Construction de grilles grossières pour les méthodes de décomposition de domaine

Construction de grilles grossières pour les méthodes de décomposition de domaine Frédéric Nataf nataf@ann.jussieu.fr, www.ann.jussieu.fr/ nataf Laboratoire J.L. Lions, CNRS UMR7598. Université Pierre et Marie Curie, France 1

Plan de l exposé 1. Calcul parallèle et Méthodes de décomposition de domaine 2. Grille grossière pour les problèmes réguliers 3. Grille grossière pour les problèmes très hétérogènes 4. Conclusion et perspectives 2

Calcul parallèle et méthodes de décomposition de domaine Depuis 2005 environ, Stagnation de la fréquence des processeurs Augmentation de la puissance de calcul dû presque exclusivement à la multiplication du nombre des coeurs Attendre la prochaine génération de machines ne suffit plus L algorithmique parallèle est indispensable pour tirer profit des nouvelles machines Les méthodes de décomposition de domaine sont naturellement parallèles 3

Schwarz method Two subdomain case The original Schwarz Method (H.A. Schwarz, 1870) (u) =f in Ω u = 0 on Ω. Schwarz Method : (u n 1,u n 2 ) (u n+1 1,u n+1 2 )with (u n+1 1 )=f in Ω 1 u n+1 1 = 0 on Ω 1 Ω u n+1 1 = u n 2 on Ω 1 Ω 2. (u n+1 2 )=f in Ω 2 u n+1 2 = 0 on Ω 2 Ω u n+1 2 = u n+1 1 on Ω 2 Ω 1. Parallel algorithm, converges Preconditioner for Krylov type methods 4

More subdomains The methods generalize to an arbitrary number of subdomains, it is a matter of notations. But, performance may deteriorate with large number of subdomains. Plateaus appear in the convergence of the Krylov methods. 0-1 -2 Log_10 (Error) -3-4 4x4 8x8-5 -6 M2 2x2 2x2 M2 4x4 M2 8x8-7 0 10 20 30 40 50 60 70 80 Number of iterations (GCR) Figure 1: Japhet, Nataf, Roux (1998) 5

More subdomains Iteration counts for a Poisson problem on a domain decomposed into strips. The number of unknowns is proportional to the number of subdomains (scalability). N subdomains Schwarz With coarse grid 4 18 25 8 37 22 16 54 24 32 84 25 64 144 25 6

More than two subdomains This corresponds to a few very large (or low) eigenvalues in the spectrum of the substructured problem. They are due to the lack of a global exchange of information in the preconditioner. u = f in Ω u = 0 on Ω The mean value of the solution in domain i depends on the value of f on all subdomains. A classical remedy consists in the introduction of a coarse grid problem that couples all subdomains. This is closely related to deflation technique classical in linear algebra (see Nabben and Vuik s papers in SIAM J. Sci. Comp). 7

Deflation and Coarse grid correction Let A be a SPD matrix, we want to solve Ax = b with a preconditioner M (for example the Schwarz method). Let Z be a rectangular matrix so that the bad eigenvectors belong to the space spanned by its columns. Define P := I AQ, Q := ZE 1 Z T, E := Z T AZ, Examples of coarse grid preconditioners P A DEF2 := P T M 1 +Q, P BNN := P T M 1 P +Q (Mandel, 1993) Some properties: QAZ = Z, P T Z = 0 and P T Q = 0. Let r n be the residual at step n of the algorithm: Z T r n = 0. How to choose Z? 8

Coarse grid correction for smooth problems For a Poisson like problem, Nicolaides (1987) : 10 4 SCHWARZ additive Schwarz with coarse gird acceleration 10 2 10 0 10 2 10 4 Z = 1 Ω1 0 0.. 1 Ω2 0...... 10 6 X: 25 Y: 1.658e 08 10 8 0 50 100 150 0 0 1 Ωd 9

Coarse Grid for heterogeneous problems Design a Schwarz domain decomposition method for the Darcy equation div(α u) =f that is robust with respect to the permeability field α and a many domains decomposition: either manually or via Metis. viscosity IsoValue -51.5789 27.2895 79.8684 132.447 185.026 237.605 290.184 342.763 395.342 447.921 500.5 553.079 605.658 658.237 710.816 763.395 815.974 868.553 921.132 1052.58 viscosity IsoValue -577.895 290.447 869.342 1448.24 2027.13 2606.03 3184.92 3763.82 4342.71 4921.61 5500.5 6079.39 6658.29 7237.18 7816.08 8394.97 8973.87 9552.76 10131.7 11578.9 Decomposition into 64 domains using Metis Figure 2: Permeability for multilayer and skyscraper cases Metis How to handle along the interface discontinuities in a simple and parallel manner? 10

Results with an algebraic coarse space based on a first solve Figure 3: SPE 10 test case (PhD of M. Szydlarski) Large matrix with 1,094,421 rows and 7,515,591 nnz highly heterogeneous case Run on the idataplex UPMC cluster on 256 cores 11

Coarse Space for heterogeneous problems for the first solve (N., H. Xiang, V. Dolean, 2010) In order to motivate the choice of the coarse space, consider the original Schwarz method for a domain Ω decomposed in a one-way partitioning. The errors e n i are harmonic in the subdomains and satisfy a Schwarz algorithm: Ω i 1 e n i Ω i Ω i+1 e n+1 i 1 e n+1 i+1 Ω i 1 e n i Ω i Ω i+1 e n+1 i 1 e n+1 i+1 Figure 4: Fast or slow convergence of the Schwarz algorithm. 12

Coarse Space for heterogeneous problems Let us consider at the continuous level the Dirichlet to Neumann map DtN Ωi. Let g : Ω i R, DtN Ωi (g) = v n i, Γi where v satisfies L(v) :=(η div(α ))v =0, in Ω i, v = g, on Ω i, (1) To construct the coarse space, we use the low frequency modes associated with the DtN operator: DtN Ωi (v λ i )=λv λ i with λ small. The functions vi λ subdomains. are extended harmonically to the 13

Coarse Space for heterogeneous problems We still choose Z of the form w 1 0 0.. w Z = 2 0..., (2) 0 0 w d where d is the number of subdomains. The columns of w i are the harmonic extensions of the eigenvectors associated with the smallest eigenvalues of the DtN map in subdomain Ω i. For a Poisson problem, the coarse space is made of piecewise constants as proposed by Nicolaides. It adapts automatically to heterogeneities. 14

Numerical Results Two layers with high heterogeneities, stripwise decomposition into 64 subdomains 10 6 AS P : AS + Z ADEF2 Nico P : AS + Z ADEF2 D2N RAS 10 4 P ADEF2 : RAS + Z Nico P : RAS + Z ADEF2 D2N 10 2 10 0 10 2 10 4 10 6 0 50 100 150 200 250 300 350 Z Nico (middle curve) does not work well, while taking two modes per subdomain Z D2N (left curves) gives a fast convergence. 17

Numerical results viscosity IsoValue -577.895 290.447 869.342 1448.24 2027.13 2606.03 3184.92 3763.82 4342.71 4921.61 5500.5 6079.39 6658.29 7237.18 7816.08 8394.97 8973.87 9552.76 10131.7 11578.9 Decomposition into 64 domains using Metis Figure 5: Permeability for the skyscraper case Metis decomposition uniform N N decomposition N N decomp. using Metis N 2 RAS RAS+Z Nico RAS+Z D2N RAS RAS+Z Nico RAS+Z D2N 4 112 97 24 229 203 29 16 >400 299 18 >400 >400 32 64 >400 294 18 >400 >400 23 18

Conclusion Robust, parallel and adaptive construction of a coarse space for problems with heterogeneities and jagged interfaces Condition number estimate The method is as algebraic as possible which paves the way to extension to systems of equations e.g. multiphase flows, elasticity problems, neutronic... 19

Thanks! 20