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252 J.-M. Brun and B. Mohammadi Once the model is established, the second inverse problem of interest is the identification of possible sources of an observed pollution. This region is defined where J p ′ is large. In this case, the parameter p is the location of the different sources (cultures). To solve the minimization problems, we use a semi-deterministic global optimization algorithm based on the solution of the following boundary value problem [MS03, Ivo06, IMSH06]: { p ζζ + p ζ = −J p(p(ζ)), ′ (7) p(0) = p 0 , J(p(1)) = J m =0, where ζ ∈ [0, 1] is a fictitious parameter. J m is the infimum of our inverse problems (here taken as 0). This can be solved using solution techniques for BVPs with free surface to find p(1) realizing the infimum (i.e. J(p(1)) = J m ). An analogy can be given with the problem of finding the interface between water and ice which is only implicitly known through the iso-value of zero temperature. In case a local minima is enough, the second boundary condition can be replaced by J ′ p(p(1)) = 0. This algorithm requires the sensitivity of the functional with respect to independent variables p. An interesting feature of the present low-cost modelling is that gradients are also available at very low calculation cost. Indeed, sensitivity evaluation for large dimension minimization problems is not an easy task. The most efficient approach is to use an adjoint variable with the difficulty that it requires the development of a specific software. Automatic differentiation brings some simplification, but does not avoid the main difficulty of intermediate states storage, even though check-pointing technique brings some relief [Gri01, CFG + 01]. By simplifying the solution of the transport problem, the present approach also addresses this issue. 4 Numerical Results The application of low complexity transport model to several flow condition is shown. Typical fields of 0.01 ∼ 0.1km 2 have been considered in a region of 400 km 2 . Rows are spaced by about 1.5 m. The source of the treatment moves at a speed of around 1 m/s and the injection velocity is taken at 7 to 10 m/s for a typical treatment of 100 kg/km 2 . Mono and multi sources situations (Figs. 3 and 4) are considered and examples of the constructed flow field are shown together with the wind measurement points assimilated by the model (Figs. 1 and 4). The transport-based and the Euclidean distances have been reported for a given point in Fig. 2. The impact of ground variations on the advected species is shown in Fig. 5. An example of source identification problem is shown in Fig. 6.
Reduced-Order Modelling of Dispersion 253 Fig. 1. Typical trajectory of the vehicle in a culture of 10000 m 2 and the location of this field in a calculation domain of 400 km 2 . Wind measurements based on two points have been reported together with the constructed divergence free flow field at z = H ∼ 3m. Fig. 2. Examples of symmetric Euclidean and non-symmetric travel time based distances. 5 Concluding Remarks A low-complexity model has been presented for the prediction of passive scalar dispersion in atmospheric flows for environmental and agricultural applications. The solution search space has been reduced using a priori physical information. A non-symmetric metric based on migration times has been used to generalize injection and plume similitude solutions in the context of variable flow fields. Data assimilation has been used to define the flow field and the parameters in the dispersion model. Sensitivity analysis has been used
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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44 E.J. Dean and R. Glowinski so fa
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46 E.J. Dean and R. Glowinski 2 A L
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48 E.J. Dean and R. Glowinski S:T=
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50 E.J. Dean and R. Glowinski minim
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52 E.J. Dean and R. Glowinski 6 On
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54 E.J. Dean and R. Glowinski Fig.
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56 E.J. Dean and R. Glowinski and
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58 E.J. Dean and R. Glowinski 7 Num
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60 E.J. Dean and R. Glowinski Fig.
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62 E.J. Dean and R. Glowinski Assum
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Higher Order Time Stepping for Seco
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u n+1 h Optimal Higher Order Time D
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Optimal Higher Order Time Discretiz
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96 I. Sazonov et al. To provide a p
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98 I. Sazonov et al. In the first s
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100 I. Sazonov et al. Fig. 1. An ex
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102 I. Sazonov et al. H z 1 exact F
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104 I. Sazonov et al. (a) (b) Fig.
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106 I. Sazonov et al. Scattering Wi
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108 I. Sazonov et al. 6.4 Scatterin
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110 I. Sazonov et al. (a) (b) Fig.
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112 I. Sazonov et al. [MHP96] K. Mo
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116 R. Sanders and A.M. Tesdall imp
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118 R. Sanders and A.M. Tesdall alo
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120 R. Sanders and A.M. Tesdall (a)
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122 R. Sanders and A.M. Tesdall D C
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124 R. Sanders and A.M. Tesdall 8.6
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126 R. Sanders and A.M. Tesdall 0.3
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128 R. Sanders and A.M. Tesdall [TR
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132 S. Lapin et al. Ω R γ Ω 2 Γ
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134 S. Lapin et al. ∫ ∂ 2 ∫
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136 S. Lapin et al. Ω R γ Fig. 3.
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138 S. Lapin et al. 4 Energy Inequa
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140 S. Lapin et al. 5 Numerical Exp
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142 S. Lapin et al. Fig. 6. Contour
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144 S. Lapin et al. Fig. 9. Obstacl
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Domain Decomposition and Electronic
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Domain Decomposition Approach for C
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Numerical Analysis of a Finite Elem
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so that |u| 2 1,ω h ≤ 2 Numerica
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188 A. Bonito et al. of the model a
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