Nouveaux d eveloppements sur FreeFem++ et HPDDM Fr ed eric Nataf - PowerPoint PPT Presentation

Nouveaux d´ eveloppements sur FreeFem++ et HPDDM Fr´ ed´ eric Nataf Laboratory J.L. Lions (LJLL), CNRS, Alpines Inria and Univ. Paris VI Ryadh Haferssas (LJLL-Alpines) Victorita Dolean (University of Nice) Fr´ ed´ eric Hecht (LJLL-Alpines) Pierre Jolivet (LJLL-Alpines now CNRS, IRIT, ENSEEIHT) Nicole Spillane (LJLL-Alpines now CNRS at CMAP) Pierre-Henri Tournier (LJLL-Alpines) Colloque Couplages Num´ eriques – Nice

Outline Overview 1 Restricted Additive Schwarz Methods 2 Optimized Restricted Additive Schwarz Methods 3 SORAS-GenEO-2 coarse space 4 5 Numerical Results with FreeFem++ 6 Conclusion F. Nataf SORAS 2 / 42

Framework: Scientific computing Applications: Large discretized system of PDEs flow in heterogeneous / strongly heterogeneous coefficients stochastic / layered media (high contrast, multiscale) structural mechanics electromagnetics etc. E.g. Darcy pressure equation, P 1 -finite elements: Au = f cond ( A ) ∼ κ max h − 2 κ min Goal: Parallel iterative solvers robust in size and heterogeneities 80% of the elapsed time for typical engineering applications F. Nataf SORAS 3 / 42

10 3 10 5 3 GHz 1 P 10 T 100 G 1 G 10 k 100 Frequency (Hz) # of transistors 10 9 10 7 Need to Go Parallel Since year 2004: CPU frequency stalls at 2-3 GHz due to the heat dissipation wall. The only way to improve the performance of computer is to go parallel Credits: http://download.intel.com/ pressroom/kits/IntelProcessorHistory.pdf 1971 1975 1979 1983 1987 1991 1995 1999 2003 2007 2011 2015 F. Nataf SORAS 4 / 42

A u = f ? Panorama of parallel linear solvers Multi-frontal sparse direct solver (I. Duff et al.) MUMPS (J.Y. L ’Excellent), SuperLU (Demmel, . . . ), PastiX, UMFPACK, PARDISO (O. Schenk), Iterative Methods Fixed point iteration: Jacobi, Gauss-Seidel, SSOR Krylov type methods: Conjuguate Gradient (Stiefel-Hestenes), GMRES (Y. Saad), QMR (R. Freund), MinRes, BiCGSTAB (van der Vorst) ”Hybrid Methods” Multigrid (A. Brandt, Ruge-St¨ uben, Falgout, McCormick, A. Ruhe, Y. Notay, . . . ) Domain decomposition methods (O. Widlund, C. Farhat, J. Mandel, P .L. Lions, ) are a naturally parallel compromise F. Nataf SORAS 5 / 42

Outline Overview 1 Restricted Additive Schwarz Methods 2 Optimized Restricted Additive Schwarz Methods 3 SORAS-GenEO-2 coarse space 4 5 Numerical Results with FreeFem++ 6 Conclusion F. Nataf SORAS 6 / 42

An introduction to Additive Schwarz Consider the discretized Poisson problem: Au = f ∈ R n . Given a decomposition of � 1 ; n � , ( N 1 , N 2 ) , define: the restriction operator R i from R � 1 ; n � into R N i , R T i as the extension by 0 from R N i into R � 1 ; n � . u m − → u m + 1 by solving concurrently: u m + 1 = u m 1 + A − 1 1 R 1 ( f − Au m ) u m + 1 = u m 2 + A − 1 2 R 2 ( f − Au m ) 1 2 = R i u m and A i := R i AR T where u m i . i Ω F. Nataf SORAS 7 / 42

An introduction to Additive Schwarz Consider the discretized Poisson problem: Au = f ∈ R n . Given a decomposition of � 1 ; n � , ( N 1 , N 2 ) , define: the restriction operator R i from R � 1 ; n � into R N i , R T i as the extension by 0 from R N i into R � 1 ; n � . u m − → u m + 1 by solving concurrently: u m + 1 = u m 1 + A − 1 1 R 1 ( f − Au m ) u m + 1 = u m 2 + A − 1 2 R 2 ( f − Au m ) 1 2 = R i u m and A i := R i AR T where u m i . i Ω 1 Ω 2 F. Nataf SORAS 7 / 42

An introduction to Additive Schwarz Consider the discretized Poisson problem: Au = f ∈ R n . Given a decomposition of � 1 ; n � , ( N 1 , N 2 ) , define: the restriction operator R i from R � 1 ; n � into R N i , R T i as the extension by 0 from R N i into R � 1 ; n � . u m − → u m + 1 by solving concurrently: u m + 1 = u m 1 + A − 1 1 R 1 ( f − Au m ) u m + 1 = u m 2 + A − 1 2 R 2 ( f − Au m ) 1 2 = R i u m and A i := R i AR T where u m i . i Ω 1 Ω 2 F. Nataf SORAS 7 / 42

An introduction to Additive Schwarz II We have effectively divided, but we have yet to conquer. Duplicated unknowns coupled via a partition of unity : N � R T I = i D i R i . i = 1 1 1 1 1 2 2 N N Then, u m + 1 = � R T i D i u m + 1 M − 1 � R T i D i A − 1 . RAS = R i . i i i = 1 i = 1 RAS algorithm (Cai & Sarkis, 1999) F. Nataf SORAS 8 / 42

An introduction to Additive Schwarz II We have effectively divided, but we have yet to conquer. Duplicated unknowns coupled via a partition of unity : N � R T I = i D i R i . i = 1 1 1 1 1 2 2 N N Then, u m + 1 = � R T i D i u m + 1 M − 1 � R T i D i A − 1 . RAS = R i . i i i = 1 i = 1 RAS algorithm (Cai & Sarkis, 1999) F. Nataf SORAS 8 / 42

An introduction to Additive Schwarz II We have effectively divided, but we have yet to conquer. Duplicated unknowns coupled via a partition of unity : N � R T I = i D i R i . i = 1 1 1 1 1 2 2 N N Then, u m + 1 = � R T i D i u m + 1 M − 1 � R T i D i A − 1 . RAS = R i . i i i = 1 i = 1 RAS algorithm (Cai & Sarkis, 1999) F. Nataf SORAS 8 / 42

Outline Overview 1 Restricted Additive Schwarz Methods 2 Optimized Restricted Additive Schwarz Methods 3 SORAS-GenEO-2 coarse space 4 5 Numerical Results with FreeFem++ 6 Conclusion F. Nataf SORAS 9 / 42

P .L. Lions’ Algorithm (1988) − ∆( u n + 1 ) = f in Ω 1 , 1 u n + 1 = 0 on ∂ Ω 1 ∩ ∂ Ω , 1 ( ∂ ) = ( − ∂ + α )( u n + 1 + α )( u n 2 ) on ∂ Ω 1 ∩ Ω 2 , 1 ∂ n 1 ∂ n 2 ( n 1 and n 2 are the outward normal on the boundary of the subdomains) − ∆( u n + 1 ) = f in Ω 2 , 2 u n + 1 = 0 on ∂ Ω 2 ∩ ∂ Ω 2 ( ∂ ) = ( − ∂ + α )( u n + 1 + α )( u n 1 ) on ∂ Ω 2 ∩ Ω 1 . 2 ∂ n 2 ∂ n 1 with α > 0. Overlap is not necessary for convergence. Parameter α can be optimized for. Extended to the Helmholtz equation (B. Despr` es, 1991) a.k.a FETI 2 LM (Two-Lagrange Multiplier ) Method, 1998. F. Nataf SORAS 10 / 42

ORAS: Optimized RAS P .L. Lions algorithm at the continuous level (partial 1 differential equation) Algebraic formulation for overlapping subdomains ⇒ Let B i 2 be the matrix of the Robin subproblem in each subdomain 1 ≤ i ≤ N , define M − 1 ORAS := � N i D i B − 1 i = 1 R T R i , Optimized i multiplicative, additive, and restricted additive Schwarz preconditioning, St Cyr et al, 2007 Symmetric variant ⇒ 3 M − 1 OAS := � N i B − 1 i = 1 R T R i (Natural but K.O.) 1 i SORAS := � N M − 1 i D i B − 1 i = 1 R T D i R i (O.K.) 2 i Adaptive Coarse space with prescribed targeted 4 convergence rate ⇒ ??? F. Nataf SORAS 11 / 42

ORAS: Optimized RAS P .L. Lions algorithm at the continuous level (partial 1 differential equation) Algebraic formulation for overlapping subdomains ⇒ Let B i 2 be the matrix of the Robin subproblem in each subdomain 1 ≤ i ≤ N , define M − 1 ORAS := � N i D i B − 1 i = 1 R T R i , Optimized i multiplicative, additive, and restricted additive Schwarz preconditioning, St Cyr et al, 2007 Symmetric variant ⇒ 3 M − 1 OAS := � N i B − 1 i = 1 R T R i (Natural but K.O.) 1 i SORAS := � N M − 1 i D i B − 1 i = 1 R T D i R i (O.K.) 2 i Adaptive Coarse space with prescribed targeted 4 convergence rate ⇒ ??? F. Nataf SORAS 11 / 42

ORAS: Optimized RAS P .L. Lions algorithm at the continuous level (partial 1 differential equation) Algebraic formulation for overlapping subdomains ⇒ Let B i 2 be the matrix of the Robin subproblem in each subdomain 1 ≤ i ≤ N , define M − 1 ORAS := � N i D i B − 1 i = 1 R T R i , Optimized i multiplicative, additive, and restricted additive Schwarz preconditioning, St Cyr et al, 2007 Symmetric variant ⇒ 3 M − 1 OAS := � N i B − 1 i = 1 R T R i (Natural but K.O.) 1 i SORAS := � N M − 1 i D i B − 1 i = 1 R T D i R i (O.K.) 2 i Adaptive Coarse space with prescribed targeted 4 convergence rate ⇒ ??? F. Nataf SORAS 11 / 42

ORAS: Optimized RAS P .L. Lions algorithm at the continuous level (partial 1 differential equation) Algebraic formulation for overlapping subdomains ⇒ Let B i 2 be the matrix of the Robin subproblem in each subdomain 1 ≤ i ≤ N , define M − 1 ORAS := � N i D i B − 1 i = 1 R T R i , Optimized i multiplicative, additive, and restricted additive Schwarz preconditioning, St Cyr et al, 2007 Symmetric variant ⇒ 3 M − 1 OAS := � N i B − 1 i = 1 R T R i (Natural but K.O.) 1 i SORAS := � N M − 1 i D i B − 1 i = 1 R T D i R i (O.K.) 2 i Adaptive Coarse space with prescribed targeted 4 convergence rate ⇒ ??? F. Nataf SORAS 11 / 42

ORAS: Optimized RAS P .L. Lions algorithm at the continuous level (partial 1 differential equation) Algebraic formulation for overlapping subdomains ⇒ Let B i 2 be the matrix of the Robin subproblem in each subdomain 1 ≤ i ≤ N , define M − 1 ORAS := � N i D i B − 1 i = 1 R T R i , Optimized i multiplicative, additive, and restricted additive Schwarz preconditioning, St Cyr et al, 2007 Symmetric variant ⇒ 3 M − 1 OAS := � N i B − 1 i = 1 R T R i (Natural but K.O.) 1 i SORAS := � N M − 1 i D i B − 1 i = 1 R T D i R i (O.K.) 2 i Adaptive Coarse space with prescribed targeted 4 convergence rate ⇒ ??? F. Nataf SORAS 11 / 42