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244 P. Neittaanmäki and D. Tiba 4 Conclusions The shape optimization problem (1), (2) is transformed in this way into the optimal control problem ∫ Min H(g)j(x, y(x)) dx (38) g∈C D subject to (15)–(17) which, in turn, may be approximated by (19)–(21) or, equivalently, by (36)–(37). To obtain good differentiability properties with respect to g in the optimization problem (38), one should replace H by H ε , some regularization of H, as previously mentioned. Analyzing further approximation properties and the gradient for (38) is a nontrivial task. However, the application of evolutionary algorithms is possible since it involves just the values of the cost (38) and no computation of the gradient with respect to g. As initial population of controls g for the genetic algorithm, corresponding to the finite element mesh in D, one may use the basis functions for the piecewise linear and continuous finite element basis. In case some supplementary information is available on the desired shape (for instance, coming from the constraints), this should be imposed on the initial population. Then, standard procedures specific to evolutionary algorithms [Hol75] are to be applied. References [Hol75] J. R. Holland. Adaptation in natural and artificial systems. TheUniversity of Michigan Press, Ann Arbor, MI, 1975. [Lio68] J.-L. Lions. Contrôle optimal des systèmes gouvernées par des equations aux dérivées partielles. Dunod, Paris, 1968. [MP01] B. Mohammadi and O. Pironneau. Applied shape optimization for fluids. The Clarendon Press, Oxford University Press, New York, 2001. [NPT07] P. Neittaanmäki, A. Pennanen, and D. Tiba. Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions. Reports of the Department of Mathematical Information Technology, Series B, Scientific Computing B16/2007, University of Jyväskylä, Jyväskylä, 2007. [NST06] P. Neittaanmäki, J. Sprekels, and D. Tiba. Optimization of elliptic systems. Springer-Verlag, Berlin, 2006. [NT95] P. Neittaanmäki and D. Tiba. An embedding of domains approach in free boundary problems and optimal design. SIAM J. Control Optim., 33(5):1587–1602, 1995. [Pir84] O. Pironneau. Optimal shape design for elliptic systems. Springer-Verlag, Berlin, 1984. [Roc70] R. T. Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970. [Tib92] D. Tiba. Controllability properties for elliptic systems, the fictitious domain method and optimal shape design problems. In Optimization, optimal control and partial differential equations (Iaşi, 1992), number 107 in Internat. Ser. Numer. Math., pages 251–261, Basel, 1992. Birkhäuser. [Yos80] K. Yosida. Functional analysis. Springer-Verlag, Berlin, 1980.
Reduced-Order Modelling of Dispersion Jean-Marc Brun 1 and Bijan Mohammadi 2 1 CEMAGREF/ITAP, FR-34095 Montpellier, France jean-marc.brun@cemagref.fr 2 I3M-Univ. Montpellier II, CC051, FR-34095 Montpellier, France bijan.mohammadi@univ-montp2.fr Summary. We present low complexity models for the transport of passive scalars for environmental applications. Multi-level analysis has been used with a reduction in dimension of the solution space at each level. Similitude solutions are used in a non-symmetric metric for the transport over long distances. Model parameters identification is based on data assimilation. The approach does not require the solution of any PDE and, therefore, is mesh free. The model also permits to access the solution in one point without computing the solution over the whole domain. Sensitivity analysis is used for risk analysis and also for the identification of the sources of an observed pollution. Key words: Reduced order modelling, source identification, risk analysis by sensitivity, non-symmetric geometry. 1 Introduction Air and water contamination by pesticides is a major preoccupation for health and environment. One aims to model pesticide transport in atmospheric flows with very low calculation cost making assimilation-simulation and statistic risk analysis by Monte Carlo simulations realistic. In this problem available data is incomplete with large variability and the number of parameters involved large. Solution space reduction and reduced order modelling appear, therefore, as natural way to proceed. Our contribution is to build a multi-level approach where a given level provides the inlet condition for the level above. In each level one aims to use a priori information in the definition of the search space for the solution and avoid the solution of partial differential equations. More precisely, a near field (to the injection device) search space is build using experimental observations. Once this local solution known, the amount of specie leaving the atmospheric sub-layer is evaluated. This quantity is candidate for long distance transport using similitude solutions for mixing layers and plumes [Sim97]. These are known in Cartesian metrics. An original
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244 P. Neittaanmäki <strong>and</strong> D. Tiba<br />
4 Conclusions<br />
The shape optimization problem (1), (2) is transformed in this way into the<br />
optimal control problem<br />
∫<br />
Min H(g)j(x, y(x)) dx (38)<br />
g∈C D<br />
subject to (15)–(17) which, in turn, may be approximated by (19)–(21) or,<br />
equivalently, by (36)–(37). To obtain good differentiability properties with respect<br />
to g in the optimization problem (38), one should replace H by H ε ,<br />
some regularization of H, as previously mentioned. Analyzing further approximation<br />
properties <strong>and</strong> the gradient for (38) is a nontrivial task. However,<br />
the application of evolutionary algorithms is possible since it involves just the<br />
values of the cost (38) <strong>and</strong> no computation of the gradient with respect to g.<br />
As initial population of controls g for the genetic algorithm, corresponding<br />
to the finite element mesh in D, one may use the basis functions for the piecewise<br />
linear <strong>and</strong> continuous finite element basis. In case some supplementary<br />
information is available on the desired shape (for instance, coming from the<br />
constraints), this should be imposed on the initial population. Then, st<strong>and</strong>ard<br />
procedures specific to evolutionary algorithms [Hol75] are to be applied.<br />
References<br />
[Hol75] J. R. Holl<strong>and</strong>. Adaptation in natural <strong>and</strong> artificial systems. TheUniversity<br />
of Michigan Press, Ann Arbor, MI, 1975.<br />
[Lio68] J.-L. Lions. Contrôle optimal des systèmes gouvernées par des equations<br />
aux dérivées partielles. Dunod, Paris, 1968.<br />
[MP01] B. Mohammadi <strong>and</strong> O. Pironneau. Applied shape optimization for fluids.<br />
The Clarendon Press, Oxford University Press, New York, 2001.<br />
[NPT07] P. Neittaanmäki, A. Pennanen, <strong>and</strong> D. Tiba. Fixed domain approaches in<br />
shape optimization problems with Dirichlet boundary conditions. Reports<br />
of the Department of Mathematical Information Technology, Series B,<br />
Scientific Computing B16/2007, University of Jyväskylä, Jyväskylä, 2007.<br />
[NST06] P. Neittaanmäki, J. Sprekels, <strong>and</strong> D. Tiba. Optimization of elliptic systems.<br />
Springer-Verlag, Berlin, 2006.<br />
[NT95] P. Neittaanmäki <strong>and</strong> D. Tiba. An embedding of domains approach in<br />
free boundary problems <strong>and</strong> optimal design. SIAM J. Control Optim.,<br />
33(5):1587–1602, 1995.<br />
[Pir84] O. Pironneau. Optimal shape design for elliptic systems. Springer-Verlag,<br />
Berlin, 1984.<br />
[Roc70] R. T. Rockafellar. Convex analysis. Princeton University Press, Princeton,<br />
NJ, 1970.<br />
[Tib92] D. Tiba. Controllability properties for elliptic systems, the fictitious domain<br />
method <strong>and</strong> optimal shape design problems. In Optimization, optimal<br />
control <strong>and</strong> partial differential equations (Iaşi, 1992), number 107 in<br />
Internat. Ser. Numer. Math., pages 251–261, Basel, 1992. Birkhäuser.<br />
[Yos80] K. Yosida. Functional analysis. Springer-Verlag, Berlin, 1980.
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