a quadratic penalty strategy the original constrained problem is replaced by a sequence 

of unconstrained subproblems, which are solved by a Barzilai-Borwein method. The 

results obtained on a set of case studies show the effectiveness of the proposed approach. 

Key words: Shape from shading, equality constrained minimization, quadratic penalty methods, Barzilai- 

Borwein method. 




Chania,Greece 

Multi-scale hybrid model of cell differentiation propagation 

as traveling waves 

Mohammed Benmir a , Nikolai Bessonov b , Soumaya Boujena a and Vitaly 

Volpert c 

a Faculty of Sciences, University Hassan II, 

Casablanca 20100, Maroc 

b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 

199178 Saint Petersburg, Russia 

c Institut Camille Jordan,UMR 5208 CNRS,University Lyon 1,69622 

Villeurbanne,France 

mohammed.benmir05@etude.univcasa.ma,boujena@gmail.com, 

bessonov@bess.ipme.ru,volpert@math.univ-lyon1.fr, 

Abstract 

Multi-scale and hybrid models are well adapted for the description of complex physiological 

processes. They represent an interesting class of models whose properties are not yet sufficiently 

well studied. In particular, they can show unusual nonlinear dynamics. In this work 

we will study propagation of reaction-diffusion waves in the medium composed of unmovable 

cells. Dynamics of cell population is determined by complex intracellular and extracellular regulations. 

If cell differentiation is initiated locally in space in the population of undifferentiated 

cells, it propagates as a travelling wave converting undifferentiated cells into differentiated ones. 

We suggest a model of this process which takes into account intracellular regulation, extracellular 

regulation and different cell types. They include undifferentiated cells and two types of 

differentiated cells. When a cell differentiates, its choice between two types of differentiated 

cells is determined by the concentrations of intracellular proteins. Differentiated cells can either 

stimulate differentiation into their own cell lineage or into another cell lineage. Periodic spatial 

patterns can emerge behind the propagating wave. 

Key words: multi-scale, hybrid, models, traveling waves, extracellular, intracellular, cells, differentiation. 





Chania,Greece 

Preconditioned derivative-free globally convergent 

Newton-GMRES methods for large sparse nonlinear systems 

Michele Benzi a , Iain S. Duff b and Xue-Ping Guo c 

a Department of Mathematics and Computer Science, Emory University, 

Atlanta, GA 30322, USA 

b RAL, Oxfordshire, England. CERFACS, 42 av. Gaspard Coriolis, 31057, 

Toulouse, Cedex 1, France. 

a Department of Mathematics, East China Normal University, 

Shanghai, 200241, P. R. China. 

iain.duff@stfc.ac.uk,benzi@mathcs.emory.edu,xpguo@math.ecnu.edu.cn 

Abstract 

Jocobian-free Newton GMRES(m) methods (JFNG) solve systems of nonlinear equations without 

computing matrix vector product and forming derivatives of functions. In this paper, we 

introduce the derivative-free HSS preconditioner in JFNG methods and obtain preconditioned 

derivative-free globally convergent Newton-GMRES methods (PDFNG) for solving large s- 

parse system of nonlinear equations. The convergence is also given. Finally, numerical tests are 

shown to illustrate the efficiency of PDFNG methods. 

Key words: preconditioner, derivative-free, system of nonlinear equations. 



Using an Euler type transform for accelerating convergence of 

series 

Faton Berisha a , Murat Sadiku b and Nimete Berisha c 

a Department of Mathematics, University of Prishtina, Prishtina, Kosovo 

b Faculty of Economics and Business, South East European University, 

Tetovo, FYROM 

c Faculty of Economics, University of Prishtina, Prishtina, Kosovo 

faton.berisha@uni-pr.edu,m.sadiku@seeu.edu.mk 

nimete.berisha@gmail.com 

Abstract 

Let us use the following operator of generalised difference, linear on a set of sequences: 

∆ 1 r 1 

(a n ) = ∆ r1 

(a n ) = a n+1 − r 1 a n , 

∆ m+1 

r 1 r 2 ...r m+1 

(a n ) = ∆ 1 r m+1 

(∆ m r 1 r 2 ...r m 

(a n )) (m = 1, 2, . . .), 

where {r m } ∞ m=1 is a given sequence of real numbers. 

In the present paper, we give a property of the operator ∆ k r 1 r 2 ...r k 

when applied on an alternating 

sequence {(−1) n a n } ∞ n=0 . Then, we use this property in order to establish modified Euler 

transforms for alternating number, power and trigonometric series. We present algorithm analysis 

for all cases of computing the n-th partial sum of transformed series by using the operator 

of generalised difference of order p, and prove that its order of complexity is O(p 2 n), i.e. that 

its complexity is linear with respect to the number of terms computed. Finally, we give two examples 

illustrating, for instance, that in order to calculate the approximate sum of a given series 

with an error not greater than 10 −6 we must compute the sum of the first 19 terms. To obtain 

this accuracy for the classical Euler transform we need 18 summands. Applying the modified 

transform considered in the paper, the same accuracy is obtained by computing the sum of the 

first 11 terms for p = 1, 7 terms for p = 2, and 4 terms for p = 3. 

Key words: Accelerating convergence of series, Euler transform. 

References 

[1] F. M. Berisha and M. H. Filipović, On some transforms of trigonometric series, Publ. Inst. 

Math. (Beograd) (N.S.) 61(75) (1997), 53–60. MR 1472937 (98g:42012) 

[2] I. Ž. Milovanović, M. A. Kovačević, S. D. Cvejić, and J. Klippert, A modification of the Euler- 

Abel transform for convergent series, J. Natur. Sci. Math. 29 (1989), no. 1, 1–9. MR 91g:40008 

[3] G. A. Sorokin, O nekotorykh preobrazovanyakh ryadov, Izv. Vyssh. Uchebn. Zaved. Mat. 

(1984), no. 11, 34–40, 83. MR 86f:40003 



Efficient Solution of the Two-Dimensional 

Shallow-Water Equations using GPUs 

D.E. Bobolakis ∗ , A.I. Delis and E.N. Mathioudakis 


Technical University of Crete, Chania, Greece 

∗ dbobolakis@isc.tuc.gr 

Abstract 

A parallel algorithm for solving two-dimensional shallow-flow problems that takes advantage 

of modern computing accelerators such as graphics processing units (GPUs) is presented. The 

high-resolution Godunov-type explicit scheme is used, with Roe’s approximate Riemman solver 

to create a numerical method suitable for different types of flood simulation. The performance 

of a real-world dam collapse test-case using a massive grid with more than 1.9 million cells 

is investigated. The application is developed in double precision Fortran code using the OpenACC 

standard and the realization of the algorithm takes place on a HP SL390s G7 multicore 

system with Tesla M2070 GPUs and PGI’s compilers. Numerical results reveal an impressive 

agreement with the post-event survey. Solver’s parallel algorithm is designed to perform 

the total computation on the GPU, significantly expediting simulations when compared to the 

conventional Central Processing Unit (CPU) - Open Multi-Processing (OpenMP) performance. 

Scientific computations for GPU technology at shallow-water simulations yield to performance 

acceleration, when compared to classical parallel CPU - OpenMP realizations. 

Key words: Shallow-Water equations, Roe’s solver, CPU-GPU computations, OpenMP, OpenACC. 





Chania,Greece 

Complex Statistics and Diffusion in Nonlinear Disordered 

Particle Chains 

Tassos Bountis 1 Christos Antonopoulos 2 and Haris Skokos 3,4 

1 Center of Research and Applications of Nonlinear Systems (CRANS), 

Department of Mathematics, University of Patras, 26500, Greece 

2 Institute for Complex Systems and Mathematical Biology 

University of Aberdeen, AB24 3UE Aberdeen, UK 

3 Department of Mathematics and Applied Mathematics 

University of Cape Town, Rondebosch, 7701, South Africa 

4 Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece 

tassos50@otenet.gr, chris.antonopoulos@abdn.ac.uk, haris.skokos@uct.ac.za 

Abstract 

We perform a numerical study to investigate dynamically and statistically diffusive motion in a 

Klein-Gordon particle chain in the presence of disorder. In particular, we examine a low energy 

(subdiffusive) and a higher energy (self-trapping) case and verify that subdiffusive spreading is 

always observed. We then carry out a statistical analysis of the motion in both cases in the sense 

of the Central Limit Theorem and present evidence of different chaos behaviors, for various 

groups of particles. Integrating the equations of motion for times as long as 10 9 , our probability 

distribution functions always tend to Gaussians and show that the dynamics does not relax onto 

a quasi-periodic KAM torus and that diffusion continues to spread chaotically for arbitrarily 

long times. We also discuss some particular features that concern our numerical computations. 

Key words: Complex Statistics, Hamiltonian Systems, Klein-Gordon, q-Gaussians, Tsallis Entropy, Diffusive 

Motion, Weak and Strong Chaos 





A modified predictor-corrector method for the generalized 

Burgers–Huxley equation 

A. G. Bratsos 

Department of Naval Architecture, 

Technological Educational Institution (T.E.I.) of Athens, 

Athens, Greece. 

e-mail: bratsos@teiath.gr 

URL: http://users.teiath.gr/bratsos/ 

Abstract 

A third-order in time modified predictor-corrector method is proposed for the numerical solution 

of the generalized Burgers–Huxley (BgH) equation, which is given by 

( 

u t + αu δ u x − u xx = βu 1 − u δ) ( ) 

u δ − γ ; 0 ≤ x ≤ 1 , t > 0, (1) 

u = u (x, t) is a sufficiently differentiable function, with α a real parameter, β ≥ 0, γ ∈ (0, 1), 

δ > 0, initial condition u (x, 0) = f(x); x ∈ [0, 1] and boundary conditions u x | x=0, 1 

= g(t); 

t > 0. Eq. (1) is the modified Burgers equation for β = 0, is the Huxley equation for α = 0, 

δ = 1 and is the Fitzhugh-Nagoma equation for α = 0. 

Many researchers have used various methods to solve the BgH equation. A theoretical study 

of the BgH equation was found in [1], while as far as the numerical methods among others in 

[2] etc. 

The main aim of this paper is to solve the BgH equation explicitly with a direct method. To 

this attempt, the solution of the resulting nonlinear system is given by expressing the unknown 

vector component wise and updating each component as soon as its value becomes available. 

This process, which is known as a modified predictor-corrector method (see, e.g., [3] and references 

therein), opposite to the iterative classical predictor-corrector one is always explicit and 

is applied once, has also been examined successfully with various other approximations in time 

giving an improvement in the accuracy over the classical method. 

References 

[1] X. Deng, Travelling wave solutions for the generalized Burgers-Huxley equation, Appl 

Math Comput 204 (2008) 733-737. 

[2] M. Javidi, A numerical solution of the generalized Burgers-Huxley equation by spectral 

collocation method, Appl Math Comput 178(2006) 338–344. 

[3] A. G. Bratsos, An improved second-order numerical method for the generalized Burgers- 

Fisher equation, ANZIAM Journal 54(3) (2013) 181–199. 

Key words: Burgers–Huxley; Modified Predictor-Corrector; Finite-difference method 




Chania,Greece 

Structured Strong Linearizations obtained from Fiedler Pencils 

with Repetition 

Maria Isabel Bueno a , K. Curlett b , and Susana Furtado c 

a Department of Mathematics and College of Creative Studies, University of 

California 

Santa Barbara, CA, USA 

b College of Creative Studies, University of California 

Santa Barbara CA, USA 

c Faculdade de Economia da Universidade do Porto and Centro de Estruturas 

Lineares e Combinatrias da Universidade de Lisboa 

Portugal 

mbueno@math.ucsb.edu,curlett@umail.ucsb.edu,sbf@fep.up.pt 

Abstract 

Let P (λ) be a matrix polynomial of degree k ≥ 2 whose coefficients are n × n matrices with 

entries in a field F. A matrix pencil L(λ) = λL 1 −L 0 , with L 1 , L 0 ∈ M kn (F), is a linearization 

of P (λ) if there exist two unimodular matrix polynomials (i.e. matrix polynomials with constant 

nonzero determinant), U(λ) and V (λ), such that 

U(λ)L(λ)V (λ) = 

[ 

I(k−1)n 0 

0 P (λ) 

Beside other applications, linearizations of matrix polynomials are used in the study of the polynomial 

eigenvalue problem. For each matrix polynomial P (λ), many different linearizations can 

be constructed but, in practice, those sharing the structure of P (λ) are the most convenient from 

the theoretical and computational point of view, since the structure of P (λ) often implies some 

symmetries in its spectrum. 

In this talk we present nk × nk matrix pencils obtained from the family of Fiedler pencils 

with repetition, introduced by S. Vologiannidis and E. N. Antoniou (2011), preserving the 

structure of P (λ), when P (λ) is symmetric, skew-symmetric or T-alternating. Under certain 

conditions, these pencils are strong linearizations of P (λ). These linearizations are companion 

forms in the sense that, if their coefficients are viewed as k-by-k block matrices, each n × n 

block is either 0 n , ±I n , or ±A i , where A i , i = 0, . . . , k, are the coefficients of P (λ). 

] 

. 

Key words: Structured linearization, Fiedler pencils with repetition, matrix polynomial, companion form, 

polynomial eigenvalue problem. 



Unified Semi-Analytical, Semi-Numerical Approach 

to Stability Analysis of Nonparallel Unsteady Flows 

Georgy I. Burde a , Ildar Sh. Nasibullayev b and Alexander Zhalij c 

a Jacob Blaustein Institutes for Desert Research, Ben-Gurion University, 

Sede-Boker Campus, Israel 

b Institute of Mechanics, Russian Academy of Sciences, 

Ufa, Russia 

c Institute of Mathematics of the Academy of Sciences of Ukraine, 

Kyiv, Ukraine 

georg@bgu.ac.il,ildar@bgu.ac.il,zhaliy@imath.kiev.ua 

Abstract 

Stability of some unsteady nonparallel three-dimensional flows (exact solutions of the Navier- 

Stokes equations) is studied via separation of variables using a semi-analytical, semi-numerical 

approach. In this approach, a new coordinate system, which allows solution with separated variables, 

is defined together with the solution form. This part of the method involves complicated 

analytical calculations which can be implemented only using symbolic manipulating programs. 

The resulting eigenvalue problems are solved numerically with the help of the spectral collocation 

method based on Chebyshev polynomials. Such computational synthesis of analytical and 

numerical calculations allows to extend the physically significant concept of normal modes to 

the case of non-steady nonparallel basic flows for which this concept in its traditional form is not 

applicable at all. This may provide a basis for a well-grounded discussion of some problematic 

points of hydrodynamic stability analysis and a useful test for methods used in the hydrodynamic 

stability theory, in general. The basic flows whose stability is studied in the paper are 

of significant interest for fluid dynamics theory and have received considerable attention in the 

literature due to their relevance in a number of engineering applications. 

Key words: Linear stability, Nonparallel unsteady flows, Separation of variables, Direct method. 




Chania,Greece 

A recursive multilevel approximate inverse-based 

preconditioner for solving general linear systems 

Yi-Ming Bu a,b and Bruno Carpentieri b 

a School of Mathematical Sciences, University of Electronic Science and 

Technology of China, Chengdu, Sichuan 611731, China 

b Institute of Mathematics and Computer Science, University of Groningen, 

Groningen, The Netherlands 

y.bu@rug.nl,b.carpentieri@gmail.com 

Abstract 

We consider multilevel approximate inverse-based preconditioning techniques for solving systems 

of linear equations 

Ax = b (1) 

where A ∈ R n×n is a typically large, sparse, nonsymmetric matrix arising from the discretization 

of partial differential equations. Approximate inverse methods directly approximate A −1 

as the product of sparse matrices, so that the preconditioning operation reduces to forming one 

(or more) sparse matrix-vector product. Due to their inherent parallelism and numerical robustness, 

this class of methods is receiving renewed consideration for iterative solutions of large 

linear systems on emerging massively parallel computer systems. In practice, however, some 

questions need to be addressed. First of all the computed preconditioner could be singular. In 

the second place, these techniques usually require more CPU-time to compute the preconditioner 

than ILU-type methods. Third, the computation of the sparsity pattern of the approximate 

inverse can be problematic, as the inverse of a general sparse matrix is typically fairly dense. 

This leads to prohibitive computational and storage costs. 

We present an algebraic recursive multilevel inverse-based preconditioner, based on a distributed 

Schur complement formulation, that attempts to remedy these problems. The proposed 

solver uses recursive combinatorial algorithms to preprocess the structure of A and to produce 

a suitable permutation of the linear system that can maximize sparsity in the approximate inverse. 

An efficient tree-based recursive data structure is generated to compute and apply the 

approximate inverse fast and efficiently. We report on numerical experiments on matrix problems 

arising in different application areas to illustrate the potential of the proposed solver to 

reduce significantly the number of iterations of Krylov methods at low memory costs, also compared 

to other sparse approximate inverses and multilevel Schur-complement based incomplete 

LU factorization methods and software. Finally, we discuss block generalizations of our method 

that can exploit available block structure in the matrix to maximize computational efficiency. 

Key words: Krylov subspace methods, Approximate inverse preconditioners, combinatorial algorithms. 





Chania,Greece 

A local anisotropic adaptive algorithm 

to solve time-dependent dominated convection problems 

Jaime Carpio, Juan Luis Prieto 

Departamento de Ingeniería Energética y Fluidomecánica 

Universidad Politécnica de Madrid, Spain 

jaime.carpio@upm.es,juanluis.prieto@upm.es 

Abstract 

In this work we present a local, anisotropic adaptive algorithm useful to solve scientific and 

engineering time-dependent problems encompassing multiple scales. The algorithm is derived 

in the context of semi-Lagrangian schemes within a finite element framework, being suitable 

for higher-order finite elements. Convection-dominated equations, like those present in Fluid 

Dymanics, are ideal to employ anisotropic refinement due to the ‘directional features’ present 

in flows such jets, mixing layers, vortices... The size, shape and orientation of the anisotropic 

elements which define the optimal triangulation are provided by a metric tensor based on an a 

posteriori error indicator of the local or truncated error incurred at each time step. 

We illustrate the good performance of the algorithm with a convection-dominated problem 

taken from the Combustion field of knowledge. Simulation in 2D and 3D is also considered to 

address the interaction between a diffusion flame and a vortex generated by a turbulent flow. 

Finally, we include a comparison with actual experimental data. 

Key words: Semi-Lagrangian schemes, finite element method, a posteriori error indicator, local anisotropic 

refinement, combustion problems. 





Chania,Greece 

CPU-GPU computations for MultiGrid techniques coupled with 

Fourth-Order Compact Discretizations for Isotropic and 

Anisotropic Poisson problems 

N. E. Charalampaki and E. N. Mathioudakis 


Technical University of Crete, Chania, Greece 

ncharalabaki@isc.tuc.gr manolis@amcl.tuc.gr 

Abstract 

A CPU-GPU parallel algorithm for a fourth-order compact finite difference scheme with unequal 

mesh size in different coordinate directions, is designed to discretize a two dimensional 

isotropic or anisotropic Poisson equation in a rectangular domain. A multigrid technique with 

partial semi-coarsening strategy is used to iteratively solve the sparse linear system derived. 

Numbering the unknowns and equations according to the line red-black fashion, the coefficient 

matrix obtains a block structure suitable for parallel computations. These blocks consist of 

Toeplitz matrices with known inverses, allowing the efficient solution of inner linear systems 

on parallel computing environments with accelerators. The realization of the algorithm takes 

place on a HP SL390s G7 multicore system with Tesla M2070 GPUs and the application is 

developed in double precision Fortran code using the OpenACC standard with PGI’s compilers. 

The performance investigation reveals that the solution of fine discretization problems can be 

accelerated, although multigrid techniques usually yield poor efficiency on parallel computing 

architectures due to solution approximations of decreased size problems. 

Key words: Multigrid techniques, Compact finite difference schemes, GPU computations, OpenACC 




Chania,Greece 

Nonlinear Model Reduction with Localized Basis for 

Two-Phase Miscible Flow in Porous Media 

Saifon Chaturantabut a 

a Department of Mathematics and Statistics, Thammasat University, 

Pathumthani, Thailand 

saifon@mathstat.sci.tu.ac.th 

Abstract 

This work presents an application of a model reduction approach to substantially decrease 

the simulation time for two-phase nonlinear miscible flow in porous media. Since this type of 

flow often contains detailed features in fingering displacement, the approach considered here 

employs the localized basis sets from Proper Orthogonal Decomposition (POD) in the Galerkin 

projection procedure to accurately capture the important dynamics of the system. Discrete Empirical 

Interpolation Method (DEIM) with corresponding localized basis is then applied to efficiently 

compute the projected nonlinear terms in the POD reduced system. The related theoretical 

aspect of this approach is discussed. This work also proposes an adaptive scheme based 

on an error estimate indicator to choose a subdivision of the localized basis, together with an 

efficient procedure for updating each localized basis during the online simulation. This localized 

model reduction approach is shown to construct a system that can accurately capture the 

characteristics of the original miscible flow in the 2D finite-difference discretized setting, with 

the dimension reduced by a factor of O(10 2 ) and the CPU time decreased by a factor of O(10 3 ). 

Key words: Nonlinear Model Reduction, Proper Orthogonal Decomposition, Empirical Interpolation 

Methods, Nonlinear Partial Differential Equations, Miscible Viscous Fingering in Porous Media. 



On positivity preservation for finite element based methods for 

the heat equation 

Panagiotis Chatzipantelidis a 

a Department of Mathematics and Applied Mathematics, University of Crete, 

Greece 

chatzipa@math.uoc.gr 

Abstract 

We consider the model initial–boundary value problem 

u t − ∆u = 0, in Ω, u = 0, on ∂Ω, for t ≥ 0, u(0) = v, in Ω, (1) 

where Ω is a bounded convex polygonal domain in R 2 . By the maximum-principle, we have 

v ≥ 0 in Ω implies u(t, ·) ≥ 0 in Ω, for t ≥ 0. (2) 

Our purpose is to discuss analogues of this property for some finite element methods, based on 

piecewise linear finite elements, including, in particular, the Standard Galerkin (SG) method, 

the Lumped Mass (LM) method, and the Finite Volume Element (FVE) method. 

We consider the analog semidiscrete problem of (1), where we discretize space using either 

the SG, LM or FVE method. It is known that for the semidiscrete SG the analog of (2) does 

not hold for all t ≥ 0. However, in the case of the LM method, this holds if and only if the 

triangulation is of Delaunay type. For the FVE method we will show here that the situation is 

the same as for the SG method. 

However, when the solution is not positive for all t > 0, it may be positive for all t sufficiently 

large. We shall study this and approximate a corresponding minimum t 0 such that for 

all t > t 0 the solution of the semidiscrete problem is positive. Also we consider similar results 

for the corresponding fully discrete problems, when we discretize time with the backward Euler 

method. Finally we provide numerical results in 1 and 2 dimensions. 

Key words: positivity, finite element method, finite volume method, lumped mass method, Delaunay 

triangulation. 





Chania,Greece 

Block Hybrid Numerical Integrators for the 

Solution of Stiff Equations 

J. P. Chollom and G.M.Kumleng 

Department of Mathematics 

University of Jos 

Jos, Nigeria 

chollomp@gmail.com 

Abstract 

Stiff equations occur in a wide variety of applications including springs, damping systems 

and chemical reactions. The stiffness occurs due to the great difference among the reaction 

constants.Due to step size restriction, it becomes necessary to search for numerical methods 

with large regions of absolute stability . In this paper, new block hybrid linear integrators of the 

Adams Moulton class for the solution of stiff systems are constructed. This is achieved through 

the multistep collocation approach which yielded discrete schemes used simultaneously in block 

form as block integrators. This approach eliminates the use of starting values and overlap of 

pieces of solutions. The stability analysis of the new methods carried shows that they A-stable, 

a property desirable of any numerical method suitable for the solution of stiff systems. The 

new methods are tested on circular reaction equations, conserved systems, Robertson problem 

and a chemical reaction problem. The results shows that the new methods are efficient as they 

compare favorably with the state of the art Mat lab ode solver, ode23s. 

Key words: Block Linear integrators ,multi step collocation, A-stability, stiff systems, chemical reactions. 



Efficient GPU pricing of interest rate derivatives: 

PDE formulation and ADI methods 

Duy-Minh Dang a , Christina Christara b and Kenneth Jackson b 

a Department of Computer Science, University of Waterloo, 

Waterloo, Ontario, Canada 

b Department of Computer Science, University of Toronto, 

Toronto, Ontario, Canada 

dm2dang@uwaterloo.ca,ccc@cs.toronto.edu,krj@cs.toronto.edu 

Abstract 

We study the parallel implementation on a Graphics Processing Unit (GPU) of Alternating Direction 

Implicit (ADI) time discretization methods for solving three-dimensional time-dependent 

parabolic Partial Differential Equations (PDEs) with mixed spatial derivatives, and investigate 

the performance of the resulting parallel methods in pricing foreign exchange (FX) interest rate 

hybrids, namely Power Reverse Dual Currency (PRDC) swaps with various exotic features. 

A model for pricing PRDC swaps involves three stochastic factors, namely the FX rate, and 

the interest rates in the two currencies. By certain financial and mathematical arguments, the resulting 

model is a three-dimensional in space parabolic PDE, which includes all cross-derivative 

terms and, assuming a local volatility model, has variable coefficients. We use standard centered 

Finite Differences (FDs) for the space discretization and the Hundsdorfer-Verwer (HV) ADI 

method for the timestepping. We discuss the parallelization on a GPU of the computational 

requirements of the ADI method, such as the multiple tridiagonal solutions along each of the 

problem’s spatial dimensions and the matrix-vector products, with special attention to coalesced 

memory access. 

Furthermore, we consider the highly popular Target Redemption (TARN) feature for PRDC 

swaps, which adds path-dependency and, therefore, complexity to the PDE problem. The pricing 

of the FX-TARN PRDC swap is handled by breaking it down into several independent 

pricing subproblems over each period of the tenor structure. Each of the subproblems is solved 

on an individual GPU, with communication at the end of each period of the tenor structure taken 

care by MPI. 

We present numerical experiments that indicate considerable speedup, when comparing the 

CPU versus the GPU implementations, as well as the implementations on one versus multiple 

GPUs. 

Key words: Power Reverse Dual Currency (PRDC) swaps, local volatility, Target Redemption (TARN), 

three-dimensional parabolic PDE, Hundsdorfer-Verwer ADI method, Graphics Processing Unit (GPU), MPI. 





Dynamical and statistical behavior of the Fermi-Pasta-Ulam 

model with long-range interactions 

H. Christodoulidi, L. Cirto, T. Bountis and C. Tsallis 

Center for Research and Applications of Nonlinear Systems (CRANS), 

Department of Mathematics, University of Patras, GR–26500, Patras, Greece 

hchrist@master.math.upatras.gr 

Abstract 

In this talk we describe the results of a recent study on a long-range-interaction generalisation of 

the one-dimensional Fermi-Pasta-Ulam (FPU) β− model. In particular, we have used a coupling 

constant of the quartic interactions that decays as 1/r α and controls the range of interaction 

(α ≥ 0). We demonstrate that: (i) For α ≥ 1 the maximal Lyapunov exponent remains finite 

and positive for increasing number of oscillators N whereas, for 0 ≤ α < 1, it asymptotically 

decreases as N −κ(α) ; (ii) The distribution of time-averaged velocities is Maxwellian for α large 

enough, whereas it is well approached by a q-Gaussian, with the index q(α) monotonically 

decreasing from about 1.5 to 1 (Gaussian) when α increases from zero to close to one. To 

achieve these results for very large numbers of particles and very long integration times we made 

use of a number of numerical methods and strategies which will be discussed in the present talk. 

Key words: 

q–Gaussian Distributions, Lyapunov Exponents, Hamiltonian Lattices, Long Range Dynamics, Symplectic 

Integrators 





Chania,Greece 

Two numerical implementations of the Fokas method 

for elliptic equations in a polygon 

Kevin Crooks a 



k.m.crooks@maths.cam.ac.uk 

Abstract 

We consider the Dirichlet boundary value problem for the Helmholz and modified-Helmholz 

equations in a convex polygonal domain. Recent work has used the Fokas method to derive a 

Dirichlet to Neumann map for Laplace’s equation on the polygon: given Dirichlet data this map 

recovers our unknown Neumann data. These data are coupled by an integral equation known as 

the global relation. By reformulating the global relation as a linear operator equation of the form 

T Φ = Ψ, it was shown that T is a semi-Fredholm operator between Banach spaces. Analogous 

results may be obtained for the class of Helmholz equations by considering the resulting linear 

operators as perturbations, T β , of T . 

We analyse two numerical implementations: a Galerkin method and a pointwise method. 

For β not an eigenvalue for a domain, we use these approaches to solve the Dirichlet to Neumann 

map. Secondly, we demonstrate their use to search for eigenvalues, with the pointwise method 

seen to be particularly effective. Finally we discuss the numerical accuracy and difficulty of 

these implementations. 

Key words: Fokas Method, Helmholz, Boundary Value Problems, Numerical Approach. 



Solving Wiener-Hopf problems without kernel factorisation 

Darren G. Crowdy and Elena Luca 

Department of Mathematics, Imperial College London, UK 

d.crowdy@imperial.ac.uk, el1710@imperial.ac.uk 

Abstract 

We present a new approach for solving Wiener-Hopf problems by showing its implementation 

in two typical examples from fluid mechanics. The new method adapts various mathematical 

ideas underlying the so-called unified transform method due to A.S. Fokas and collaborators in 

recent years. The method has the key advantage of avoiding what is usually the most challenging 

part of the usual Wiener-Hopf approach: the factorisation of kernel functions into sectionally 

analytic functions. We show that the new approach leads naturally to fast and accurate schemes 

for evaluation of the solutions. 

Key words: Fokas transform method, Wiener-Hopf, complex analysis 



Numerical evaluation of hypersingular integrals on the semiaxis 

Maria Carmela De Bonis a and Donatella Occorsio a 

a Department of Mathematics, Computer Science and Economics, 

University of Basilicata, 

Potenza, Italy 

mariacarmela.debonis@unibas.it, donatella.occorsio@unibas.it 

Abstract 

We consider hypersingular integrals of the following type 

H p (g, t) = = 

∫ +∞ 

0 

g(x) 

dx, 

(x − t) p+1 

where 0 < t < +∞, p ≥ 1 is an integer and the function g behaves like x α for x → 0 and has 

an exponential or algebraic decay for x → +∞, i.e., it can be written in one of the following 

forms 

• g(x) = f(x)w α (x), w α (x) = x α e −x , α ≥ 0; 

• g(x) = f(x)u α,β (x), u α,β (x) = xα 

(1+x) β , α, β ≥ 0. 

They appear in different contexts and, in particular, in some problems of mathematical theory 

of elasticity (see [?]) and in hypersingular integral equations coming from Neumann twodimensional 

elliptic problems defined on a half-plane by using a Petrov-Galerkin infinite BEM 

approach as discretization technique (see [?]). To our knowledge, the literature dealing with the 

approximation of hypersingular integrals on unbounded intervals is very poor. 

In this talk we propose some numerical procedures for the pointwise approximation of the 

integrals H p (g, t) that are based on Gaussian-type quadrature formulas. We prove that such 

procedures are stable and convergent in suitable weighted uniform spaces and, for each of them, 

we give error estimates. Finally, we show that the theoretical results are confirmed by the 

numerical tests. 

Key words: hypersingular integrals, Gaussian-type quadrature rules. 

References 

[1] A. Aimi, M. Diligenti, Numerical integration schemes for hypersingular integrals on the real 

line, Communications to SIMAI Congress, doi: 10.1685/CSC06003, ISSN 1827-9015, Vol. 2 

(2007). 

[2] A.I. Kalandya, Mathematical methods of two-dimensional elasticity, Mir Publisher, Moskow, 

1975. 






A Characterization Theorem for the Discrete Best L 1 Monotonic 

Approximation Problem 

Ioannis C. Demetriou 

Department of Economics, University of Athens, 

Athens, Greece 

demetri@econ.uoa.gr 

Abstract 

Let n measurements of a real valued function of one variable be given. If the function is monotonic 

but the data have lost monotonicity due to measuring errors, then the least sum of the 

moduli of the errors that provides nonnegative first divided differences may be required. A 

characterization theorem is obtained for the solution of this problem in terms of Lagrange multipliers. 

Key words: first divided differences, monotonic approximation, L 1 -norm. 




Chania,Greece 

Numerical techniques for sliding motion 

in Filippov discontinuous systems 

Luca Dieci a , Cinzia Elia b and Luciano Lopez a 

a School of Mathematics, Georgia Tech Institute, Atlanta, GA 30332-0160, USA 

b Department of Mathematics, University of Bari, 70125 Bari, Italy 

dieci@math.gatech.edu,cinzia.elia@uniba.it,luciano.lopez@uniba.it 

Abstract 

In this talk we present numerical techniques to approximate the solution of a discontinuous 

differential system of Filippov type during sliding motion. Namely, for a given surface Σ defined 

as the 0-set of a smooth scalar function h: Σ = {x : h(x) = 0}, we have the problem 

x ′ = f(x), where f(x) = f 1 (x) when h(x) < 0, and f(x) = f 2 (x) when h(x) > 0. Further, 

(∇h) T f 1 > 0 and (∇h) T f 2 < 0, on and near Σ, so that trajectories are attracted to Σ and must 

remain there. 

So the main steps of a numerical procedure for solving such kind of problems are: reach Σ 

at x 0 (by an event location procedure) and, starting with x 0 ∈ Σ, solve the differential system 

x ′ = (1 − α)f 1 + αf 2 , where α has to be found so that x ′ is tangent to Σ: sliding motion. 

Here we propose an event location procedure, which determines the event point on Σ in a 

finite number of steps, and compare different numerical procedures to integrate our piecewise 

differential system during the sliding motion. 

It is well understood that when one integrates the differential system on Σ typically the numerical 

solution does not remain on Σ. Thus, the main feature of an effective numerical method 

is to require that the the numerical solution also remains on Σ. To achieve this, projection 

techniques can be used. 

We will consider the following three different flavors of projection techniques. 

(i) Classical projection of the numerical solution obtained by any method, say an explicit 

method. In particular, we will discus two ways to perform this projection. 

(ii) A change of variable technique, in case one can write explicitly h(x) = 0 ⇔ x k = 

g(x 1 , . . . , x k−1 , x k+1 , . . . , x n ). 

(iii) Reverse projection technique, whereby rather than starting with x 0 on Σ we seek a perturbed 

initial condition ˜x 0 ≈ x 0 such that the value of x 1 computed by one step of a 

numerical method starting at ˜x 0 is on Σ. Even for this technique, we will present different 

implementations. 

We will compare the above projection techniques on several examples. 

Key words: Discontinuous ODEs, Filippov systems, sliding motion, one-step methods, projection methods. 



A new filled function method applied to unconstrained 

global optimization 

T. M. El-Gindy, M. S. Salim and A. I. Ahmed 

Department of Mathematics, Faculty of Science 

Assiut University 

Assiut, Egypt 

Taha.elgindy@gmail.com, salim@yahoo.com, ibrahim@yahoo.com 

Abstract 

The filled function method is an efficient approach to find the global minimizer of multidimensional functions. 

A number of filled functions were proposed recently, most of which have one or two adjustable 

parameters. The idea behind the filled function methods is to construct an auxiliary function that allows 

us to escape from a given local minimum of the original objective function. It consists of two phases: local 

minimization and filling. So a global optimization problem can be solved via a two-phase cycle: 

In phase 1, we start from a given point and use any local minimization method to find a local minimizer 

x1 ∗ of f (x). 

In phase 2, we construct a filled function at x1 ∗ and minimize the filled function in order to identify a point 

x ′ with f (x ′ ) < f (x1 ∗). 

If such a point x ′ is found, x ′ is certainly in a lower basin than B ∗ 1 . Then we can use x′ as the initial 

point in phase 1 again, and hence we can find a better minimizer x2 ∗ of f (x) with f (x∗ 2 ) < f (x∗ 1 

). This 

process repeats until the time when minimizing a filled function does not yield a better solution. The 

current local minimum will be taken as a global minimizer of f (x). 

In this paper, we Consider the following unconstrained optimization problem: 

(P) 

min f (x) 

s.t. x ∈ R n . 

(1) 

let L(P) stands for the set of local minimizers of f (x). The new two-parameters filled function for problem 

(P) at the local minimizer x1 ∗ has the following form: 

( 

F(x, x1 ∗ , r, q) = 1 

|f (x) − f (x 

∗ ) 

1 + ‖x − x1 ∗‖ arctan 1 

) + r| 

|f (x) − f (x1 ∗)| + q , (2) 

where 0 < q < r and r satisfies 

0 < r < max 

x ∗ , x ∗ 1 ∈L(P) 

f (x ∗ )


References 

[1] T.M. El-Gindy, M.S. Salim, Abdel-Rahman Ibrahim, A Modified partial quadratic interpolation method for unconstrained 

optimization, Journal of Concrete and Applicable Mathematics−JCAAM 11(1) (2013) 136−146. 

[2] C. Wang, Y. Yang, J. Li, A new filled function method for unconstrained global Optimization, J. Comput. Appl. 

Math. 225 (2009) 68−79. 

[3] Y. Yang, Y. Shang, A new filled function method for unconstrained global Optimization, Appl. Math. Comput. 173 

(2006) 501−512. 





Chania,Greece 

Multiresolution analysis for 3D scattered data sets 

L. Fernández a , M. A. Fortes a and M. L. Rodríguez a 



lidiafr@ugr.es,mafortes@ugr.es,miguelrg@ugr.es 

Abstract 

In [1], the authors develop a multiresolution analysis in a one-dimensional context based on 

Harten’s multiscale representation (see e.g. [2, 3]). In the present work we propose to generalize 

[1] to a three-dimensional context in order to handle with clouds of 3D datasets: we obtain 

the decomposition and reconstruction algorithms associated to different interpolatory schemes, 

such as the one considering just function values, or the one considering function and first derivative 

values. Different interpolatory schemes will lead to consider different interpolatory spaces 

where to develop the algorithms. As an application of the developed theory, we will consider 

some examples regarding data compression and discontinuities detection. 

Key words: Multiresolution analysis, decomposition-reconstruction algorithms, compression data, discontinuities 

detection. 

References 

[1] R. M. Beam and R. F. Warming, Discrete multiresolution analysis using Hermite interpolation: 

Biorthogonal multiwavelets, SIAM J. Sci. Comput. 22(4), (2000) 1269–1317. 

[2] A. Harten, Discrete multi-resolution analysis and generalized wavelets, Applied Numerical 

Mathematics 12 (1993), 153–192. 

[3] A. Harten, Multiresolution representation and numerical algorithms: A brief review, ICASE 

Report No. 94-59, 1994. 



Exploring the Performance of Out-of-Core Linear Algebra 

Algorithms in Flash based Storage 1 

Athanasios Fevgas, Panagiota Tsompanopoulou and Panayiotis Bozanis 

Department of Electrical and Computer Engineering, University of Thesssaly, 

Volos, Thessaly, Greece 

fevgas@inf.uth.gr,yota@inf.uth.gr,pbozanis@inf.uth.gr 

Abstract 

In the recent years, flash memory has been widely utilized as storage medium to mobile 

and embedded systems, laptops and servers. The outstanding efficiency of flash based storages 

motivated us to study the performance of out-of-core linear algebra algorithms in flash SSDs. 

Flash memory is a non-volatile electronic storage that can be electrically erased and reprogrammed. 

There are two types of flash, NOR and NAND with the later utilized as mass storage 

medium. In the rest of this document the term flash denotes the NAND flash. Storages based on 

flash lack of mechanical and moving parts, providing low power consumption, shock resistance 

and high read/write performance. Flash consists of cells which store one or more bits. Cells are 

organized to pages and pages to blocks. Reads and writes are performed at page level, while 

erases at block level. Write operations are slower than reads and erases are even slower. Moreover, 

pages have to be erased before are re-written and flash endurance is limited by a finite 

number of write/erase cycles (wear out). Solid state drives (SSDs) are block devices compatible 

with traditional hard disk drives (HDDs) relying in flash memories. The main components of an 

SSD are the flash memory chips and a controller which emulates the block interface using FTL 

(Flash Translation Layer). FTL remaps logical addresses, used by the upper layers, to physical 

addresses in flash chips. It incorporates out-of-place-updates, wear leveling and garbage collection 

mechanisms aiming to improve write performance and prevent wear out. All the mentioned 

flash characteristics make data structures and algorithms designed for hard disks not performing 

well in it. Many recent studies, mostly in databases, aim to design new approaches suitable for 

flash. Some of them are using deltas instead of performing expensive page rewrites while others 

deferring operations in the future in order to reduce random writes. 

The development of efficient external memory (out-of-core) algorithms for solving linear equations 

systems or calculating eigenvalues of large matrices has been a popular research topic. 

Several algorithms have been proposed aiming to accelerate calculations by efficiently partitioning 

and managing large disk-resident datasets (matrices) into main memory blocks (submatrices). 

Alternative approaches require clusters with distributed memory, large enough for 

the entire dataset, and high bandwidth interconnections. Nowadays, the emergence of multiprocessor, 

multi-core and GPU accelerated computers provides high processing power at low 

cost. On the other hand, flash storages are capable to accelerate the storage layer. Considering 

the specifics of the flash memory, we present a study of the performance of few out-of-core 

algorithms for numerical linear algebra problems in flash based storages. 

Key words: Out-of-Core algorithms, linear algebra, scientific data, flash memory, SSD 




Social Fund. 



A comparative study on the effect of the ordering schemes for 

solving sparse linear systems, based on factored approximate 

sparse inverse matrix methods 

Christos K. Filelis-Papadopoulos, George A. Gravvanis 


School of Engineering, Democritus University of Thrace, 

University Campus, Kimmeria, GR 67100 Xanthi, Greece 

cpapad@ee.duth.gr,ggravvan@ee.duth.gr 

Abstract 

Preconditioned iterative schemes have been used extensively in many scientific disciplines, during 

the last decades for solving sparse linear systems. The effectiveness of the Preconditioning 

methods relies on the construction and use of efficient preconditioners, in the sense that are 

close approximants to the coefficient matrix of the linear system, suitable for modern computer 

systems. 

Recently, a class of Generic Approximate Inverses has been proposed that can handle any sparsity 

pattern of the coefficient matrix. A class of Generic Approximate Sparse Inverse matrix in 

conjunction with approximate inverse sparsity patterns, based on powers of sparsified matrices, 

has been proposed, that presented improved convergence behavior than existing Generic Approximate 

Banded Inverses schemes. Moreover, a factored approach, namely Generic Factored 

Approximate Sparse Inverse has been proposed, that further improved the convergence rate and 

further reduces the computational complexity and memory requirements. The Modified Generic 

Factored Approximate Sparse Inverse is a column wise variant that increases the performance 

by reducing the searches for nonzero elements required in the row-wise approach. The reordering 

schemes have been used to reduce fill-in for computing the decomposition factors of the 

coefficient matrix. Additionally, the reordering schemes have been used to increase the quality 

of incomplete factorization used in conjunction with preconditioned iterative methods. The various 

reordering schemes, namely Approximate Minimum Degree, the Reverse Cuthill-McKee 

and the Block Breadth First Search, affect the number of nonzeros and the quality of Modified 

Generic Factored Approximate Sparse Inverse. Moreover, the reordering schemes affect 

the sparsity pattern of the resulting approximate sparse inverse preconditioners and the convergence 

behavior of the proposed schemes. 

Finally, we examine the effectiveness and applicability of the various ordering schemes on the 

computation of the Modified Generic Factored Approximate Sparse Inverse (MGenFAspI) matrix 

in conjunction with the preconditioned Bi-Conjugate Gradient STABilized method for solving 

various problems from Matrix Market collection and numerical results are given, which are 

comparatively better than existing ones. 

Key words: Modified Generic Factored Approximate Sparse Inverses, Reordering schemes, Preconditioned 

iterative methods, Sparsity patterns. 




Chania,Greece 

Tsunami hazard and inundation 

for the northern coast of Crete 

Flouri Evangelia a,b , Vassilios Dougalis a , and Costas Synolakis b 

a Institute of Applied and Computational Mathematics, 

Foundation for Research and Technology Hellas, Heraklion, Crete 

b Department of Environmental Engineering, 

Technical University of Crete, Chania, Crete 

flouri@iacm.forth.gr 

Abstract 

Tsunamis are rare events compared to other natural hazards, but population growth along shorelines 

has increased their potential impact. Tsunamis are usually generated by an earthquake– 

induced dislocation of the seabed which displaces a large mass of water. They can be simulated 

effectively as long waves whose propagation and inundation are modeled by the nonlinear shallow 

water equations. 

In this work, we present a systematic assessment of earthquake-generated tsunami hazards 

for the northern coastal areas of the island of Crete. Our approach is based on numerical hydrodynamic 

simulations, including inundation computations, with the model MOST, using accurate 

bathymetry and topography data of the study area. MOST implements a splitting method 

in space to reduce the hyperbolic system of shallow water equations in two successive systems, 

one for each spatial variable, and uses a dispersive, Godunov–type finite difference method to 

solve the equations in characteristic form. 

In the present study we consider hypothetical, but credible, ‘worst case’ scenarios based on 

the unit sources methodology of NOAA, and, present inundation results, associated with seismic 

events of magnitude 8.5 originated in the Hellenic Arc, and 7.5 due to the seismic sources of the 

central Aegean sea. We also implement a probabilistic scenario in which we assess the influence 

of the epicenter location on the tsunami hazard, for time windows of 100, 500 and 1000 years. 

Our results include calculations of the maximum inundation and the maximum wave elevation 

for the two largest cities of the northern coast of Crete, Chania and Heraklion. We illustrate our 

findings superimposed on satellite images as maps indicating the estimated maximum values. 

Key words: tsunami hazard, inundation, Crete. 





Chania,Greece 

Eigenvalues and eigenfunctions for the Laplace Operator 

Athanassios S. Fokas a and Konstantinos Kalimeris b 


Cambridge, UK 

b RICAM, Austrian Academy of Sciences, 

Linz, Austria 

konstantinos.kalimeris@ricam.oeaw.ac.at 

Abstract 

The eigenvalues of the Laplace operator for the Dirichlet, Neumann and Robin problems 

in the interior of an equilateral triangle were first obtained by Lamé. Here, we present a simple, 

unified approach for deriving the relevant eigenvalues for several types of Boundary Value 

Problems (BVPs). Among these results the most general one consists of a system of explicit algebraic 

equations which give the eigenvalues for the Poincaré type BVP. These formulae for the 

Poincaré eigenvalues yield, via appropriate limits, the relevant formulae for the oblique Robin, 

Robin, Neumann and Dirichlet eigenvalues. The latter three give exactly the above mentioned 

results of Lamé. The method introduced here is based on the analysis of the so-called global 

relation, which as shown recently in the literature provides an effective tool for the study of 

BVPs. Moreover, we illustrate results considering the relevant eigenfunctions and some ideas 

related to other convex and bounded regular domains 

Key words: Eigenvalues, Laplace operator, global relation. 





Chania,Greece 

Filling holes with geometric constraints 

M. A. Fortes a , P. González a , A. Palomares a and M. Pasadas a 



mafortes@ugr.es,prodelas@ugr.es,anpalom@ugr.es,mpasadas@ugr.es 

Abstract 

Let D ⊂ R 2 be a polygonal domain, H be a subdomain of D and f : D − H −→ R be 

a function. In this paper we propose a method to reconstruct the ‘hole’ of f over H using a 

technique based on the minimization of an energy functional. More precisely, we construct a 

C 1 -Powell-Sabin spline function f ∗ over the whole D that approximates f outside H, and fills 

the hole of f inside H by respecting some geometric constraints.We present some graphical and 

numerical examples. 

Key words: Filling, approximation, finite element, Powell-Sabin, minimal energy. 




Chania,Greece 

Matrix-free resolution of PDEs using the Powell-Sabin FE 

Miguel A. Fortes a , Pedro González a , Antonio Palomares a and Miguel 

Pasadas a 



mafortes@ugr.es,prodelas@ugr.es,anpalom@ugr.es,mpasadas@ugr.es 

Abstract 

Let consider the following general boundary-value problem of second or fourth-order (depending 

on the values of τ 1 , τ 2 ∈ R, not vanishing simultaneously), 

⎧ 

⎪⎨ 

⎪⎩ 

−l ∂ (l) 

t u − τ 1 ∆u + τ 2 ∆ 2 u = f, t > 0 in Ω 

u (l t, ·) = ϕ (l t, ·) , τ 2 

∂u 

∂n (l t, ·) = τ 2 ψ (l t, ·) , t ≥ 0 on Γ 

l u (0, ·) = l u 0 (·) , (l − 1) ∂ ∂t u (0, ·) = (l − 1)u 1 (·) , on Ω 

where, depending on the value of l ∈ {0, 1, 2}, ∂ (l) 

t u will denote just u (for l = 0), ∂u 

∂t (for 

l = 1) or ∂2 u 

(for l = 2) and the problem considered may be elliptic, parabolic or hyperbolic. 

∂t 2 

For solving numerically any of these problems, depending on whether it is a transient 

parabolic or hyperbolic PDE problem, in a finite temporal interval [0, T ] ⊂ R (with T > 0), or 

just a stationary elliptic one (independent of time, or just considered only for t = 0), we will 

apply a general Galerkin procedure to their corresponding variational formulation. 

In this work we present a procedure to obtain a C 1 -surface on a polygonal domain Ω ⊂ R 2 , 

depending or not on time, that also solves the corresponding Galerkin variational formulation 

of a transient or stationary PDE problem up to fourth-order (1). The approximation space is that 

of C 1 -quadratic splines constructed from the Powell-Sabin subtriangulation associated with an 

α-triangulation of Ω. 

The main idea also is to try to avoid the resolution of any large linear system, or even to consider 

matrix-free formulations of the problems, using special triangulations with not too many 

(or even without any) nodes on the interior of the domain, and by using the appropriate interpolation 

conditions over some points in the boundary in order to take into account the boundary 

conditions for each of these problems. We study the actual feasibility of such procedure for these 

prototype problems, and give some numerical and graphical examples to assess their efficiency 

and reliability. 

(1) 

Key words: Powell-Sabin FE, interpolating PS-splines, matrix-free formulation. 





Chania,Greece 

Numerical Solution for Sparse Linear Systems that occur from 

the discretization of Boussinesq-type equations 

Maria Gaitani a , Maria Kazolea b and Argiris Delis a 

a School of Production Engineering and Management, Technical University of 

Crete, Chania, Crete, Greece 

b School of Environmental Engineering, Technical University of Crete, Chania, 

Crete, Grece 

mgaitani1@isc.tuc.gr,mkazolea@isc.tuc.gr, adelis@science.tuc.gr 

Abstract 

This work investigates preconditioned iterative techniques for the solution for sparse linear systems 

that occurs from the discretization of Boussinesq-type (BT) models using a finite volume 

scheme on unstructured meshes. The past few years enhanced Boussinesq-type (BT) models 

and their numerical solutions have evolved as predictive tools in the modeling of wave propagation 

and transformations. Recently, a novel high-order FV scheme on unstructured meshes 

for the extended 2D BT equations of Nwogu was developed. The equations of Nwogu are recasted 

in the form of a system of balance laws and are then numerically solved using a novel 

high-order well-balanced FV numerical method in unstructured meshes. In each time step the 

solution of a large sparse linear system (with a mesh depended matrix, M that occurs from the 

discretization of the dispersion terms) is mandatory to recover the velocity field. Matrix M is 

sparse, un-symmetric and often ill-conditioned. The properties of the matrix also vary on the 

physical situation of the problem examined. Various preconditioned and reordering strategies 

are investigated, including the ILU factorization the ILUT factorization and the CMK and RCM 

reordering techniques. Two iterative methods, BicGstab and GMRES, are tested for the solution 

process. A detailed comparison of the methods is given and their strengths and limitations of 

each are discussed. Furthermore, the performance of the various strategies is tested versus the 

most important parameters of the problem examined. 

Key words: sparse matrix, finite volumes, Boussinesq-type equations. 





Approximations Using Radon Projection Data in the Unit Disc 

Irina Georgieva a , Clemens Hofreither b and Rumen Uluchev c 

a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 

Sofia, Bulgaria 

b “Computational Mathematics”, Johannes Kepler University Linz 

Linz, Austria 

c Department of Mathematics and Informatics, University of Transport 

Sofia, Bulgaria 

irina@math.bas.bg, chofreither@numa.uni-linz.ac.at, rumenu@vtu.bg 

Abstract 

Noninvasive methods using line integrals for 2D object reconstruction have their theoretical 

foundation in the work of Johann Radon in the early twentieth century and have important 

practical applications in medicine, geology, radiology, astronomy, etc. 

In our survey we present recent results on various approximation problems where the basic 

information consists of line integrals in the unit disc. For instance, in 2D computing tomography, 

the data on which the reconstruction is based, comes as Radon projections along fixed directions. 

More generally, sometimes our data includes function values on the unit circle, in addition to 

line integrals. Our methods stay nondestructive in such a case and analytical as well. 

For Radon projection data we have studied: 

• interpolation and fitting by bivariate polynomials; 

• interpolation by quadratic bivariate splines; 

• interpolation and fitting by harmonic polynomials; 

• interpolation problem for the Poisson equation; 

• cubatures for harmonic functions. 

Questions under consideration were to determine sets of chords on the unit disc for which 

the relevant problem has a unique solution, developing numerical algorithms, error estimation, 

etc. 

Key words: Interpolation, fitting, cubature, bivariate polynomial, harmonic function, Poisson equation, 

Radon transform. 




Chania,Greece 

Splitting methods based on Approximate Matrix Factorization 

and Radau-IIA formulas for the time integration of advection 

diffusion reaction PDEs. 

Severiano González-Pinto a , Domingo Hernández-Abreu a 

a Departamento de Análisis Matemático. Universidad de La Laguna, 

Santa Cruz de Tenerife, Spain. 

spinto@ull.es,dhabreu@ull.edu.es 

Abstract 

A family of methods for the time integration of evolutionary Advection Diffusion Reaction 

Partial Differential Equations (PDEs) semi-discretized in space is introduced. The methods are 

obtained by combining a splitting J h = ∑ d 

j=1 J h,d of the Jacobian matrix J of the resulting 

ODE -where h is a small positive parameter related to the spatial resolution, such as the meshwidth- 

and a number of inexact Newton Iterations applied to the two-stage Radau IIA method. 

The overall process reduces the storage and the algebraic cost involved in the numerical solution 

of the multidimensional linear systems to the level of one-dimensional linear systems with small 

bandwidths. 

The local error of AMF-Radau methods when applied to semi-linear equations is described. 

From here, since the order in time is at most three, some specific methods considering up to 

three inexact Newton Iterations are selected. Furthermore, linear stability properties for the 

selected methods are established, in such a way that the wedges of stability depend on the 

number of terms d considered in the splitting J h . In particular, A(α d )-stability is shown for 

1 ≤ d ≤ 4, where α d := min{ π 2 , π 

2(d−1) 

}, and A(0)-stability for any d ≥ 1. 

Numerical experiments on 2D and 3D problems are presented, and they show that the methods 

compare well with standard classical methods in parabolic problems and can also be successfully 

used for advection dominated problems whenever some diffusion or stiff reactions are 

present. 

Key words: Evolutionary Advection-Diffusion-Reaction Partial Differential Equations, Approximate Matrix 

Factorization, Runge-Kutta Radau IIA methods, Stability. 



On the numerical modelling and solution of multi-asset 

Black-Scholes equation based on Generic Approximate Sparse 

Inverse Preconditioning 

Eleftherios-Nektarios G. Grylonakis, Christos K. Filelis-Papadopoulos, 

George A. Gravvanis 


School of Engineering, Democritus University of Thrace, 

University Campus, Kimmeria, GR 67100 Xanthi, Greece 

elevgryl@ee.duth.gr,cpapad@ee.duth.gr,ggravvan@ee.duth.gr 

Abstract 

One of the most important topics in the area of financial mathematics is the study of the multiasset 

Black-Scholes equation for the pricing of options. While there is a closed-form solution 

in one dimension for pricing European vanilla options, in higher dimensions the finite difference 

method allows the consideration of a wider range of parameters (coefficients of the partial 

differential equation, initial and boundary conditions). Hence, research efforts have been directed 

towards finding accurate prices for options with two or more underlying assets. In this 

paper, we present a fourth order accurate discretization scheme for the numerical solution of 

Black-Scholes equation in two space variables. 

The purpose of this work is to derive fourth order accurate option pricing methods while maintaining 

low computational complexity. For the space discretization we use a fourth order finite 

difference scheme combined with Richardson’s extrapolation method while for the time integration 

high order Backward Differences along with fourth order Gauss-Legendre Runge-Kutta 

scheme was used. The resulting sparse linear system of algebraic equations is solved by preconditioned 

iterative techniques based on generic approximate sparse inverses. Herewith, the 

Preconditioned Induced Dimension Reduction (PIDR(s)) method in conjunction with Generic 

Approximate SParse Inverse (GenAspI) is used for the efficient solution of the sparse linear 

systems. The GenAspI is computed through an incomplete factorization of the coefficient matrix 

to a predefined sparsity pattern acquired from Powers of Sparsified Matrices (PSM’s), thus 

handling any sparsity pattern. 

Numerical results are presented along with discussions for the proposed schemes in order to 

highlight the applicability and efficiency for solving the Black-Scholes equation in two space 

variables. The implementation issues of the proposed method are also discussed. 

Key words: Multi-Asset Black-Scholes equation, high order finite difference schemes, sparse linear systems, 

generic approximate sparse inverses, preconditioned induced dimension reduction method. 



The Error Analysis of the Indirect Padé Method for Matrix Exponential 1 

Chuanqing Gu, Ke Zhang 

Department of Mathematics, Shanghai University, Shanghai 200444, China 

cqgu@staff.shu.edu.cn,xznuzk123@126.com 

Abstract 

One of the most frequently discussed matrix function is the matrix exponential. Compared with 

other methods for computing the matrix exponential, the scaling and squaring method highlights 

itself not least for its implementation in MATLAB function expm. Najfeld and Havel in [Adv. in 

Appl. Math., 16 (1995)] presented an efficient algorithm using the scaling and squaring method 

as well as indirect Padé approximation which is a little different from the traditional method 

for the computation of the matrix exponential. This method is known for its lower computation 

cost, however, as pointed out by Higham in [SIAM J.Matrix Anal. Appl., 26 (2005)], it is lack 

of sufficient error analysis. In this paper we give an analysis of the sensitivity and conditioning 

of the matrix polynomial function H(B) by Najfeld and Havel given and conclude that H(B) 

is a well-conditioned matrix. We also present the relative error bounds of the approximating 

function (H 2m (B) + B)(H 2m (B) − B) −1 and exploit the impact of the condition number of 

H(B)−B and the scaling times d on the relative error bounds in a heuristic way. Our numerical 

result shows that the algorithm given by Najfeld and Havel generally provides accuracy almost 

the same as the MATLAB 7.6 functions expm and funm with a lower cost and proves to be 

stable. 

Key words: Matrix exponential, error analysis, Padé approximation, scaling and squaring, overscaling, 

MATLAB. 

1 The work are supported by National Natural Science Foundation (11371243), by Innovation major project of Shanghai 

Municipal Education Commission (13ZZ068) and by Key Disciplines of Shanghai Municipality (S30104). 




Chania,Greece 

Optimization of pre-recontruction restoration filtering for 

filtered back projection reconstruction (FBP) 

R. Guedouar 1 , A. Bouzabia 2 , B. Zarrad 3,4 

1 Biophysics department, faculty of pharmacy, University of Monastir, Tunisia 

2 Electronics and automatics department, High Institute of Informatics and Math, 

University of Monastir, Tunisia 

3 Biophysics and medical imaging department, Higher school of health sciences 

and technicals, University of Monastir, Tunisia 

4 Biophysics laboratory, Higher institute of medical technologies, University of 

Tunis-Elmanar, Tunisia 

raja guedouar@yahoo.fr 

Abstract 

Tomographic reconstruction is the technique underlying nearly all of the key diagnostic imaging 

modalities since they generated 2D/3D representation from a set of 2D projections acquired 

by a topographic system. For a long time, and despite the advantages of iterative algorithms, 

the FBP algorithm was preferred because it was computationally faster and more practical for 

routine use. However, in clinical practice, the FBP algorithm performances depend on several 

parameters effecting seriously final reconstruction results: The use of a smoothing filter to reduce 

the noise results in a loss of resolution and the choice of an optimal pre-reconstruction filter 

is necessary to provide the best trade-off between image noise and image resolution. A significant 

improvement in the quality of SPECT images has been demonstrated through the use of 2D 

pre-reconstruction restoration filtering of the projection images with FBP techniques. However, 

these filters should be designed to account for the image blurring, the noise level, and the imaged 

object to obtain the maximum restoration of image quality. In this work, we propose a userfriendly 

Interface for interactive optimization of FBP pre-construction filtering with emphases 

on Image-dependent restoration filters. The framework proposes an interactive visual algorithm 

for implementation of digital smoothing (hanning and Butterworth) and semi-automatically implementation 

of Metz restoration filters in frequency domain. To more objectively determine the 

optimum cut-off frequency, the user is assisted in visual feed-back optimization, by displaying 

the calculated of the power spectrum of both projection and estimated noise, and the filtered 

reconstructed images. A comparative study using 6464 2D-noiseless-numerical simulated and 

myocardial perfusion SPECT data, was conducted to investigated the performance of optimized 

filters on a pre-reconstruction task using FBP in terms of visual assessment, mean standard deviation 

and contrast. The reconstruction was done by a conventional FBP method using a ramp 

filter with no attenuation or scatter correction. Results show that optimized Metz restoration filtering 

provides reconstructed data with reduced noise without unduly penalizing resolution. It is 

also giving more improvement in clinical SPECT image contrast than Butterworth and Hanning. 





Chania,Greece 

On the Solution of the Linear Complementarity 

Problem by the Generalized Accelerated 

Overrelaxation Iterative Method 1 

Apostolos Hadjidimos a and Michael Tzoumas b 

a Department of Electrical and Computer Engineering 

University of Thessaly, 382 21 Volos, Greece 

b Department of Mathematics, University of Ioannina 

451 10 Ioannina, Greece 

hadjidim@inf.uth.gr,mtzoumas@sch.gr 

Abstract 

In the present work, we determine intervals of convergence for the various parameters involved 

for what is known as the Generalized Accelerated Overrelaxation iterative method for the solution 

of the Linear Complementarity Problem. The convergence intervals found constitute 

sufficient conditions for the Generalized Accelerated Overrelaxation method to converge and 

are better than what have been known so far. 

1 J. Optim. Theory Appl., in press, DOI 10.1007/s10957-014-0589-4 





Chania,Greece 

Strong-stability-preserving additive linear multistep methods 

Yiannis Hadjimichael a and David I. Ketcheson a 

a Computer, Electrical and Mathematical Sciences and Engineering Division, 

King Abdullah University of Science and Technology (KAUST), 

P.O. Box 4700, Thuwal 23955, Saudi Arabia 

yiannis.hadjimichael@kaust.edu.sa, david.ketcheson@kaust.edu.sa 

Abstract 

Semi-discretization of a variety of partial differential equations results in ordinary differential 

systems containing terms with different stiffness properties. In such cases additive methods can 

be used to make the most of the special structure of the resulting system. We study the monotonicity 

properties of additive linear multistep methods. We show that for a fixed number of 

steps and order of accuracy, optimal strong-stability-preserving (SSP) additive methods attain 

the same time-step restriction as the optimal SSP linear multistep methods, regardless of the 

stiffness of the problem. The concept of SSP linear multistep methods is also extended to problems 

for which the upwind- and downwind-biased operators have different stiffness properties. 

Key words: strong-stability-preservation (SSP), monotonicity, linear multistep methods, time integration 




Chania,Greece 

Fokas method and Kelvin transformation applied to potential 

problems in non convex unbounded domains. 

Maria Hadjinicolaou 

School of Science and Technology, Hellenic Open University, 

Patras, Greece 

hadjinicolaou@eap.gr 

Abstract 

In this presentation Fokas integral method is combined with Kelvin transformation to develop 

a new method for solving Dirichlet or Neumann problems in non-convex unbounded 

domains. A key aspect in Fokas method is the coupling of all boundary values in one equation, 

which has been termed global relation. Through this, any missing data on a boundary value 

problem can be derived, as Dassios and Fokas have shown. On the other hand, Kelvin transformation 

preserves harmonicity, and thus, by applying it to an exterior potential problem, the 

solution of the equivalent interior problem can be established, in the domain which is the Kelvin 

image of the original exterior one. In the present work, these two methods have been employed 

in order to derive integral representations for the Dirichlet and the Neumann problem in a nonconvex 

domain which is the Kelvin image of an equilateral triangle. The proposed methodology 

for the case of a Neumann exterior problem is given below. Physically, this could be explained 

as the construction of a potential for a vector field of which the effect of its normal derivative is 

known along its boundary that is assumed to be the image of an equilateral triangle under the 

Kelvin transformation. 

First, we apply the Kelvin inversion and thus the corresponding Neumann data on the boundary 

of the equilateral triangle are obtained. By then employing the Neumann to Dirichlet map, 

the Dirichlet data on the perimeter of the triangle are extracted. Subsequently, an integral representation 

of the solution of the Neumann problem in the interior of the triangle is accomplished. 

Applying again the Kelvin transformation to the attained Dirichlet data we derive the 

corresponding Dirichlet data on the initial boundary. By employing Kelvins 2-D theorem, we 

eventually obtain an integral representation of the solution of the Neumann problem in the given 

exterior non convex domain. Furthermore, we derive the Neumann to Dirichlet map for every 

Fourier component of some arbitrary data. This way, we provide a basis for representing the 

Dirichlet data on the boundary and thus we can obtain an integral representation of the solution 

of a large class of potential problems regarding non convex domains, encounted in many fields 

of science and engineering, that they would not be possible otherwise. 

Alternatively, in the case of a Dirichlet problem, we pursue the proposed methodology modified 

appropriately to obtain analogous results. 

Key words: Fokas method, Kelvin inversion integral representation , potential problems 





Chania,Greece 

Construction and approximation of surfaces by smoothing 

meshless methods. 

A. Hananel a , M. Pasadas a , and M. L. Rodríguez a 



ahananel@ugr.es, mpasadas@ugr.es, miguelrg@ugr.es 

Abstract 

In Earth science, especially Geology and other Sciences and Technologies, the reconstruction 

of surfaces from some scattered data set is a commonly encountered problem. 

In this work, under a generic schema, we enrich the theory of the discrete variational spline 

functions by minimizing some quadratic functional in a suitable space which can be a fairness 

functional, the flexion energy of a thin plate or others. 

It is essential to consider a finite dimension space of functions, where the minimization 

problem can be solved, and then a variational problem will be formulated. The discrete finite 

dimension space of functions that we propose, in this case, is a parametric finite dimensional 

space generated by a radial function basis. Then, we describe a smoothing meshless method of 

surfaces. The convergence of the problem is shown and finally, we analyze some numerical and 

graphical examples. 

Key words: Surfaces, approximation, smoothing, meshless methood. 





Chania,Greece 

The definitive estimation of the neuronal current via 

EEG and MEG using real data 

P. Hashemzadeh and A.S Fokas 


University of Cambridge, UK 

hashemzadeh@damtp.cam.ac.uk T.Fokas@damtp.cam.ac.uk 

Abstract 

The medical significance of Electroencephalography (EEG), and Magneto-Electroencephalography 

is well established, see for examples [1, 2, 3, 5]. EEG and MEG are considered two of 

the most important imaging techniques for real time brain imaging. In order to generate images 

of the brain activation using either EEG or MEG, it is necessary to analyse certain mathematical 

inverse problems. The definitive answer to the inverse source problem for the case of EEG 

and MEG was finally obtained by [4]. Here, we present reconstructions of the current using real 

data via the formulation proposed by [4]. The data was provided by the medical research council 

(MRC) Cambridge, UK. It involves both auditory and visual stimulus. We show comparisons of 

the reconstructed irrotational component of the neuronal current using EEG measurements and 

the radial component of the neuronal current using MEG measurements. Based on the results, 

we argue that EEG imaging technology has the potential to become the dominant real time, low 

cost brain imaging tool. 

References 

[1] Ribary U Ionannides A A Singh K D Hasson R Bolton J P R Lado F Mogilner A and LLinas 

R. Magnetic field tomography of coherent thalamocortical 40-hz oscillations in humans. Proc. 

Natl Acad. Sci. USA, 8,11 037-11 041, 1991. 

[2] Hauk O Rockstroth B Eulitz C. Gapheme monitoring in picture naming: an electrophysiological 

study of language production. Brain Topogr., 14:3–13, 2001. 

[3] Papanicolaou A C. The amensias: a clinical textbook of memory disorders. Oxford, UK: Oxford 

University Press., 2006. 

[4] A S Fokas. Electro-magneto-encephalography for a three-shell model: distributed current in 

arbitrary, spherical and ellipsoidal geometries. J.R.Soc. Interface, 6:479–488, 2009. 

[5] Langheim F J Leuthold A C Georgopolous A P. Synchronous dynamic brain networks revealed 

by magnetoencephalography. Proc, 103:455–459, 2006. 





Chania,Greece 

Numerical Solution of the Unified Transform for 

Linear Elliptic PDEs in Polygonal Domains 

P. Hashemzadeh and A.S Fokas 


University of Cambridge, UK 

hashemzadeh@damtp.cam.ac.uk T.Fokas@damtp.cam.ac.uk 

Abstract 

Integral representations for the solution of linear elliptic partial differential equations (PDEs) 

can be obtained using Green’s theorem. A new transform method for solving BVPs for linear 

and integrable nonlinear PDEs usually referred to as the Unified Transform or (Fokas Transform) 

was introduced by the second author [1]. The numerical implementation of this method has led 

to new numerical techniques for both evolution and elliptic PDEs, see for example [4, 5, 2, 3]. 

Here, we consider Laplace, Helmholtz, and modified Helmholtz equations in polygonal domains 

with a Robin boundary condition. We validate and compare the numerical solution obtained by 

Unified Transform to the solution obtained via the finite element method (FEM). We present a 

simple rule for choosing collocation points-i.e points in the Complex Fourier plane where the 

so called global relations are evaluated which guarantees a low condition number the matrix of 

the associated linear system. 

References 

[1] Fokas A S. A unified transform method for solving linear and certain nonlinear PDEs. Proc. 

R. Soc. A, 453:1411–1443, 1997. 

[2] Bengt Fornberg and Nathasha Flyer, A numerical implementation of Fokas boundary integral 

approach: Laplace’s equation on a polygonal domain, Proc. R. Soc.A, 467:2083–3003, 2011. 

[3] C.I. Davis and Bengt Fornberg, A spectrally accurate numerical implementation of the Fokas 

transform method for Helmholtz-type PDEs. Complex Variables and Elliptic Equations, 59: 

Issue 4:564–577, 2014. 

[4] Sifalakis A.G, Papadopoulou E.P and Saridakis Y G, Numerical study of iterative methods 

for the solution of the Dirichlet-Neumann map for linear elliptic PDEs on regular polygon 

domains, Int. J. Appl. Math. Comput. Sci , 4:173-178, 2007. 

[5] Sifalakis A.G, Fulton S.R, Saridakis Y.G. Direct and iterative solution of the generalized 

Dirichlet-Neumann map for elliptic PDEs on square domains. J. Comput. Appl. Math, 

227:171-184, 2009. 





Chania,Greece 

Symmetric Key Cryptography Algorithms Based on Numerical 

Methods 

Youssef Hassoun a and Hiba Othman b 

a,b Department of Mathematics, American University of Science and Technology, 

Beirut, Lebanon 

yhassoun@aust.edu.lb,hothman@aust.edu.lb 

Abstract 

Cryptography is used to protect information content communicated over a network from being 

accessed by adversaries. This is achieved by transforming (encrypting) plaintext before transmission 

in such a way that its contents can only be disclosed upon application of a reverse 

transformation (decryption). Both transformations involve a secret component which can be 

either the transformations themselves or some key used in the process. This paper focuses on 

implementing symmetric-key cryptography algorithms based on numerical methods. An empirical 

study is performed investigating the correlation of encryption and decryption efficiency of 

different root-finding numerical methods to the size of the plaintext and to key parameters. 

There are two categories of key-based cryptographic algorithms 1 , symmetric-key and asymmetrickey 

or public-key algorithms [1, 2, 3]. In the first category, sender and recipient share a private 

key known only to both; the same key is used for encryption and decryption. By contrast, in 

public-key cryptography two keys are used, one key is made publicly available to senders for 

encrypting plaintexts while a second key is kept secret and is used by the recipient for decrypting 

ciphered texts. 

Depending on the plaintext chunks on which an algorithm operates, symmetirc encryption algorithms 

are classified as stream and block ciphers. Stream ciphers operate on individual characters 

one at a time using time-varying encryption transformation. Block ciphers, on the other 

hand, operate on blocks of characters (n ≥ 64 bits) using fixed encryption transformation. 

Menzes et. al [3] defines a block cipher as an encryption function which maps n-bit plaintext 

block into an n-bit ciphertext block, where n represents the blocklength- a substitution cipher 

with a large character size. The function is bijective and is parametrized by a k-bit key. DES is 

an example of a 64-bit block cipher with a 56-bit key. Caesar cipher can be classified as a block 

cipher with one character block and a shift of k characters as key. 

We propose a symmmetric-key encryption algorithm based on solving a system of linear equations. 

It is a block cipher that maps n-characters of plaintext into n-characters of ciphertext. The 

1 Hash functions are one-way and do not fall into these categories, since there is no decryption 



key consists of (n×1) vector b j and an (n×n) matrix (a ij ). Encrypting a block of n characters, 

represented by (n × 1) vector (c j ), is achieved by solving the systems of linear equations: 

n∑ 

a ij x j = b j − c j (I) 

i=1 

Provided that det(a ij ) ≠ 0, the solution vector (x ∗ j ) exists; it represents the cipher text and is of 

dimension (n × 1). The condition on (a ij ) guarantees that encryption function is bijective and, 

consequently, has an inverse- the decryption function. Decrypting the ciphered text is achieved 

by substituting solution vectors into equation (I) giving rise to c j = ∑ n 

i=1 a ij x ∗ j − b j. 

Another algorithm proposed in [4] and based on solving non-linear equations can also be classified 

as a 1-character block cipher. Any non-linear function with one variable can be defined 

as a key 2 . The encryption function is defined as finding the solution of the equation: 

f(x) − c i = 0 

Here, (c i ) represents the numerical code of the i th character in the plaintext, e.g., the ascii-code, 

and f(x) is an arbitrary non-linear function, polynomial or otherwise. To guarantee that encryption 

function has an inverse, numerical encoding of plaintext together with f(x) must be 

chosen in such a way that equation (II) has at least one real root. The set of resulting roots {x ∗ i } 

represents the ciphertext. On the recipient side, each entry (x ∗ i ) in the ciphered text is decrypted 

by substituting it into f(x) giving rise to c i = f(x ∗ i ). 

To conclude, we propose symmetric encryption algorithms based on solving a system of linear 

equations as well as solving non-linear equations using numerical methods [5]. The proposed 

cryptosystems are block ciphers with matrices and non-linear functions as private keys. To make 

encryption functions invertible, and thereby guarantee decryption, the keys are constrained to 

satisfy some conditions. We implement the algorithms and perform an empirical study investigating 

the efficiency of encryption and decryption functions in terms of the functions’ parameters 

and in terms of plaintext message size. 

References 

[1] N. Ferguson, B. Schneier and T. Kohno (2010). Cryptography Engineering. John Wiley & 

Sons. ISBN: 9780470474242 

[2] B. Schneier (1996). Applied Cryptography: Protocols, Algorithms, and Source Code in C. 

John Wiley & Sons. ISBN: 0471117099 

[3] A. J. Menezes, P. C. van Oorschot and S. A. Vanstone (1996). Handbook of Applied Cryptography. 

CRC Press. ISBN: 0849385237 

[4] A. Ghosh and A. Saha (2013). A Numerical Method Based Encryption Algorithm with 

Steganography. R. Bhattacharyya et al. (Eds) : ACER 2013, pp. 149157. CS & IT-CSCP 

[5] R. L. Burden and J. D. Faires(2010). Numerical Analysis, 9 th Edition. Cengage Learning. 

ISBN-13: 9780538733519 

(II) 

Key words: Cryptography, Encryption, Numerical Methods, Symmetric-Key. 

2 The authors in [4] proposed a polynomial function of degree 3 as a key 

2 





Chania,Greece 

The Fokas Method and Initial-Boundary Value Problems 

for Multidimensional Integrable PDEs 

Iasonas Hitzazis 


Rion, Greece 

hitzazis@math.upatras.gr 

Abstract 

The Fokas Method, or Unified Transform Method, introduced in (A. S. Fokas, Proc. Roy. 

Soc. Lond. A 453 (1997), 1411-1443), is the appropriate generalization of the classical inverse 

scattering method which renders it applicable to the far more rich context of initial-boundary 

value problems (IBVPs) for integrable evolution PDEs in one spatial dimension. 

It is, however, interesting that there do also exist physically significant evolution equations 

in 2 or more spatial dimensions which share the property of integrability, i.e. that of admitting 

a Lax pair formulation. The most well-known integrable nonlinear PDEs in 2+1 (2 spatial and 

1 temporal) dimensions are the so-called Davey-Stewartson (DS) and Kadomtsev-Petviashvili 

(KP) equations. Recently ( A. S. Fokas, Commun. Math. Phys. 289 (2009), 957-993) took 

the first step towards the extension of his method to the case of multidimensions. In particular, 

the problem treated therein was the IBVP for the DS equation - as well as for its linearized 

version - posed on the half-plane. Soon thereafter, the half-plane case of the KP equation was 

also analyzed. 

In the present work we attempt to generalize this methodology so as to cover cases of more 

general domains, again in the context of (2+1)-dimensional PDEs. The linearized version of 

the DS equation is used as a prototypical example. In particular, we analyze the following two 

problems:(i) the quarter-plane IBVP, and (ii) the IBVP in a rectangular domain, both problems 

under smooth, temporally-decaying, non-homogeneous boundary conditions. 

The approach is totally based on the Lax pair formulation of the given PDE, and thus is the 

first step towards the construction of a formalism for the nonlinear case, i.e., for the DS equation 

itself, in each one of the geometries (i) and (ii). It is shown how both two eigenvalue equations 

constituting the Lax pair can undergo a simultaneous spectral analysis associated to any of the 

given domains (i) and (ii). Thus, in any one of the two cases, we achieve an appropriate d-bar 

problem for a sectionally non-analytic (in fact, sectionally generalized-analytic) function, i.e., 

for a function that has different generalized-analytic representations in different regions of the 

complex plane. 

In the presentation, if time permits, we will also briefly refer to the nonlinear case, i.e., to 

the DS equation itself in each one of the geometries (i) and (ii). 

Key words: Lax pairs, initial-boundary value problems, multidimensions. 




Chania,Greece 

Some new perturbation bounds of generalized polar 

decomposition 

X.-L. Hong, L.-S. Meng and B. Zheng 

School of Mathematics and Statistics, Lanzhou University 

Lanzhou, Gansu Province, P.R.China 

hongxiaoli2007@163.com,menglsh07@lzu.edu.cn,bzheng@lzu.edu.cn 

Abstract 

Let A, Ã = A + E ∈ Cm×n have the (generalized) polar decompositions 

A = QH and Ã = ˜Q ˜H, (1) 

where Q is subunitary and H is Hermitian positive semi-definite. We present the following 

new bounds of the positive (semi-)definite polar factor and the (sub) unitary polar factor for 

the (generalized) polar decomposition under the general unitarily invariant norm ∥ · ∥ and the 

spectral norm ∥ · ∥ 2 , which are stated as in the following theorem. 

Theorem. Let A, Ã = A + E ∈ Cm×n r have the (generalized) polar decompositions in (1). 

(1). When r and N ∈ C n×n 

> . Hence, all perturbation bounds in the above theorem 

can be naturally extended to the case of the weighted polar decomposition of A, which also 

improved the known perturbation bounds for the weighted polar decomposition. 

Key words: Perturbation bounds; Positive semi-definite polar factor; Subunitary polar factor; Generalized 

polar decomposition; Weighted polar decomposition; Unitarily invariant norm; Spectral norm. 





Chania,Greece 

On block preconditioners for PDE-constrained optimization 

problems 

Yu-Mei Huang a and Xiao-Ying Zhang a 

a School of Mathematics and Statistics, Lanzhou University, 

Lanzhou 730000, Gansu Province, P.R. China 

huangym@lzu.edu.cn, aiqian21921@163.com 

Abstract 

Recently, Bai proposed a block-counter-diagonal and a block-counter-triangular preconditioning 

matrices to precondition the GMRES method for solving the structured system of linear 

equations arising from the Galerkin finite-element discretizations of the distributed control problems 

in (Computing 91 (2011) 379-395). He analyzed the spectral properties and derived explicit 

expressions of the eigenvalues and eigenvectors for the preconditioned matrices. By applying 

the special structures and properties of the eigenvector matrices of the preconditioned matrices, 

we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give 

the asymptotic convergence factors for the preconditioned GMRES methods with the blockcounter-diagonal 

and the block-counter-triangular preconditioners. Experimental results show 

that the convergence analyses match well with the numerical results. 

Key words: PDE-constrained optimization, the GMRES method, preconditioner, condition number, asymptotic 

convergence factor. 





Chania,Greece 

Modeling drug release kinetics 

George Kalosakas 

University of Patras, Materials Science Dept., Rio Gr-26504, Greece, & 

Crete Center for Quantum Complexity and Nanotechnology, Physics Dept., Univ. 

of Crete, Greece 

georgek@upatras.gr 

Abstract 

We numerically calculate drug release profiles from simple or composite spherical devices, as 

well as from slabs of inhomogeneous thickness, using Monte Carlo simulations, when diffusion 

is the dominant release mechanism. In the case of spherical matrices the numerical results are 

compared with analytical solutions of Ficks second law of diffusion. 

Release curves are accurately described in all cases by the stretched exponential function. 

The dependence of the two stretched exponential parameters on the device characteristics is 

investigated and simple analytical relations are provided. Release kinetics does not depend on 

the initial drug concentration. 

We discuss in detail the numerical procedure followed in the Monte Carlo simulations. Particular 

emphasis is given in the techniques used to simulate regions of different drug diffusion 

coefficients when composite spheres are considered, as well as to construct slabs with inhomogeneous 

boundaries. 

Key words: drug release, diffusion, Monte Carlo simulations 





Chania,Greece 

Granular Transport Dynamics: Numerics and Analysis 

Giorgos Kanellopoulos and Ko van der Weele 

Center of Research and Applications of Nonlinear Systems (CRANS), 


Rio, 26500, Greece 

kanellop@master.math.upatas.gr, weele@math.upatras.gr 

Abstract 

We study numerically the transport of granular matter on a compartmentalized conveyor belt, 

being a representative model for numerous applications both in industry and the natural environment, 

and a prime example of an open multi-particle system prone to spontaneous pattern 

formation. When the inflow rate exceeds a certain critical threshold value, a cluster is formed 

at the entrance of the conveyor belt and the flow is obstructed. This behavior can be understood 

by a dynamical flux model, in which the flow from one compartment to the next is described 

by a flux function. We show how the detailed form of the flux function can be reconstructed 

from Molecular Dynamics simulations, using a least-squares method. We then investigate the 

relation between the form of the flux function and the precise way in which the transition from 

free flow to the clustered state takes place. In particular, we find that – depending on the reconstructed 

parameter values – this transition can either take place via a reverse or a forward period 

doubling bifurcation. 

Key words: Clustering, flux function analysis, molecular dynamics simulations 



Quantitative evaluation of SRT for PET imaging: 

Comparison with FBP and OSEM 

George A Kastis a , Anastasios Gaitanis b and Athanasios S Fokas a,c 

a Research Center of Mathematics, Academy of Athens, 

Soranou Efessiou 4, Athens 11527, Greece 

b Biomedical Research Foundation of the Academy of Athens (BRFAA), 

Soranou Efessiou 4, Athens 11527, Greece 

c Department of Applied Mathematics and Theoretical Physics, University of 

Cambridge, Cambridge, CB30WA, UK 

gkastis@academyofathens.gr,agaitanis@bioacademy.gr,t.fokas@damtp.cam.ac.uk 

Abstract 

SRT is a new, fast, algorithm for PET imaging based on an analytic formula for the inverse 

Radon transform. Its mathematical formulation involves the numerical evaluation of the Hilbert 

transform of the sinogram via an approximation in terms of ‘custom made’ cubic splines. 

Here, we present a comparison between SRT, filtered backprojection (FBP) and ordered-subsets 

expectation-maximization (OSEM) with 21 subsets at various iteration numbers (1, 2, 4, 6 and 

10) using simulated and real PET projection data. 

For the simulation studies, we have simulated sinograms of an image quality (IQ) phantom 

and a Hoffman phantom with an implanted tumor of various tumor-to-background ratios. Using 

these sinograms, we have created realizations of Poisson noise at five noise levels. In addition to 

visual comparisons of the reconstructed images, we have determined contrast, bias and radioactivity 

concentration ratios for different regions of the phantoms as a function of noise level. For 

the real-data studies, sinograms of an IQ phantom has been acquired from a commercial PET 

system. We have determined RCR and contrast for the various lesions of the IQ phantom. 

In all simulated phantoms, the SRT exhibits higher contrast and lower bias than FBP and 

OSEM at 2 iterations (clinical protocol) at all noise levels. The contrast and bias of OSEM 

approach the values of SRT after 6 iterations. However, the SRT reconstructions exhibit higher 

coefficient of variations (COV). In real studies, SRT exhibits better contrast and RCR in all 

spheres over both FBP and OSEM at the clinical protocol of 21 subsets and 2 iterations. This 

improvement increases as the diameter of the relevant spheres in the phantom decrease. 

In conclusion, SRT is an analytical algorithm with clearly improved quantification characteristics 

over FBP and the clinical protocol of OSEM. Since SRT increases the noise in the 

reconstructed image, further investigations are needed to determine appropriate applications for 

the algorithm. 

Key words: Image reconstruction-analytical methods, PET, filtered backprojection, ordered-subsets expectationmaximization, 

OSEM. 



A wave breaking mechanism for an unstructured finite volume 

scheme 

Maria Kazolea a , Argiris I. Delis b and Costas E. Synolakis a 

a School of Environmental Engineering, Technical University of Crete, Chania, 

Crete, Grece 

b School of Production Engineering and Management, Technical University of 

Crete, Chania, Crete, Greece 

mkazolea@isc.tuc.gr, adelis@science.tuc.gr, costas@usc.edu 

Abstract 

Wave breaking is a natural phenomenon of fundamental significance in the near-shore and one 

of the most important issues once have to consider in the numerical modeling of non-linear wave 

transformations. In this work a new methodology is presented and incorporated to TUCWave 

code, as to handle wave breaking over complex bathymetries in extended two-dimensional 

Boussinesq-type (BT) models. In the TUCWave code the 2D BT equations of Nwogu(1993), 

are solved using a novel high-order well balancing finite volume (FV) numerical method in unstructured 

meshes following the median dual node-centered approach. The novel wave breaking 

mechanism is of a hybrid type and consists of to parts. We first estimate the location of breaking 

waves using certain explicit criteria. Once breaking waves are recognized we switch locally 

in the computational domain from BT equations to the Non-linear Shallow Water Equations 

(NSWE) by suppressing the dispersive terms in the vicinity of the wave fronts. An additional 

methodology is presented on how to perform a stable switching between the BT and the NSWE 

equations within the unstructured FV framework. Comparison with laboratory data reveals that 

the proposed mechanism can accurately predict wave’s breaking position along with wave’s 

height decay and mean water level for both regular and solitary waves propagation on sloping 

beaches and submerged shoals. 

Key words: wave breaking, unstructured, Boussinesq-type equations 




Chania,Greece 

Numerical Simulation of Flow Separation Control using 

Dielectric Barrier Discharge plasma actuator 

R. Khoshkhoo and A. Jahangirian 

Aerospace Engineering Department Amirkabir University of Technology, Tehran, 

Iran. 

r.khoshkhoo@aut.ac.ir and ajahan@aut.ac.ir 

Abstract 

A numerical simulation method is employed to investigate the effect of the steady plasma body 

force over the stalled NACA 0015 airfoil on flow field at low Reynolds number flow condition. 

The plasma body force created by a Dielectric Barrier Discharge (DBD) actuator modeled with 

a phenomenological plasma method is coupled with 2-dimensional compressible Navier- Stokes 

equations. The body force distribution is assumed to vary linearly in the triangular region, and 

the body force decreases by going away from the surface.The equations are solved using an 

implicit finite volume method on unstructured grids.The responses of the separated flow field 

to the effects of a steady body force in various angles of attack are studied; also the effect of 

single- and multi-actuator and the positioning of the actuator on the flow field are investigated. 

It is shown that the DBD have significant effect on flow separation control in low Reynolds 

number aerodynamics. 

Key words: Flow control, Plasma actuator, Numerical simulation, Low Reynolds number. 




Chania,Greece 

An effective approach on finite-difference-time-domain method 

for quasi-static electromagnetic field analysis 

Minhyuk Kim a , Hyun-Kyo Jung a and SangWook Park b 

a Department of Electrical and Computer Engineering, Seoul National University, 

Seoul, Korea 

b ICT Convergence Research Team, EMI/EMC R&D Center, Corporation Support 

& Reliability Division, Korea Automotive Technology Institute, 

Chon-Ahn, Korea 

ejnp@snu.ac.kr, hkjung@snu.ac.kr, parksw@katech.re.kr 

Abstract 

This paper deals with an effective computational electromagnetic numerical method for the 

quasi-static field problems. There are lots of numerical technique to simulate electromagnetic 

problems. Among them, the finite-difference-time-domain (FDTD) is a popular method to analyze 

huge computational complexity problems. However, it is realistically impossible to apply 

directly standard FDTD method to the near-field analysis under a few MHz because of the time 

step problem. We overcome this by approximating the current source to have quasi-static behavior 

on the arbitrary surface at first. Then, the surfaces are employed in the FDTD method as the 

source excitation by the surface equivalence theorem. The time consuming computation problems 

are treated efficiently and the results of our method are in good agreement with full-wave 

analysis electromagnetic commercial solver. 

Key words: Finite-difference-time-domain, quasi-static electromagnetic field, surface equivalence theorem. 



 

Domain decomposition method with complete radiation 

boundary conditions for the Helmholtz equation in waveguides 

Seungil Kim a and Hui Zhang b 

a Department of Mathematics, Kyung Hee University, 

Seoul, South Korea 

b Section of Mathematics, University of Geneva, 

Geneva, Switzerland 

sikim@khu.ac.kr, mike.hui.zhang@hotmail.com 

Abstract 

In this paper, we present a nonoverlapping domain decomposition algorithm with a high-order 

transmission condition for the Helmholtz equation posed in a waveguide. We introduce the 

new high-order transmission conditions based on the complete radiation boundary conditions 

(CRBCs) that have been developed for high-order absorbing boundary conditions [3, 4]. We 

verify the rapid convergence of the Schwarz algorithm in terms of the order of CRBCs. It will 

be shown that damping parameters involved in the transmission conditions can be selected in an 

optimal way for enhancing the convergence of the Schwarz algorithm. This algorithm can also 

be employed efficiently for a preconditioner in GMRES implementations as recently developed 

sweeping preconditioners [1, 2, 5]. Finally, numerical examples confirming the theory will be 

presented. 

Key words: Helmholtz equation, domain decomposition, complete radiation boundary condition. 

References 

[1] Z. Chen and X. Xiang. A source transfer domain decomposition method for Helmholtz equations 

in unbounded domain. SIAM J. Numer. Anal., 51(4):2331–2356, 2013. 

[2] B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: moving perfectly 

matched layers. Multiscale Model. Simul., 9(2):686–710, 2011. 

[3] T. Hagstrom and T. Warburton. Complete radiation boundary conditions: minimizing the long 

time error growth of local methods. SIAM J. Numer. Anal., 47(5):3678–3704, 2009. 

[4] S. Kim. Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide 

with complete radiation boundary conditions. J. Math. Anal. Appl., 410(1):275–291, 2014. 

[5] C. C. Stolk. A rapidly converging domain decomposition method for the helmholtz equation. 

Journal of Computational Physics, 241(0):240 – 252, 2013. 




September 2–5, 2014 


Generalizations and Modifications of Iterative Methods for 

Solving Large Sparse Indefinite Linear Systems 

David R. Kincaid a , Jen-Yuan Chen b , and Yu-Chien Li b 

a Institute for Computational Engineering and Sciences, 

The University of Texas at Austin, Austin, Texas 78712 USA 

b Department of Mathematics, National Kaohsiung Normal University, 

Kaohsiung, TAIWAN, 

kincaid@cs.utexas.edu jchen@nknucc.nknu.edu.tw 

Abstract 

An overview of generalizations and modifications of iterative methods for solving large sparse 

indefinite linear systems with both symmetric and nonsymmetric coefficients matrices. 

Keywords: iterative methods, large sparse indefinite linear systems, generalizations and modifications of 

iterative methods, Arnoldi process, GMRES, SYMMLQ/SYMMQR, Lanczos SYMMLQ/SYMMQR. 

Frequently, when computing numerical solutions of partial differential equations, we need to 

solve systems of very large sparse linear algebraic equations of the form 

Ax = b 

where A is a given n × n matrix, b the given righthand side vector, and we seek a numerical 

solution vector x or a good approximation of it. Particularly for large linear systems arising from 

partial differential equations in three dimension, well-known direct methods based on Gaussian 

elimination may become prohibitively expensive in terms of both computer memory and computer 

time. On the other hand, iterative methods may avoid these difficulties. 

While the conjugate gradient (CG) method (and variations of it) may work well, for linear systems 

with a symmetric positive definite (SPD) coefficient matrices A, the choice of a suitable iterative 

method is not at all clear, when the linear system has a symmetric indefinite coefficient matrix. 

We discuss a variety of iterative methods that are based on the Arnoldi Process for solving large 

sparse symmetric indefinite linear systems. We describe the SYMMLQ and SYMMQR methods, 

as well as, generalizations and modifications of them. Then, we cover the Lanczos/MSYMMLQ 

and Lanczos/MSYMMQR methods, which arise from a double linear system. We present some 

pseudocodes for these algorithms. 

Finally, we mention some additional generalizations and modifications of iterative methods for 

solving large sparse symmetric and nonsymmetric indefinite systems of linear equations such as 

GMRES, MGMRES, MINRES, LQ-MINRES, QR-MINRES, MMINRES, and others. 



An energy market stackelberg game solved with particle swarm 

optimization 

Panagiotis Kontogiorgos, 1 Elena Sarri, 1 Michael N. Vrahatis, 2 George P. 

Papavassilopoulos 1 

1 Department of Electrical and Computer Engineering, National Technical 

University of Athens, Athens, Greece, 

2 Department of Mathematics, University of Patras, GR-26110 Patras, Greece, 

panko09@hotmail.com, elena@netmode.ntua.gr, vrahatis@math.upatras.gr, 

yorgos@netmode.ece.ntua.gr 

Abstract 

Complex interactions between stakeholders in deregulated markets are formulated using game 

theory notions. This study is motivated by energy markets and addresses Stackelberg games with 

a leader that decides first his strategy and many followers, each one with his own characteristics. 

A static Stackelberg game corresponding to a Voluntary Load Curtailment (VLC) program for 

energy consumers is formulated. This leads to a bilevel programming problem that is generally 

difficult to solve, due to nonlinearities, nonconvexities that arise and the large dimensionality of 

the problem due to the existence of many followers. In these problems metaheuristic algorithms 

become attractive. In the present study an algorithm for solving such problems is developed, 

using Particle Swarm Optimization (PSO), which is based on collective intelligent behaviors in 

nature and has gained wide recognition in recent years. Some examples are then solved using 

the proposed algorithm in order to evaluate its efficiency and examine the interactions between 

the players of the game. 

Key words: Complex Systems, Energy Market, Stackelberg, Particle Swarm Optimization 



Serial and Parallel Implementation of the 

Interface Relaxation Method GEO 1 

Aigli Korfiati a , Panagiota Tsompanopoulou b and Spiros Likothanassis a 

a Department of Computer Engineering and Informatics, 

University of Patras, Patras, Greece 

b Department of Computer and Communications Engineering, 

University of Thessaly, Volos, Greece 

korfiati@ceid.upatras.gr, yota@inf.uth.gr, 

likothan@ceid.upatras.gr 

Abstract 

Interface Relaxation (IR) methods are an interesting approach for the solution of multiphysics / 

multidomain problems. Assuming initial guesses on the interfaces of the original problem, IR 

methods iteratively solve the subproblems and relax for new values on the interfaces until convergence 

is succeed. Their main advantages are that their rates of convergence only depend on 

the parameters of the problem itself, the parameters related to its decomposition into subproblems 

and the parameters related to the operator imposed on the interfaces. In this paper a new 

implementation of an IR method named GEO is presented. GEO is based on a simple geometric 

coorrection mechanism and acts iteratively so as to relax the values of the solution on the interfaces. 

In particular, it adds to the old interface values a geometrically weighted combination of 

the normal boundary derivatives of the adjacent subdomains. 

In this paper GEO is implemented in FEniCS. The FEniCS project is a collection of free software 

for automated, efficient solution of differential equations. In order to evaluate the GEO 

implementation, it is applied on two different PDE problems with the same differential equation 

and boundary conditions and different domains. FEniCS methods are used to specify the problem’s 

subdomains properties (i.e. geometry, PDE operator and boundary/interface conditions). 

They are also used to generate and/or refine meshes (triangular elements) for each subdomain, 

solve the local PDE problems and show the computed results in the global domain and on the 

interfaces. Getting values of the solutions on the interface (boundaries of the subproblems) and 

passing the new relaxed values back to the subproblems as updated values for the boundary 

conditions is the main challenge of the IR methodology implementation and contribution of this 

paper. 

The experiments are performed for 2-dimensional elliptic partial differential model problems 

with partitions in multiple subdomains and the results are examined in terms of the method’s 

applicability and convergence. The exact solution and the computed approximations on the 

whole domain and on interface points, are depicted per iteration in appropriate graphs for applicability 

and convergence evaluation. A parallel implementation of the GEO method using 

FEniCS is also presented, as well as its performance comparison to the serial implementation. 

Key words: Interface relaxation, GEO method, multiphysics problems, parallel implementation, FEniCS. 




Social Fund. 





Chania,Greece 

A special class of integrable Lotka-Voltera systems and their 

Kahan discretization 

Theodoros Kouloukas 

Department of Mathematics, La Trobe University, Melbourne VIC 3086, Australia 

T.Kouloukas@latrobe.edu.au 

Abstract 

We present a family of integrable systems associated with a special set of polynomials z (n) 

i . 

The quadratic vector fields associated with z (n) 

3 are closely related to a class of Lotka-Voltera 

systems. We prove that they are superintegrable when n is odd and non-commutative integrable 

(of rank 2) when n is even. We also apply the Kahan-Hirota-Kimura discretization (a special 

Runge-Kutta method) to these quadratic vector fields, restricted to a subspace, and show that 

Liouville integrability and superintegrability are preserved. In the more general case of full, 

non-restricted, quadratic vector fields, numerical computations indicate integrability as well 

and therefore generate new questions for further investigation. 

Key words: Integrable systems, Kahan-Hirota-Kimura discretization 




Chania,Greece 

Constraint handling for gradient-based optimization of 

compositional reservoir flow 

Drosos Kourounis a 

a Institute of Computational Science, Università della Svizzera italiana 

CH-6904 Lugano, Switzerland drosos.kourounis@usi.ch 

Abstract 

The development of adjoint gradient-based optimization techniques for general compositional 

flow problems is much more challenging than for oil-water problems due to the increased complexity 

of the code and the underlying physics. An additional challenge is the treatment of non 

smooth constraints, an example of which is a maximum gas rate specification in injection or 

production wells, when the control variables are well bottom-hole pressures. Constraint handling 

through lumping is a popular and efficient approach. It introduces a smooth function that 

approximates the maximum of the specified constraints over the entire model or on a well-bywell 

basis. However, it inevitably restricts the possible solution paths the optimizer may follow 

preventing it to converge to feasible solutions exhibiting higher optimal values. A simpler way 

to force feasibility, when the constraints are upper and lower bounds on output quantities, is to 

satisfy these constraints in the forward model. This heuristic treatment has been demonstrated 

to be more efficient than lumping and at the same time it obtained better feasible optimal solutions 

for several models of increased complexity. In this work a new formal constraint handling 

approach is presented. Necessary modifications of the nonlinear solver used at every timestep 

during the forward simulation are also discussed. All these constrained handling approaches are 

applied in a gradient-based optimization framework for exploring optimal CO2 injection strategies 

that enhance oil recovery for a realistic offshore field, the Norne field. The new approach 

recovers 4% more oil than the best of the optimal solutions obtain by its competitors. 

Key words: Constraint-handling, production optimization, recovery-optimization, discrete adjoint. 



Combining Discontinuous Galerkin and Finite Differences 

Methods for Simulation of Seismic Wave Propagation 

Vadim Lisitsa a and Vladimir Tcheverda a 

a Institute of Petroleum Geology and Geophysics of SB RAS, 

Novosibirsk, Russia 

lisitsavv@ipgg.sbras.ru,tcheverdava@ipgg.sbras.ru 

Abstract 

In this paper we present an original approach to combination of standard staggered grid finitedifference 

scheme with interior penalty discontinuous Galerkin (DG) method for simulation of 

seismic wave propagation in presence of sharp interfaces with complex topography. The approach 

takes the advantages of the two numerical techniques. Discontinuous Galerkin methods 

is applied in the vicinity of the free-surface or sea-bed ensuring accurate approximation of the 

surface by the triangular (tetrahedral) mesh. However, DG is typically more computationally 

intense than finite differences (FD). So, in the major part of the model the FD are applied to 

reduce the overall computational cost of the algorithm. As the result the designed approach 

combines high accuracy of the DG with computational intensity of the FD. 

The idea of the approach is to combine the two approaches via transition zone where P0 discontinuous 

Galerkin method on a regular rectangular grid is used. This formulation is equivalent 

to the conventional (nonstaggered grid) scheme, approximating the elastic wave equation with 

a second order. So, the coupling of FD with the DG on arbitrary triangular mesh is reduced to 

the two independent problems. First, standard staggered grid scheme should be combined with 

a conventional scheme. This is done on the base of approximation of the reflection coefficients, 

either physical and artificial (corresponding to the spurious modes). Second, the DG on an arbitrary 

mesh is coupled with DG on a regular rectangular grid (conventional finite difference 

scheme). This is done on the base of hp-adaptivity of the DG method. 

Key words: discontinuous Galerkin, finite differences, elastic wave quation. 



Decreasing Computational Load by Using Similarity for 

Lagrangian Approach to Gas-solid Two-phase Flow 

Zhihong Liu a , Yoshiyuki Yamane a , Takuya Tsuji b and Toshitsugu Tanaka b 

a Heat and Fluid Dynamics Department, Research Laboratory, 

IHI Corporation, Yokohama,235-8501 Japan 

b Department of Mechanical Engineering, Osaka University, 

Osaka, 565-0871 Japan 

shikou ryuu@ihi.co.jp,yoshiyuki yamane@ihi.co.jp 

tak@mech.eng.osaka-u.ac.jp,tanaka@mech.eng.osaka-u.ac.jp 

Abstract 

Gas-solid two-phase flow constitutes a continuous-discrete phase system. It is natural to employ 

the Lagrangian approach to numerically analyze the discrete phase (solid particles); However 

the computational load becomes very heavy, when the number of particles increases. In this paper, 

we employ similarity rules to decrease the computational load. An imaginary system with 

imaginary gas and particles is used to describe the real gas-particle system. Each imaginary 

particle with a diameter K times of that of the real particle, replaces a group of real particles. 

By dimensionless analysis of the conservation equations of gas-solid two- phase flow, it is show 

that the solution of the imaginary system is similar to that of the real system, if the physical 

properties of the imaginary system are adjusted such that Reynolds number Re and Archimedes 

number Ar euqal those of the real system. Enlarging the imaginary particles by a factor K can 

decrease the number of particles and hence the computational load by a factor of K −4.5 . 

In order to validate the similarity rules, the behavior of a bubble in a fluidized bed is simulated 

with various factors K. It is shown that the similarities show reasonable accuracy. 

Nomenclature 

Re = |V − U|ρ f εD p 

µ f 

Ar = D3 pρ f (ρ p − ρ f )g 

µ 2 f 

D p particle diameter ε void fraction 

g gravity acceleration ρ f gas density 

U particle velocity ρ p particle density 

V gas velocity µ f gas viscosity 

Key words: Similarity, Lagrangian approach, computation load, two-phase flow simulation 



Curvilinear Grids for Five-Axis Machining 

Stanislav Makhanov 

School of Information and Computer Technology, Sirindhorn International 

Institute of Technology, Thammasat University, Thailand 

makhanov@siit.tu.ac.th 

Abstract 

Machining large complex industrial parts with a high accuracy often requires tens, hundreds of 

thousands or even millions of cutter location points and hundreds hours of machining. That 

is why reducing the machining time is one of the most important topics in the optimization of 

CNC codes for five axis milling machines. 

We propose and analyze a new method of constructing curvilinear tool paths which partly or 

even entirely align with the direction of the maximum material removal rate. The alignment 

based on the curvilinear elliptic grid generation allows to minimize the machining time while 

keeping the convenient zigzag-like topology of the path. The method is applicable to a variety of 

cost functions such as the length of the path, the machining speed, the material removal rate, the 

kinematic error, etc., generating different machining strategies. The method has been combined 

with a new version of the adaptive space filling curves. 

The approach has been tested against the standard iso-parametric zigzag, MasterCam X5 and 

the conventional space filling curves. The material removal rate cost function has been tested 

against the tool path length criteria. The numerical experiments, the real machining as well the 

accuracy measurements demonstrate a considerable advantage of the proposed method. 

Key words: numerical grid generation, milling machine, error minimization, tool path planning. 



Numerical Solution Of Optimization Problems for Semilinear 

Elliptic Equations with Discontinuous Coefficients and Solutions 

Aigul Manapova a , Fedor Lubyshev b 

a Department of Mathematics and IT, Bashkir State University, 

Ufa, Republic of Bashkortostan, Russian Federation 

b Department of Mathematics and IT, Bashkir State University, 

Ufa, Republic of Bashkortostan, Russian Federation 

aygulrm@yahoo.com 

Abstract 

Mathematical models of optimization for systems with distributed parameters (described by 

equations of mathematical physics) are the most difficult class of problems in optimization, 

especially for nonlinear optimal control problems. By ”non-linear optimization problems” for 

equations of mathematical physics we understand those in which the mapping g → u(g) from 

the set of admissible controls U to the space of states W is a nonlinear. Particular formulations 

of optimization problems for distributed parameter systems depend substantially on whether the 

controls enter into the free terms of the equations of state or in the equation coefficients and on 

whether linear or nonlinear PDEs describe the states of the control systems. Linear control systems 

with sufficiently smooth input data and control state functions have been most thoroughly 

studied and nonlinear optimization problem have been least studied to this day. Especially interesting 

from theoretical and practical points of view are optimal control problems in which the 

states are described by nonlinear PDEs with discontinuous coefficients and the solutions can be 

discontinuous due to the character of the physics process under study. 

Before solving optimal control problems numerically, they have to be approximated by problems 

of a simpler nature, specifically, by ”finite-dimensional problems”. One of the most convenient 

and universal techniques for finite-dimensional approximation as applied to optimization problems 

is the grid method. Also relevant is the question of the development of efficient numerical 

methods for solving constructed finite-dimensional grid optimal control problems, which requires 

effective procedures for calculating the gradient of the minimized functional. 

In this work we consider optimization problems for processes described by semilinear partial 

differential equations of elliptic type with discontinuous coefficients and solutions (with imperfect 

contact matching conditions), with controls involved in the coefficients. Finite difference 

approximations of optimization problems are constructed. For the numerical implementation of 

finite optimization problems differentiability and Lipshitz-continuity of the grid functional of 

the approximating grid problems are proved. Effective procedures for calculating gradients of 

minimized functionals using the solutions of direct problems for the state and adjoint problems 

are obtained. 

Key words: semilinear elliptic equations, optimization problem, numerical method. 





Chania,Greece 

A MultiGrid accelerated high-order pressure correction 

compact scheme for incompressible Navier-Stokes solvers 

V.G. Mandikas a , E.N. Mathioudakis a , G.V. Kozyrakis b,c , 

J.A. Ekaterinaris d and N.A. Kampanis b 

a Applied Mathematics and Computers Laboratory, Technical University of Crete, 

University Campus, 73132 Chania, Hellas 

b Institute of Applied and Computational Mathematics, Foundation for 

Research and Technology - Hellas, 70013 Heraklion, Crete, Hellas 

c Department of Marine Sciences, University of the Aegean, 

University Hill, Mytilene 81100, Hellas 

d Department of Aerospace Engineering, Daytona Beach College of Engineering, 

Embry-Riddle Aeronautical University, 600 S. Clyde Morris Blvd., Daytona 

Beach FL 32114, USA 

bmandikas@science.tuc.gr, manolis@amcl.tuc.gr 

gkoz@iacm.forth.gr, ekaterin@iacm.forth.gr 

kampanis@iacm.forth.gr 

Abstract 

A high-order accurate compact finite-difference numerical scheme, based on multigrid techniques, 

is constructed on staggered grids in order to develop an efficient incompressible Navier- 

Stokes solver. The enforcement of the incompressibility condition by solving a Poisson-type 

equation at each time step is commonly accepted to be the most computationally demanding 

part of the global pressure correction procedure of a numerical method. Since the efficiency 

of the overall algorithm depends on the Poisson solver, a multigrid acceleration technique coupled 

with compact high-order descretization scheme is implemented to accelerate the iterative 

procedure of the pressure updates and enhance computational efficiency. The employment of 

geometric multigrid techniques on staggered grids has an intrinsic difficulty, since the coarse 

grids do not constitute part of the finer grids. Appropriate boundary closure formulas are developed 

for the cell-centered pressure approximations of the boundary conditions. Performance 

investigations demonstrate that the proposed multigrid algorithm can significantly accelerate the 

numerical solution process, while retaining the high order of accuracy of the numerical method 

even for high Reynolds number flows. 

Key words: Global pressure correction, Poisson type equation, Incompressible Navier-Stokes equations, 

High-order compact schemes, staggered grids, Geometric MultiGrid techniques. 



A Fourier Collocation Method for the Nonlocal Nonlinear Wave 

Equation 

Gulcin M. Muslu a and Handan Borluk b 

a Istanbul Technical University, Department of Mathematics, 

Maslak 34469, Istanbul, Turkey. 

b Isik University, Department of Mathematics, 

Sile 34980, Istanbul, Turkey. 

gulcin@itu.edu.tr, hborluk@isikun.edu.tr 

Abstract 

We consider a general class of nonlinear nonlocal wave equation arising in one-dimensional 

nonlocal elasticity [1]. The model involves a convolution operator with a general kernel function 

whose Fourier transform is nonnegative. We propose a Fourier collocation numerical method for 

the nonlinear nonlocal wave equation. We first test our scheme for some well-known examples 

of nonlinear nonlocal wave equation, such as Boussinesq-type equations which arise from the 

suitable choices of the kernel function. To understand the structural properties of the solutions 

of nonlocal nonlinear wave equation, we present some numerical results illustrating the effects 

of both the smoothness of the kernel function and the strength of the nonlinear term on the 

solutions. 

This work has been supported by the Scientific and Technological Research Council of Turkey 

(TUBITAK) under the project MFAG-113F114. 

References 

[1] N. Duruk, H. A. Erbay, A. Erkip, Global existence and blow-up for a class of nonlocal nonlinear Cauchy 

problems arising in elasticity, Nonlinearity 23, 107–118, (2010). 

Key words: Nonlocal nonlinear wave equation, Boussinesq-type equations, Solitary waves. 





Chania,Greece 

Perturbation Theory of Dark-Bright solitons 

in Bose-Einstein condensates 

I.K.Mylonas a V.M.Rothos a , P.G.Kevrekidis b and D.J.Frantzeskakis c 

a Department of Mechanical Engineering, Faculty of Engineering Aristotle 

University of Thessaloniki GR54124 Thessaloniki, Greece 

b Department of Mathematics and Statistics, University of Massachusetts, 

Amherst, Massachusetts 01003-4515, USA 

c Department of Physics, University of Athens, Panepistimiopolis, Zografos, 

Athens 157 84, Greece 

imylon1986@gmail.com 

Abstract 

We develop a direct perturbation theory for a coupled dark-bright solitons and we derive the 

equation of motion for the soliton parameters. In this method, we solve the linearized wave 

equation around the solitons by expanding its solution into a set of complete eigenfunctions of 

the linearization operator. Suppression of secular growth in the linearized solution gives the 

evolution equations of soliton parameters. This method does not rely explicitly on the inverse 

scattering transform but its connection to the integrable theory is still visible since these eigenfunctions 

of the linearized equation are simply the squared eigenfunctions of the underlying 

scattering operator. Moreover, we study the stability of dark-bright solitons. Our analytical results 

for the small-amplitude oscillations of solitons is in good agreement with results obtained 

via a Bogoliubov-de Gennes analysis and compares very well with direct numerical computations. 

Key words: Solitons, Near-Integrable PDEs, BEC 





Efficient Unconstrained Optimization Multistart Solvers 

Using a Self-Clustering Technique 

Ioannis A. Nikas 

Department of Tourism Management, 

TEI of Western Greece, 

Patras, Greece 

nikas@teipat.gr 

Abstract 

One of the commonly occurring drawbacks in multistart solvers in unconstrained optimization 

problems is their inability to determine the quality and the quantity of local minima regions of 

attraction. Thus, when a sample of random points is generated it is not feasible the correspondence 

of each point to a single region of attraction, and consequently to a single local minimum. 

This results into a repeated application of a local search method on all sample points, finding, 

in general, multiple times the same local minima. 

In this work, motivated mostly on the above mentioned weakness of multistart algorithms 

and having in mind the local minimum definition, a new technique is proposed to correspond 

each sample point to a single candidate region of attraction. Specifically, each point of the 

sample is moved towards the best nearest neighbor point, which has the best functional value in 

this neighborhood. Through this process the sample points are concentrated around these best 

points, creating clusters that constitute candidate region of attractions. Then, it is assumed that 

each point inside a candidate region of attraction will drive a multistart algorithm to the same 

local minimum. For this reason, a new set of points is created. The new set will contain the best 

points inside each candidate region of attraction, namely the center of each self-clustered area, 

and these points will feed the multistart algorithm. 

It is noted that the number of created clusters depends on the sample size and the overall 

morphology of the objective function, that is the actual number of local minima. Furthermore, 

the proposed technique is a first-order process, that is, only functional values are necessary to 

determine the clusters and their corresponding centers. 

Finally, the proposed technique is utilized in a classic multistart solver, using a local search 

algorithm, and is tested on a set of well-known, one-dimensional test function. The results of 

these experiments show that in most cases the proposed technique makes the classic multistart 

algorithm efficient in finding uniquely many local minima. In addition, the experimental results 

showed that in most of the cases the global minimum is also found and as the grid size grows 

up, the number of clusters tends rapidly to the total number of local and global minima of the 

objective function. 

Key words: multistart algorithm, unconstrained optimization, self-clustered technique. 





Chania,Greece 

Essential spectral equivalence via multiple step preconditioning 

and applications to ill conditioned Toeplitz matrices 

Dimitrios Noutsos a , Stefano Serra-Capizzano b and Paris Vassalos c 

a Department of Mathematics, University of Ioaninna, 

Ioannina, Greece 

b Department of Science and high Technology, 

University of Iunsubria, Como, Italy. 

c Department of Informatics, Athens University of Economics and Business, 

Athens,Greece. 

dnoutsos@uoi.gr, stefano.serrac@uninsubria.it, pvassal@aueb.gr 

Abstract 

We are concerned with the fast solution of Toeplitz linear systems with coefficient matrix T n (f), 

where the generating function f is nonnegative and has a unique zero at zero of any real positive 

order θ. As preconditioner we choose a matrix τ n (f) belonging to the so-called τ algebra, 

which is diagonalized by the sine transform associated to the discrete Laplacian. In previous 

works, the spectral equivalence of the matrix sequences {τ n (f)} n and {T n (f)} n was proven 

under the assumption that the order of the zero is equal to 2: in other words the preconditioned 

matrix sequence {τn 

−1 (f)T n (f)} n has eigenvalues, which are uniformly away from zero and 

from infinity. Here we prove a generalization of the above result when θ < 2. Furthermore, 

by making use of multiple step preconditioning, we show that the matrix sequences {τ n (f)} n 

and {T n (f)} n are essentially spectrally equivalent for every θ > 2, i.e., for every θ > 2, 

there exist m θ and a positive interval [α θ , β θ ] such that all the eigenvalues of {τn 

−1 (f)T n (f)} n 

belong to this interval, except at most m θ outliers larger than β θ . Such a nice property, already 

known only when θ is an even positive integer greater than 2, is coupled with the fact that the 

preconditioned sequence has an eigenvalue cluster at one, so that the convergence rate of the 

associated preconditioned conjugate gradient method is optimal. As a conclusion we discuss 

possible generalizations and we present selected numerical experiments. 

Key words: Toeplitz systems, τ algebra, preconditioning. 



Nyström methods for two-dimensional 

Fredholm integral equations on unbounded domains 

Donatella Occorsio1 a , and Maria Grazia Russo a 

a Department of Mathematics, Computer Science and Economics, 

University of Basilicata, Potenza, Italy 

donatella.occorsio@unibas.it, mariagrazia.russo@unibas.it 

Abstract 

We investigate the numerical solution of two-dimensional Fredholm integral equations defined 

on the set S = [a, b] × [c, d], −∞ ≤ a 

∫ 

f(x, y) − µ k(x, y, s, t)f(s, t)w(s, t) ds dt = g(x, y), (x, y) ∈ S, (1) 

S 

where w(x, y) := w 1 (x)w 2 (y) and w 1 , w 2 are suitable weight functions defined on [a, b], [c, d] 

respectively, µ is a real number. k and g are given functions defined on ([a, b] × [c, d]) 2 and 

[a, b]×[c, d] respectively, which are sufficiently smooth on the open sets but can have (algebraic) 

singularities on the finite boundaries and an exponential growth at ±∞ at most w.r.t. each 

variable. f is the unknown function. 

Therefore S is intended to be an unbounded domain, for instance a quarter of the plane, a 

strip etc. 

We introduce some Nyström methods based on cubature formulas obtained as tensor products 

of two Gaussian quadrature formulas w.r.t. the weights w 1 , w 2 . Due to the “unboundedness” 

of the domain we need to “truncate” the quadrature rules. The convergence, stability and 

well conditioning of the methods are proved in suitable weighted spaces of continuous functions. 

Some numerical examples illustrate the efficiency of the methods. 

Key words: Fredholm integral equation, Nyström method, spectral methods 



Numerical stability of block direct methods for solving 

symmetric saddle point problem 

Felicja Okulicka-Dłużewska, Alicja Smoktunowicz 

Faculty of Mathematics and Information Science, Warsaw University of 

Technology, ul. Koszykowa 75, Warsaw, 00-662, Poland 

F.Okulicka@mini.pw.edu.pl, A.Smoktunowicz@mini.pw.edu.pl 

Abstract 

We study the numerical properties of some block direct methods for solving the following saddle 

point problem (quasidefinite case) 

( ) ( ) 

A B x 

Mz = f ⇔ 

B T −C y 

= 

( 

b 

c 

) 

, (1) 

where A ∈ R m×m , C ∈ R n×n are symmetric positive definite and B ∈ R m×n , n ≤ m. 

Then M is nonsingular and there is a unique solution (x ∗ , y ∗ ) of (??). Such problems arise in 

many applications, e.g., in optimization, in the solution of PDEs, weighted least squares (image 

restoration), FE formulations of consolidation problem. The matrices A and B are usually 

large, sparse and ill-conditioned. Structure of the problem leads to the application of block 

methods which operate on groups of columns of M and allow to apply the BLAS-3 compatible 

algorithms. It is known that the block LU methods are not numerically stable, in general. The 

block factorization 

( ) ( 

A B 

M = 

B T = 

−C 

I 0 

B T A −1 I 

) ( A 0 

0 −(C + B T A −1 B) 

) ( I A −1 ) 

B 

0 I 

is commonly used to solve the equation (??). We analyze the methods avoiding computing the 

Schur complement S = −(C + B T A −1 B). If C is ill-conditioned then the computed Schur 

complement S may be singular in working precision. We propose and analyze algorithms for 

solving symmetric saddle point problem which are based upon the block Cholesky decomposition 

and the block Gram-Schmidt method. In particular, we prove that the algorithm BCGS2 

(Reorthogonalized Block Classical Gram-Schmidt) using Householder Q-R decomposition implemented 

in the floating point arithmetic is backward stable, under a mild assumption on the 

matrix M. 

Extensive numerical testing was done in MATLAB to compare the performance of some 

direct methods for solving linear system of equations of special block matrices. 

Key words: symmetric quasidefinite (sqd) systems, saddle point problem, QR decomposition, numerical 

stability, condition number, iterative refinement. 



Robust numerical simulation of reaction-diffusion models 

arising in Mathematical Ecology 

Kolade M. Owolabi a and Kailash C. Patidar b 

a Department of Mathematics and Applied Mathematics, University of the Western 

Cape, Private Bag X17, Bellville 7535, South Africa 

b Department of Mathematics and Applied Mathematics, University of the Western 

Cape, Private Bag X17, Bellville 7535, South Africa 

Speaker: Kailash C. Patidar, E-mail: kpatidar@uwc.ac.za 

Abstract 

In this work, we consider numerous types of reaction-diffusion models arising in mathematical 

ecology. Using local analysis theory applied to ecological modeling, we study four important 

ecological systems describing some prey-predator models. We address the interaction between 

two species in terms of predator-prey systems and the biological system displaying the formation 

of chaotic spatiotemporal patterns arising from a community of three competitive species. 

Then we design and analyze robust time-integration techniques to simulate these models. Two 

competing exponential time-differencing methods that are of order-four are used as the major 

time stepping methods. We justify the supremacy of these two schemes when applied to above 

mentioned dynamical systems and compared our results with those obtained by other existing 

multistep exponential integrators of orders four, five and six. 

Key words: Reaction-diffusion models, Mathematical Ecology, Exponential time-differencing methods. 




Chania,Greece 

Unified Tranforms and classical spectral theory of operators 

Beatrice Pelloni a and David A Smith b 

a Department of Mathematics, University of Reading, 

Reading, UK 

b Department of Mathematics, University of Cincinnati, 

Cincinnati, OH, USA 

b.pelloni@reading.ac.uk,david.smith2@uc.edu 

Abstract 

We describe how the application of the unified method of Fokas to the study of linear evolution 

PDEs sheds light on the spectral theory of non self-adjoint linear differential operators. 

Key words: unified transform, fokas method, spectral theory. 



MATLAB : Parallel and Distributed Computing 

using CPUs and GPUs 

K. Petsounis 

Mentor Hellas Ltd 

Greece 

costas@mentorhellas.com 

Abstract 

MATLAB is a high level structured language and an interactive development environment for 

technical computing and algorithm development. It has enabled scientists and engineers to efficiently 

process and analyze data, develop and deploy algorithms and applications. Furthermore, 

Parallel and Distributed Computing capabilities in MATLAB, allow users to solve computationally 

and data intensive problems by taking advantage of the latest multiprocessing systems: 

multicore desktops, computer clusters, GPUs, grid and cloud computing services. It is now possible 

to interactively prototype and develop distributed and parallel applications, briefly touch 

upon the parallel data structures, such as distributed arrays, and programming constructs such 

as parallel for loops, parallel numeric algorithms and message passing functions. Using typical 

numerical computing problems as examples, this workshop describes how to use MATLAB 

parallel tools to take full advantage of the performance enhancements offered by multicore / 

multiprocessor computing environments. In addition, you will learn how you can leverage the 

computing power of NVIDIA CUDA-enabled GPUs to accelerate your MATLAB applications 

with minimal programming effort using GPU arrays and GPU enabled MATLAB functions. 

Sample codes, differences in CPU and GPU implementations as well as benchmark results for 

some typical numerical computing problems will be presented. 



Method for solving nonlinear singular problems 

Agnieszka Prusińska a and Alexey A. Tretýakov b,c 

a Department of Mathematics and Physics, Siedlce University of Natural Sciences 

and Humanities, Siedlce, Poland 

b System Research Institute, Polish Academy of Sciences 

Warsaw, Poland 

c Computing Center, Russian Academy of Sciences, 

Moscow, Russia 

aprus@uph.edu.pl, tret@uph.edu.pl 

Abstract 

The aim of our work is to investigate conditions for existence of local solutions to nonlinear 

equations of the form 

F (x) = 0, F : R n → R m , (1) 

in degenerate case, i.e. when Im F ′ (x 0 ) ≠ R m and x 0 is a chosen initial point. We propose an 

algorithm of the numerical method that is convergent to a solution point in the mentioned case. 

In our approach we use p-factor operator and some elements of p-regularity theory [1]. Main 

result of this theory is a description of the tangent cone to the solution set in degenerate case. 

We assume that F is a p + 1 times differentiable mapping and for some h ∈ R n consider 

the sequence 

x k+1 = x k − Λ −1 

h (f 1(x 0 + ωh + x k ), . . . , f p (x 0 + ωh + x k )) , k = 1, 2, . . . , (2) 

( 

where 0 < ω < 1/2. An operator Λ h = f 1 ′(x 0), f 2 ′′(x 

1 

0)[h], . . . , 

(p−1)! f p (p) 

) 

(x 0 )[h, . . . , h] for 

x ∈ R n is called p-factor operator. To construct this operator we decompose R m into a direct 

sum Y 1 ⊕· · ·⊕Y p and define auxiliary mappings f i : R n → Y i , f i (x) = P Yi F (x), i = 1, . . . , p, 

such that f (k) 

i (x 0 ) = 0, k = 1, . . . , i − 1 and P Yi : R m → Y i – the projection operator, where 

Y i is closed subspace of Y (see [1]). For a linear operator Λ h we define its right inverse Λ −1 

h 

and Λ −1 y is an element x ∈ Rn such that ‖x‖ = min {‖z‖ : Λ h (z) = y}. By the “norm” of 

Λ −1 

h 

h 

we mean the number ‖Λ−1 

h ‖ = sup ‖y‖=1 inf{‖x‖ : Λ h x = y, x ∈ R n }. If Im Λ h = R m , 

then the sequence (2) is convergent to the solution of (1). 

We illustrate the basic idea of the method with some numerical examples. 

Key words: p-regularity, singularity, contracting mapping, multimapping, p-factor operator. 

References 

[1] Tretýakov A. A., Marsden J. E.: Factor-analysis of nonlinear mappings: p-regularity theory, 

Commun. Pure Appl. Math. 2, 425–445 (2003) 






Solving the Fredholm integral equation of the second kind by 

global spline quasi-interpolation of the kernel 

P. Sablonnière a and D. Barrera b 

a INSA & IRMAR, Rennes, France 

b Department of Applied Mathematics, University of Granada, 


Paul.Sablonniere@insa-rennes.fr,dbarrera@ugr.es 

Abstract 

For solving the linear Fredholm integral equation of the second kind 

u(x) = f(x) + 

∫ b 

a 

k(x, t)u(t)dt, 

we propose to approximate the kernel by tensor products or blending sums of univariate spline 

quasi-interpolants (abbr. QI). These QIs have the general form Qf := ∑ j∈J c j(f)B j , where 

the B j are B-splines defined on some partition of [a, b] and the coefficients c j (f) are linear 

functionals based on values of the function f on some finite subset S := {s j , j ∈ J} of the 

interval I := [a, b]. Thus, the kernel will be approximated 

1. either by the tensor product of two univariate quasi-interpolants Q 1 and Q 2 in the variables 

x and t: 

k(x, t) ≈ (Q 1 ⊗ Q 2 )k(x, t) = ∑ K i,j B i (x)B j (t), 

i,j 

where the coefficients K i,j are linear combinations of values k(s i , s j ), for (i, j) ∈ J × J, 

2. or by the continuous blending sum of the two univariate quasi-interpolants Q 1 and Q 2 : 

k(x, t) ≈ (Q 1 ⊕ Q 2 )k(x, t) := (Q 1 ⊗ Id + Id ⊗ Q 2 − Q 1 ⊗ Q 2 )k(x, t) 

= ∑ i 

˜ki (t)B i (x) + ∑ j 

k j (x)B j (t) − ∑ i,j 

K i,j B i (x)B j (t), 

where the functions ˜k i (t) = c i (k(., t)) (resp. k j (x) = c j (k(x, ·))) are linear combinations 

of left sections k(s k , t) (resp. right sections k(x, s l )) of the kernel. 

When substituting these approximate (degenerate) kernels in the Fredhom equation, we get 

two types of approximate solutions: 

• u(x) = f(x) + ∑ X i B i (x) in the tensor-product case, 

• u(x) = f(x) + ∑ X i B i (x) + ∑ Y j k j (x) in the continuous blending case, 

The vectors of variables X i and Y j are then solutions of systems of linear equations. The two 

methods can be used with any type of spline QIs, although only C 1 quadratic and C 2 cubic 

splines will be considered. 

Key words: Fredholm equation, quasi-interpolation, tensor product, boolean sum. 




Chania,Greece 

On the comparison between fitted and unfitted finite element 

methods for the approximation of void electromigration 

Andrea Sacconi a , 

a Department of Mathematics, Imperial College London, 

London, United Kingdom 

a.sacconi11@imperial.ac.uk 

Abstract 

Microelectronic circuits usually contain small voids or cracks, and if those defects are large 

enough to sever the line, they cause an open circuit. Two fully practical finite element methods 

for the temporal analysis of the migration of voids in the presence of surface diffusion and 

electric loading are presented. We simulate a bulk-interface coupled system, with a moving 

interface governed by a fourth-order geometric evolution equation and a bulk where the electric 

potential is computed. A fitted approach (where the interface grid is always extracted from the 

boundary of the bulk grid) and an unfitted approach (where there is no perfect matching between 

the two grids) are analysed. A comparison between the two methods, in terms of experimental 

order of convergence (when the exact solution to free boundary problem is known), CPU time, 

and coupling operations (e.g., smoothing/re-meshing of the grids, intersection between elements 

of the two grids), is presented in detail. Several numerical simulations are performed in order to 

test the accuracy of the methods. 

Key words: void electromigration, fitted, unfitted, finite element methods, re-meshing, smoothing. 





Chania,Greece 

A Numerical Mesh-Less Method for Solving Unsteady 

Compressible Flows 

Samad Sattarzadeh a and Alireza Jahangirian a 

a Department of Aerospace Engineering, Amirkabir University of Technology, 

Tehran, Tehran, Iran 

sattarzadeh@aut.ac.ir,ajahan@aut.ac.ir 

Abstract 

A dual-time implicit mesh-less method for unsteady compressible flows is presented. Polynomial 

least-square (PLS) method is used to estimate the spatial derivatives at each node. The 

points distributed are moved based on the boundary movement. The unsteady flows over stationary 

and moving objects at different flow conditions are solved. Results indicate the computational 

efficiency of the method in comparison with the similar explicit and finite volume 

approaches. 

In the recent years researchers have tried to solve the numerical computation of flow with 

complex stationary and/or moving boundaries by using Euler and Navier-Stokes equations in 

different regimes. One of the main problems in the Computational Fluid Dynamics (CFD) for 

numerical flow simulation around complex geometries is the quality of the mesh. Mesh-less 

methods are more advantageous, especially in the moving and large deformations. The reason 

is that replacing and moving points are much simpler than changing or replacing the edges and 

volumes. Another attractive property of mesh-less methods is the ability of adding and subtracting 

nodes from the pre-existing nodes. Several mesh-less methods have been used with 

different privileges and drawbacks. So, it should be noted that choosing one method depends on 

the desired applications. In these methods, the approximation of the characteristics or derivatives 

is based on a group of nodes which can be nominated as neighbors. Mesh-less methods 

need more nodes in comparison with the finite difference method to achieve the same order of 

accuracy. As a result, the Navier-Stokes equations are solved by great bandwidth of the matrix 

in these methods. Therefore, the computational memory is unavoidably extensive. Nowadays 

by increasing the power of computational instruments, this problem is solved automatically. 

To show the ability of the method, computational results are compared using experimental 

and other reliable numerical data. 

Key words: Mesh-less method, unsteady flow, compressible flow. 





Chania,Greece 

A Numerical Adaptive Mesh-Less Method for Solution of 

Compressible Flows 

Samad Sattarzadeh a , Alireza Jahangirian a and Mehdi Ebrahimi a 

a Department of Aerospace Engineering, Amirkabir University of Technology, 

Tehran, Tehran, Iran 

sattarzadeh@aut.ac.ir,ajahan@aut.ac.ir,Mebrahimi@aut.ac.ir 

Abstract 

Nowadays mesh-less methods for numerical simulation of fluid flows has attracted much interest. 

This is because of the advantages of the method in comparison with alternative ones. At 

first, this method is used to solve boundary problems such as heat transfer and solid mechanics. 

Its advantages such as being less sensitive to the location of points; easier to change the cloud of 

points; to generate the point easily specially in the complex geometries or in the critical zones. 

It is shown that the time of point generating is less than generating the grid in the field. In 

the other hand, as it is obvious, the results depend on the quality of the grid especially in the 

critical zones, such as shock waves. There are different ways to increase the quality of the point 

cloud. One way is to use more points which may decrease the efficiency and performance. The 

other way to increase the quality of the grid and the performance is adaptation methods. There 

are several works on adapting the mesh-based methods but a few works have been done on 

mesh-less methods. 

In this paper, an adaptation mesh-less method is developed to solve the compressible flow 

equations. The comparison shows that the results are in good agreement with experimental and 

other reliable numerical data. In addition it has better convergence behavior than un-adapted 

point distribution. 

Key words: Mesh-less, Adaptive method, Compressible flow. 




Chania,Greece 

Solving CT reconstruction with a particle physics tool (RooFit) 

Enrico Jr. Schioppa a , Wouter Verkerke a , Jan Visser a and Els Koffeman a 

a Nikhef, Dutch national institute for subatomic physics 

Amsterdam, The Netherlands 

e.schioppa@nikhef.nl 

Abstract 

Spectral X-ray CT makes use of novel detector technologies that provide energy information. 

This information can be naturally included in the reconstruction phase when the algorithm 

is based on a statistical formulation. In this framework, the full dataset can be described in 

terms of a likelihood function whose expectation values are the density vectors of the different 

materials present in the sample. The problem of image reconstruction is thus translated into a 

problem of multivariate maximization. 

From the formal point of view, the same type of problem is often encountered in the analysis of 

large amounts of data in particle physics. For this purpose, the RooFit tool was developed during 

the years by the high energy physics community. In this work, we present first studies and 

results on the possibility to employ RooFit to implement a spectral CT reconstruction algorithm. 

Key words: Spectral X-ray CT, maximum likelihood reconstruction. 



Ziggurat Algorithm for Sampling 

from Bivariate Distributions 

Efraim Shmerling a 

a Department of Computer Science and Mathematics, Ariel University, 

Ariel 40700, Israel 

efraimsh@yahoo.com,efraimsh@ariel.ac.il 

Abstract 

It is shown that the Ziggurat algorithm designed for sampling from monotone decreasing univariate 

distributions can be extended to continuous bivariate distributions with necessary modifications. 

A bivariate version of the Ziggurat algorithm is presented. Results of experimental 

sampling from a bivariate Gamma distribution utilizing a RNG implementing the algorithm are 

given. 

Key words: Ziggurat algorithm, bivariate gamma distribution, random number generation. 



Fokas transform method for classes of advection-diffusion 

IBVPs 1 

A.G. Sifalakis ∗ , M.G. Papadomanolaki, E.P. Papadopoulou and 

Y.G. Saridakis 

Applied Mathematics and Computers Laboratory (AMCL) 


Chania 73100, Greece 

∗ sifal@science.tuc.gr 

Abstract 

It is now well established that Fokas transform approach for the solution of linear PDE problems, 

yields novel integral representations of the solution in the complex plane that, for appropriately 

chosen integration contours, decay exponentially fast and converge uniformly at the 

boundaries. Motivated by these method-inherent advantages and the fact that their coupling with 

simple quadrature integration rules produce practical, powerful and efficient methods, recently 

we considered applying them for the solution of discontinuous advection-diffusion equations 

that model the evolution of aggressive forms of primary brain tumors in heterogeneous brain 

tissue. The purpose of the present work is two-folded: 

• To review our recent results on the Fokas method for multi-domain linear advectiondiffusion 

equations with discontinuous diffusivity for brain tumor models 

• To examine the behavior of the Fokas method for classes of advection-diffusion equations 

with linear in t diffusivity in the real half-line, as the first step of extending the above 

results in non-constant diffusitivy models. 




Social Fund. 





Chania,Greece 

Approximate algorithm for single valued nonexpansive and 

multi-valued strictly pseudo contractive mappings in Hilbert 

spaces 

Wutiphol Sintunavarat a 

a Department of Mathematics and Statistics, Faculty of Science and Technology, 

Thammasat University Rangsit Center, 

Pathumthani 12121, Thailand. 

wutiphol@mathstat.sci.tu.ac.th, poom teun@hotmail.com 

Abstract 

In this talk, we introduce and study a new one-step iterative process to approximate common 

fixed points for single valued nonexpansive and multi-valued strictly pseudo contractive mappings 

in Hilbert spaces. Also, we give the strong convergence theorems of the purposed process 

under some appropriate additional conditions. 

Key words: Single valued nonexpansive mappings, Multi-valued strictly pseudo contractive mappings, 

Common fixed points, Strong convergence. 




Chania,Greece 

Application of an image registration method 

based on maximization of mutual information 

C. Spanakis a 1,a 2 

, K. Marias b and N. A. Kampanis c 

a 1 

Institute of Computer Science, Foundation 

of Research and Technology, Greece 

a 2 

Department of Sciences, Technical University of Crete, 

Chania, Crete, Greece 

b Institute of Computer Science, Foundation 

of Research and Technology, Greece 

c Institute of Applied and Computational Mathematics, 

Foundation of Research and Technology, Greece 

kspan@ics.forth.gr, kmarias@ics.forth.gr, kampanis@iacm.forth.gr 

Abstract 

Image Registration is the process of transforming sets of data acquired at different time-points, 

sensors and viewpoints into a single coordinate system. It is widely used in computer vision, 

medical imaging and satellite image analysis. Although it has been a central research topic in 

computer vision and medical image analysis for a long time, there are still unresolved issues and 

success rates seem to be data-dependent. There are many categories of methods that that are able 

to align images, but usually they are either specialized and accurate for specific types of data 

or more generic and error-prone frequently stumbling upon pitfalls. In this work, we describe 

our implementation and results on Maes method (Maes et al. 1997). By using three different 

variants of mutual information (used as the similarity measure), we present indicative results 

from different imaging domains and discuss the drawbacks/pitfalls of the method especially 

with regard to initial transformation selection and the initial direction vectors. The results are 

quite accurate with translation and rotation when dealing with images of good quality. However, 

the choice of a starting point and the initial direction vectors proved to be two critical factors 

for the success of the method, since different starting point or/and different initial direction 

vectors may lead to different optimal alignment registration results between the images. In 

order to solve this problem, we propose an extension of this method by enhancing its global 

optimization scheme by means of stochastic optimization. 

Key words: Image Registration, Mutual Information, Genetic Algorithms. 





Chania,Greece 

Successive approximations for optimal control in some nonlinear 

systems with small parameter 

Alexander Spivak 

Department of Computer Science, HIT, Holon Institute of Technology, 

52 Golomb str., Holon, ISRAEL 

spivak@hit.ac.il 

Abstract 

An important class of nonlinear control systems is bilinear systems. Such systems are linear on 

phase coordinates when the control is fixed, and linear on the control when the coordinates are 

fixed. The first point for the study of bilinear systems is to investigate the dynamic processes of 

nuclear reactors, kinetics of neutrons, and heat transfer. Further investigations show that many 

processes in engineering, biology, ecology and other areas can be described by the bilinear 

systems . It is shown that bilinear systems may be applied to describe some chemical reactions 

and many physical processes in the growth of the human population. 

In this work we consider nonlinear stochastic systems that can be described in the form 

ẋ(t) = ɛf 1 (t, x) + B(t)x(t)u(t) + σẇ(t), 

x(0) = x 0 , 0 ≤ t ≤ T. (1) 

Here the vector x(t) is from the Euclidean space E n , the control u(t) ∈ E m , the matrices σ 

and B have continuous and bounded elements, ɛ ≥ 0 is a small parameter, and the initial vector 

x 0 ∈ E n and the constant T ≥ 0 are given. The function f 1 (t, x) ∈ E n is continuous in the 

totality of its arguments, and satisfies some constraints, w(t) is standard Wiener process. The 

matrix σ(t) in (1) is such that σ(t)σ ′ (t) is positive definite. We understand the equation (1) in 

the sense of Ito. Note that if ɛ = 0, initial system (1) is stochastic bilinear, that is, it contains a 

nonlinearity of the form x(t)u(t). The problem is to find a control u minimizing the functional 

J(0, u), where 

[ 

J(t, u) = M 

x ′ (t)H 1 x(t) + 

∫ T 

t 

( x ′ (s)H 2 (s)x(s)+ 

] 

+ u ′ (s)H 3 (s)u(s) + f(s, x(s))) ds . (2) 

Here H i , i = 1, 2, 3 are given matrices, so that H 1 , H 2 (t) are non-negative defined, H 3 (t) is 

positive defined in the interval [0, T ], and the matrices H 2 (t) and H 3 (t) are measurable and 

bounded. The vector f(t, x) is determined later. 

When ɛ = 0, the optimal control synthesis is found in an exact analytic form. Successive 

approximations to the optimal control are constructed with the help of the perturbation method. 

Error estimates of the suggested method are presented. 

Key words: Successive Approximations, Small Parameter, Perturbation Theory. 



Inverse moment problems with applications in 

shape reconstruction 

Nikos Stylianopoulos 

Department of Mathematics and Statistics, University of Cyprus, 

Nicosia, Cyprus 

nikos@ucy.ac.cy 

Abstract 

Let µ be a finite positive Borel measure with compact support in the complex plane, and let 

{p n (µ, z)} ∞ n=0 denote the sequence of the orthonormal polynomials, with positive leading coefficients, 

defined by the inner product 

∫ 

⟨f, g⟩ µ := f(z)g(z)dµ(z). 

The purpose of the talk is to report on some recent developments regarding the asymptotics of 

{p n (µ, z)} ∞ n=0 , in cases when µ belongs to a special class of measures that includes area-type 

measures and arc-length measures. This leads to algorithms for recovering the shape of the 

support of µ, from a finite set of the moments 

∫ 

µ i,j := z i z j dµ(z) i, j = 0, 1, . . . , n, 

and thus, via the Radon transform, to applications in 2D geometric tomography. 

Key words: Orthogonal Polynomials, Inverse Moment Problems, Shape Reconstruction, Geometric Tomography. 



Method for solving degenerate sub-definite nonlinear equations 

Ewa Szczepanik a , Alexey Tretyakov b 

a Department of Computer Science, Faculty of Sciences, 

Siedlce University of Natural Sciences and Humanities, Siedlce, Poland 

b Department of Mathematics and Physics , 

Siedlce University of Natural Sciences and Humanities, Siedlce, Poland 

System Research Institute of the Polish Academy of Sciences, Warsaw, Poland 

Dorodnicyn Computing Center of the Russian Academy of Sciences, 

Moscow, Russia 

ewa.szczepanik@ii.uph.edu.pl,tret@uph.edu.pl 

We consider system of nonlinear equations 

⎡ 

F (x) = 

Abstract 

⎢ 

⎣ 

f 1 (x) 

... 

f m (x) 

⎤ 

⎥ 

⎦ = 0, 

where x ∈ R n , m ≤ n, and F ′ (x ∗ ) is degenerate at the solution point x ∗ . 

Denote the feasible set as follows M(x ∗ ) = {x ∈ R n |F (x) = 0}. 

It is known that in nondegenerate case ( when F ′ (x ∗ ) is nondegenerate) the Newton-Gauss 

method converges to some point ˜x ∗ ∈ M(x ∗ )([1] pp. 228), and the rate of convergence is 

quadratic. 

In the complete system of nonlinear equations one of the main result of the p- regularity 

theory is p-factor method. Scheme of this method is as follows 

x k+1 = x k − {ψ ′ p(x k )} −1 ψ p (x k ), 

where p-factor operator ψ p (x k ) has following form 

ψ p (x k ) = P 1 F ′ (x k ) + P 2 F ′′ (x k )[h] + ... + P p F (p) (x k )[h] [p−1] . 

Here P 1 is ortoprojection onto Im(F ′ (x ∗ )) ⊥ and P i , i = 2, ..., p also ortoprojection on the same 

defined sets and the element h(‖h‖ = 1) we construct in such a way that p-factor matrix 

P 1 F ′ (x ∗ ) + P 2 F ′′ (x ∗ )h + ... + P p F (p) (x ∗ )[h] [p−1] 

is nonsingular at the solution point x ∗ = 0 (p-regular along h). 

However for degenerate sub-definite nonlinear equations has been considered only the case 

for p = 2. In this paper we present the generalization of the p-factor method for sub-definite 

case and p ≥ 2. We will also describe a numerical algorithm of the p-factor method and will 

give numerical results. 

References 

[1] A.F. IZMAILOV, A. A.TRETYAKOV, Factor-analysis of nonlinear mappings, Nauka, 

Moscow, 1994.[in Russian] 

Key words: nonlinear equation, p-factor operator, 1-regularity, singularity, necessary and 

sufficient condi-tions, nonregular constraints, 1-factor methods 



Stochastic optimization for a problem of saltwater intrusion in 

coastal aquifers with heterogeneous hydraulic conductivity 1 

P. N. Stratis a , G.P. Karatzas b , E.P. Papadopoulou a and Y.G. Saridakis a 

a Applied Mathematics and Computers Laboratory 

b Environmental Engineering Department 


73100 Chania, Greece 

pstratis@science.tuc.gr 

Abstract 

In the present study we implement the stochastic optimization technique ALOPEX, in order to 

control the problem of saltwater intrusion in coastal aquifers. The objective is to maximize the 

total volume of freshwater pumped by the wells of the aquifer, while protecting the aquifer from 

salt water intrusion. Extending previous results, we examine some cases of non-homogeneous 

aquifers, divided in rectangular areas with different values for the hydraulic conductivity parameter. 

At the same time, appropriate penalties strategies are used to produce different management 

policies. Numerical experimentation with several test cases of non-homogeneous aquifers and 

two real case coastal aquifers (Vathi and Hersonissos aquifers in Greece) are presented. 

Key words: ALOPEX stochastic optimization, non-homogeneous coastal aquifers, saltwater intrusion, 

pumping management, hydraulic conductivity 




Social Fund. 



Numerical simulations for 1+2 dimensional coupled nonlinear 

Schrödinger type equations 

Thiab Taha 

Department of Computer Science, University of Georgia, 

Athens, GA, USA 

thiab@cs.uga.edu 

Abstract 

The coupled nonlinear Schrödinger equation is of tremendous importance in both theory and 

applications. Coupled nonlinear Schrödinger equation (CNLS) is the vectorial version of the 

nonlinear Schrödinger equation (NLS). The NLS equation is the main governing equation in 

the area of optical solitons. In this paper, we perform numerical simulations of two dimensional 

CNLS equation using the finite difference methods: a) the Explicit Finite Difference method and 

b) the Implicit Finite Difference method (Alternating Directions Implicit method). The methods 

are implemented. Our preliminary numerical results have shown that these methods give good 

results. 

Key words: Parallel Algorithms, MPI, Finite Difference. 





Chania,Greece 

A Numerical Model for the prediction of flooding in Water 

Rivers 

Katerina Tsakiri a , Antonios Marsellos b . 

a School of Computing, Engineering and Mathematics, University of Brighton, 

BN2 4GJ, United Kingdom 

b School of Environment and Technology, University of Brighton, BN2 4GJ, 


k.tsakiri@brighton.ac.uk,a.marsellos@brighton.ac.uk 

Abstract 

A numerical model is presented for the explanation and prediction of the daily water discharge 

time series derived by three locations nearby Mohawk River, New York during the period 

2005-2013. For the analysis of the model, we use daily water discharge time series, daily data 

of ground water level and the climatic variables in Mohawk River, New York. A methodology is 

used for the decomposition of the time series of all the variables into different components (long, 

seasonal and short term component). The long term component describes the fluctuations of a 

time series defined as being longer than a given threshold; the seasonal component describes the 

year-to-year fluctuations, while the short term component describes the short term variations. 

The Kolmogorov-Zurbenko (KZ) filter is used for the decomposition of the time series. The KZ 

filter, which separates the long term variations from the short term variations in a time series, 

provides a simple design and the smallest level of interferences between the scales of a time series. 

The application of the KZ filter in an example of Schoharie Creek (nearby Mohawk River) 

has improved the prediction of the water discharge up to 81%. This methodology has been also 

applied for Sussex Rivers in United Kingdom. 

Key words: Flooding Prediction, decomposition of time series, KZ filter. 



Interface Rexation Methods for the solution of Multi-Physics 

Problems 1 

Panagiota Tsompanopoulou 

Department of Computer and Communications Engineering, 

University of Thessaly, Volos, Greece 

yota@inf.uth.gr 

Abstract 

Multi-domain multi-physics problems simulate real world problems demanding efficient and 

high accuracy solutions. Domain Decomposition methods are well known methods that treat 

such kind of problems, but they first discretize the global problem (even if it is already partitioned 

by its physics) and then decompose it at the linear algebra level. Several techniques, 

mainly iteratively, are used to solve the set of the strongly coupled systems of linear equations 

that arise. 

Interface Relaxation (IR) methodology is a different and relatively new way to study such problems. 

The idea behind IR is to confront the global problem as closer as possible to its nature by 

realizing and utilizing its basic properties and behavior. Subproblems arise either by the physics 

of the original problem or by computational and parallelization issues. These “small” problems 

are studied independently of each other and appropriate methods (FEM, FD etc.) are used for 

their solution. However these subproblems are coupled on the common interfaces so as to satisfy 

the conditions resulting from the global problem’s properties (e.g., continuity and smoothness 

of the solution of the global problem). Initial guesses are considered on the interfaces, passed as 

boundary conditions to the “small” problems. These are solved concurrently and the resulting 

approximations are used by an IR method to relax the value and/or the derivative to get better 

estimates of the solution on the interfaces. These new estimates are passed again as boundary 

conditions to the small problems and the procedure iterates until convergence is achieved. 

When studying IR methods, one should consider issues from both mathematical analysis, computational 

complexity and software/hardware viewpoint. Mathematical analysis is often derived 

for model problems representatives of the original multiphysics applications since it is not possible 

and practical to get analysis for the realistic problems. A variety of software packages 

for the solution of simple non-multiphysics problems exist but they have to be combined under 

suitable software and hardware environments. Thus software reuse is of great importance when 

implementing IR methods. 

In this review we consider the Interface Relaxation methods proposed for the solution of multidomain 

multi-physics problems from both theoretical and implementation perspectives. 

Key words: Interface relaxation, multiphysics problems, Elliptic PDEs, Parabolic PDEs, software reuse. 




Social Fund. 





Chania,Greece 

An order 19-rational integrator 

L. A. Ukpebor 

Department of Mathematics, 

Ambrose Alli University, Ekpoma, Nigeria 

lukeukpebor@gmail.com 

Abstract 

In this research paper, an order 19-rational integrator was developed for solving stiff initial-value 

problems of differential equations of the form: 

y ′ = f(x, y); y(x 0 ) = y 0 

The consistency and convergence of this method were established following the steps of Lambert 

J.D [?], where he stated that a one-step numerical integrator of the form 

y n+1 = y n + h n φ(x n , y n , h n ) 

is convergent if and only if it is consistent. 

The application of this method to some selected stiff problems showed that it compared favourably 

with existing methods in terms of efficiency and accuracy. 

Key words: initial value problems, rational integrator, consistency, convergence, stiff problems. 





Chania,Greece 

A Meshfree Method with Fundamental Solutions for 

Inhomogeneous Elastic Wave Problems 

Svilen S Valtchev a,b , Carlos J S Alves a,c and Nuno F M Martins a,d 

a CEMAT, ULisbon, Portugal, 

b ESTG, Polytechnic Institute of Leiria, Portugal 

c Department of Mathematics, ULisbon, Portugal 

d Department of Mathematics, FCT, Universidade Nova de Lisboa, Portugal 

ssv@math.ist.utl.pt,carlos.alves@math.ist.utl.pt,nfm@fct.unl.pt 

Abstract 

We consider the numerical solution of the inhomogeneous Cauchy-Navier equations of elastodynamics, 

assuming time-harmonic variation for the displacement field U(x, t) = u(x)e −iωt 

of an isotropic material with Lamé constants λ and µ and density ρ. The resulting elliptic PDE, 

posed in a bounded simply connected domain Ω is coupled with Dirichlet boundary conditions 

and solved trough a meshfree method, based on the Method of Fundamental Solutions (MFS). 

{ µ∆u + (λ + µ)∇(∇ · u) + ρω 2 u = f in Ω 

u = g 

In particular, an extension, from the scalar [1, 2] to the vector case, of the MFS is applied and the 

displacement field u is approximated in terms of a linear combination of fundamental solutions 

(Kupradze tensors) of the corresponding homogeneous PDE with different source points and test 

frequencies. The applicability of the numerical method is justified in terms of density results [3]. 

The high accuracy and the convergence of the proposed method will be illustrated through 2D 

numerical simulations. Convex and non-convex domains and different sets of boundary data and 

body forces will be considered. Interior elastic wave scattering problems will also be addressed. 

Key words: Method of Fundamental Solutions, Inhomogeneous BVP, Elastic Wave Propagation. 

on Γ 

References 

[1] C.J.S. Alves and C.S. Chen, A new method of fundamental solutions applied to nonhomogeneous elliptic 

problems. Adv. Comput. Math., 23, 125(18pp), 2005. 

[2] C.J.S. Alves and S.S. Valtchev, A Kansa type method using fundamental solutions applied to elliptic 

PDEs. Advances in meshfree techniques, Computational Methods in Applied Sciences Series, vol. 5, 

241(16pp), Springer, 2007. 

[3] C.J.S. Alves, N.F.M. Martins and N. C. Roberty, Identification and reconstruction of elastic body forces. 

Inverse Problems, 30, 055015(18pp), 2014. 



On the Numerical Solution of Power Flow Problems 1 

Manolis Vavalis and Dimitris Zimeris 


Volos, Greece 

{mav,dzimeris}@uth.gr 

Abstract 

The power generation, transmission and distribution system has been widely recognized as one 

of the most complex man-made systems and the related power flow analysis as the main ingredient 

of many related studies. 

The power flow problem is modeled through a system of non-linear equations that relate 

the bus voltages to the power generation and consumption. Its solution is used to access the 

stability of the power system and perform contingency analysis. It is also required by other 

related and relatively new problems, for example the optimal power flow problem, the financial 

transmission rights mechanisms and many others. 

Several recent advances (the liberalization of the energy markets, the emerging of smart grid 

technologies, the stochasticity in the power production due to utilization of renewable energy 

sources, the decentralization of the energy production) have recently increased the complexity 

of the power flow problems significantly. 

The envisioned interconnection of national power systems with global energy markets will 

be based on truly large scale, continent-wide power flow simulations where the efficiency of the 

numerical solution of the power flow equations is expected to be a vital component. 

This paper consists an up-to-day review of the various numerical methods that have been 

very recently proposed for the solution of power flow equations and several other related problems. 

These methods are examined from both the theoretical (convergence analysis) and the 

practical (efficiency, robustness, numerical stability, implementation) viewpoint. 

We also propose several new research directions which, we believe, have the potential to 

lead us to next generation power grid simulation engines. Engines that are capable to support 

operational large scale modern power grid systems associated with open energy markets, paying 

special attention to the information flow in addition to the power flow. 

Key words: Numerical linear algebra, Newton’s method, power flow equations, GMRES, preconditioning, 

basic iterative methods. 




Social Fund. 





Chania,Greece 

Towards robust parallel solvers for tridiagonal 

systems for multiGPUs 

Ioannis E. Venetis a , Alexandros Kouris a , Nikolaos Nikoloutsakos a , 

Alexandros Sobczyk a and Efstratios Gallopoulos a 

a Computer Engineering & Informatics Department, University of Patras, 

Patras, Achaia, Greece 

venetis@ceid.upatras.gr,kouris@ceid.upatras.gr, 

nikoloutsa@ceid.upatras.gr,sobczyk@ceid.upatras.gr, 

stratis@ceid.upatras.gr 

Abstract 

We recently ([2]) presented an algorithm for nonsingular tridiagonal systems that is robust in 

that it can handle arbitrary partitionings of the matrix, even when the resulting diagonal blocks 

are exactly singular. We also showed an implementation on an NVIDIA GPU card that demonstrated 

performance that is very close to state of the art solvers, e.g. [1]. We extend here this 

work to clusters of GPUs using a combination of CUDA with MPI. The algorithm is based on 

ideas first presented by Sameh and Kuck in [3]; it is based on Givens rotations and requires no 

pivoting, which makes the algorithm simpler and more robust than existing ones for the GPU 

and for multiGPUs. 

References 

[1] L.-W. Chang, J.A. Stratton, H.S. Kim, and W.-M.W. Hwu. A scalable, numerically stable, 

high-performance tridiagonal solver using GPUs. In Proc. Int’l. Conf. High Performance 

Computing, Networking Storage and Analysis, SC ’12, pages 27:1–27:11, Los Alamitos, 

CA, USA, 2012. IEEE Computer Society Press. 

[2] I. E. Venetis, A. Kouris, A. Sobczyk, E. Gallopoulos and A.H. Sameh. Revisiting the 

Spike-based framework for GPU banded solvers: A Givens rotation approach for tridiagonal 

systems in CUDA. Parallel Matrix Algorithms and Applications Workshop, Lugano, 

July 2014. 

[3] A.H. Sameh and D.J. Kuck. On stable parallel linear system solvers. J. Assoc. Comput. 

Mach., 25(1):81–91, January 1978. 

Key words: Tridiagonal systems, Givens rotations, Spike, GPU, CUDA, MPI. 



Some new perturbation bounds of generalized polar 

decomposition 

X.-L. Hong, L.-S. Meng and B. Zheng 

School of Mathematics and Statistics, Lanzhou University 

Lanzhou, Gansu Province, P.R.China 

hongxiaoli2007@163.com,menglsh07@lzu.edu.cn,bzheng@lzu.edu.cn 

Abstract 

Let A, Ã = A + E ∈ Cm×n have the (generalized) polar decompositions 

A = QH and Ã = ˜Q ˜H, (1) 

where Q is subunitary and H is Hermitian positive semi-definite. We present the following 

new bounds of the positive (semi-)definite polar factor and the (sub) unitary polar factor for 

the (generalized) polar decomposition under the general unitarily invariant norm ∥·∥and the 

spectral norm ∥·∥ 2 , which are stated as in the following theorem. 

Theorem. Let A, Ã = A + E ∈ Cm×n r have the (generalized) polar decompositions in (1). 

(1). When r and N ∈ C n×n 

> . Hence, all perturbation bounds in the above theorem 

can be naturally extended to the case of the weighted polar decomposition of A, which also 

improved the known perturbation bounds for the weighted polar decomposition. 

Key words: Perturbation bounds; Positive semi-definite polar factor; Subunitary polar factor; Generalized 

polar decomposition; Weighted polar decomposition; Unitarily invariant norm; Spectral norm. 



Chebyshev accelerated preconditioned MHSS iteration methods 

for a class of block two-by-two linear systems 

Zeng-Qi Wang a , 

a Department of Mathematics, Shanghai Jiao Tong University , 

Shanghai, China 

wangzengqi@sjtu.edu.cn 

Abstract 

The preconditioned modified Hermitian and skew-Hermitian iteration method [1] is efficient 

for solving the the following block two-by-two systems of linear equations 

( ) ( ) ( ) 

W −T y p 

Ax ≡ 

= ≡ g. 

T W z q 

It could be written as the following procedure: 

⎧ ( ) ( ) 

αV + W 0 y (k+ 1 2 ) 

⎪⎨ 0 αV + W z (k+ 1 2 ) 

( ) ( ) 

αV + T 0 y 

(k+1) 

⎪⎩ 0 αV + T z (k+1) 

= 

= 

( ) ( ) ( αV T y 

(k) p 

−T αV z (k) + 

q 

( ) ( αV −W y (k+ 1 2 ) 

W 

αV 

z (k+ 1 2 ) ) 

) 

, 

( q 

+ 

−p 

) 

, 

where α is a given positive constant and V ∈ R n×n is a prescribed symmetric positive definite 

matrix. The Chebyshev semi-iteration method is fulfilled for accelerating the above iteration 

method. It could be verified that the Chebyshev accelerated PMHSS iteration method is a parameter 

free method. It converges unconditionally. The new method is utilized on solving the 

distributed control problems. Numerical experiments shows that the performance of the Chebyshev 

accelerated PMHSS iteration method is independent on not only the mesh size and the 

regularization parameter of the cost functional. 

Key words: Chebyshev semi-iteration, PMHSS iteration, PDE-constrained optimization, block two-bytwo 

matrices. 

References 

[1] Bai Z-Z, Benzi M, Chen F, Wang Z-Q (2013) Preconditioned MHSS iteration methods for 

a class of block two-by-two linear systems with applications to distributed control problems. 

IMA Journal of Numerical Analysis 33:343-369 




Chania,Greece 

The WR-HSS Methods for Non-Self-Adjoint Positive Definite 

Linear Differential Equations and Applications to the Unsteady 

Discrete Elliptic Problem 

Xi Yang a , 

a Dept. Math., Nanjing University of Aeronautics and Astronautics, 

Nanjing 210016, Jiangsu, P.R. China 

yangxi@lsec.cc.ac.cn 

Abstract 

We consider the numerical methods for non-self-adjoint positive definite linear differential equations, 

L(x) = B ẋ + A x = q, x(0) = x 0 , (1) 

with B being Hermitian and A being non-Hermitian positive definite, and their corresponding 

applications to the unsteady discrete elliptic problem, which is derived from spatial discretization 

of the unsteady elliptic problem with Dirichlet boundary condition, i.e., 

{ ∂u 

∂t − ∇ · [a(x)∇u(x)] + ∑ d ∂ 

j=1 ∂x j 

(p(x)u(x)) = f(x), u(x, 0) = u 0 (x), x ∈ Ω 

Dirichlet Boundary Condition. 

(2) 

Taking into account the idea of Hermitian/skew-Hermitian splitting (HSS) in [1], we establish 

a class of waveform relaxation iteration methods based on the HSS splitting of the non-selfadjoint 

positive definite linear operator L, i.e., WR-HSS methods. We analyze these WR-HSS 

methods with the help of Fourier Transform. Similarly to the HSS methods for solving linear 

algebraic equations, we find that the WR-HSS methods are unconditionally convergent to the 

solution of (1). In addition, we derive the upper bound of the contraction factor of the WR-HSS 

methods which is only dependent on the Hermitian part of L. Finally, the applications of these 

WR-HSS methods to the unsteady discrete elliptic problem demonstrate their effectiveness and 

the corresponding theoretical results. 

Key words: elliptic problem, Hermitian/skew-Hermitian splitting, waveform relaxation. 

References 

[1] Z.-Z. Bai, G.H. Golub and M.K. Ng, Hermitian and skew-Hermitian splitting methods for 

non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24(2003), 603- 

626. 




Chania,Greece 

Sensitivity of the Domain Decomposition Method to 

Perturbation of the Transmission Conditions 

Anastasiya Zaitseva a and Vadim Lisitsa a,b 

a Novosibirsk State University, 


b Institute of Petroleum Geology and Geophysics of SB RAS, 


zaf1990@mail.ru, lisitsavv@ipgg.sbras.ru 

Abstract 

Nowadays Domain Decomposition (DD) method is one of the common tools to construct preconditioner 

to solve 3D Helmholtz equation, especially in geophysical applications. There are 

numerous papers devoted to construction of optimal transmission conditions to improve convergence 

of the DD. However, these researches are focused on the differential statements and no 

perturbation is typically assumed. Whereas solution of the 3D Helmholtz equation requires the 

use of the numerical methods such as finite differences of finite elements, thus a numerical error 

is introduced in the transmission conditions as a result of numerical approximation. Moreover, 

in some cases it is worth using different numerical methods in adjoint subdomains, which makes 

the considered perturbations nonsymmetric. 

In this paper, a simplest 1D Helmholtz equation was considered and the perturbation of 

the Dirichlet-to-Neumann map based transmission conditions are considered. It was proved, 

that if the perturbation is symmetric (the same numerical methods and discretizations are used 

in the adjoint subdomains) the numerical solution converges to the true solution for almost 

all practically meaningful cases. However, if the perturbation is nonsymmetric, i.e. different 

numerical methods are used, different discretizations are applied, different approximations of 

the boundary operators are utilized, the numerical solution converges, but not to the solution of 

the original problem. In this case, an irreducible error presents, which linearly depends on the 

perturbation of the transmission conditions. 

The research was done under financial support of the Russian Foundation for Basic Research 

grants no. 13-05-00076, 13-05-12051, 14-05-00049, 14-05-93090, 14-01-31340, fellowship 

SP-150.2012.5 of the President of the Russian Federation, and integration projects of SB RAS 

127 and 130. 

Key words: Domain decomposition, Helmholtz equation. 



An improved model of heart rate kinetics 

Maria Zakynthinaki 




marzak@science.tuc.gr 

Abstract 

The heart rate in response to movement (exercise) is modeled as a dynamical system and its 

temporal evolution is given as the solution of a system of two coupled differential equations. 

The model assumes the heart rate kinetics to be a function of exercise intensity (which can also 

be time-dependent), blood lactate and the current cardiovascular condition of the individual. 

By means of numerical optimization the model can be fit to experimental heart rate time series 

data and provide important information regarding an individual’s cardiovascular condition. 

Numerical simulations can also provide predictions for any given exercise intensity, even those 

that no data exist for. This is of great importance, not only for efficiently designing training 

sessions for healthy subjects, but also for providing a complete means of heart rate analysis in 

population groups for which direct heart rate recordings at intense exercises are not possible or 

not allowed, such as elderly or pregnant women. Examples of successful fit of the proposed 

model to recorded heart rate time series data, as well as heart rate kinetics simulations will be 

presented. 

Key words: Cardiac dynamics, numerical models, numerical optimization, numerical simulation. 





Chania,Greece 

Normalizations of the Proposal Density 

in Markov Chain Monte Carlo Algorithms 

Antoine Zambelli a 

a Anderson School of Management, University of California Los Angeles, 

Los Angeles, CA, USA 

antoine.zambelli.2014@anderson.ucla.edu 

Abstract 

We explore the effects of normalizing the proposal density in Markov Chain Monte Carlo algorithms, 

in the context of a nonlinear inverse problem. Our problem is that of reconstructing the 

conductivity term K in the 2-dimensional heat equation 

u xx + u yy = 2H 

Kδ u (1) 

given temperatures at the boundary points, given by d. A Metropolis-Hastings MCMC algorithm 

is implemented to do so. Markov Chains produce a probability distribution of possible 

solutions conditional on the observed data. We generate a candidate solution K ′ and solve the 

forward problem, obtaining d ′ . In this way, at step n the probability of setting K n+1 = K ′ is 

given by 

{ 

α(K ′ |K n ) ≡ min 

1, P (K′ |d)g(K n |K ′ ) 

P (K n |d)g(K ′ |K n ) 

For our given proposal density g, this is initially computed as 

α = min 

{ 

1, e 

∑ 

−1 n,m 

2σ 2 i,j=1 

[(d ij −d ′ ij) 2 −(d ij −d nij ) 2] } 

} 

(2) 

{ 

= min 1, e −D} (3) 

We identify certain issues with this construction, stemming from large and fluctuating values of 

D. Using this framework, we develop normalization terms z 0 , z and parameters λ that preserve 

the inherently sparse information at our disposal, rewriting (3) as 

α = z 0 e −λzD (4) 

We examine the results of this variant of the MCMC algorithm on the reconstructions of several 

2-dimensional conductivity functions. 

Key words: Ill-posed, Inverse Problems, MCMC, Normalization, Numerical Analysis. 




Chania,Greece 

A local preconditioned alternating direction iteration method for 

generalized saddle point problems 

Guo-Feng Zhang a and Zhong Zheng a 

a School of Mathematics and Statistics, Lanzhou University, 

Lanzhou, 730000, China 

gf zhang@lzu.edu.cn, zhengzh13@lzu.edu.cn 

Abstract 

In this paper, a local preconditioned alternating direction iteration method is presented for solving 

the generalized saddle point problems. By using the new method, we only need solver two 

linear sub-system of linear equations with symmetric and definite positive coefficient matrices 

per iteration step for solving the generalized saddle point problems. The convergence of the 

new iteration method is analyzed and some spectral properties of the preconditioned matrix are 

discussed. Numerical examples are reported to confirm the efficiency of the proposed method. 





Chania,Greece 

Katservich Algorithm Based on Spherical Detector for 

Cone-Beam CT and the Implementation on GPU 

Yan Zhang, Qian Li 

Harbin Institute of Technology Shenzhen Graduate School, 

Shenzhen, Guangdong, China, 518055 

ianzh@foxmail.com,lqiankm@foxmail.com 

Abstract 

Katsevich algorithm is an exact cone-beam reconstruction algorithm of filtered backprojection 

(FBP) type. In this paper, an implementation for a spherical detector is proposed. It reduces the 

error generated by geometric shapes such as curved or plan detector. CT image reconstruction 

using spherical detector makes the speed of reconstruction faster and the quality of image better. 

Since CT image reconstruction has a huge amount of computation, it is difficult to meet the requirements 

of both fast and accurate reconstruction using CPU. This paper takes the advantages 

of GPU which is programmable and parallelizable to accelerate the image reconstruction. 

Key words: cone-beam CT, Katservich algorithm, spherical detector, GPU, acceleration 



A Riemannian Newton Algorithm for Nonlinear Eigenvalue 

Problems 

Zhi Zhao a , Zheng-Jian Bai b and Xiao-Qing Jin a 

a Department of Mathematics, University of Macau, 

Macao, People’s Republic of China 

b School of Mathematical Sciences, Xiamen University, 

Xiamen 361005, People’s Republic of China 

zhaozhi231@163.com, zjbai@xmu.edu.cn, xqjin@umac.mo 

Abstract 

We give the formulation of a Riemannian Newton algorithm for solving a class of nonlinear 

eigenvalue problems by minimizing a total energy function subject to the orthogonality constraint. 

Under some mild assumptions, we establish the global and quadratic convergence of the 

proposed method. Moreover, the positive definiteness condition of the Riemannian Hessian of 

the total energy function at a solution is derived. Some numerical tests are reported to illustrate 

the efficiency of the proposed method for solving large-scale problems. 

Key words: nonlinear eigenvalue problem, Riemannian Newton algorithm, Stiefel manifold, Grassmann 

manifold. 




Chania,Greece 

Finite element approximations for a linear stochastic 

Cahn-Hilliard-Cook equation 

Georgios E. Zouraris a 

a Department of Mathematics and Applied Mathematics, 

University of Crete, 

Heraklion, Crete 

Greece 

zouraris@math.uoc.gr 

Abstract 

We consider an initial- and Dirichlet boundary- value problem for a linear stochastic Cahn- 

Hilliard-Cook equation driven by an additive noise. We approximate its solution using, for 

the discretization in space, a finite element method, and for the discretization in time, a timestepping 

method. For the proposed numerical method, we derive strong a priori error estimates. 

Key words: stochastic Cahn-Hilliard-Cook equation, finite element method, error estimates.

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