Adaptive Coarse Spaces and Multiple Search Directions: Tools for Robust Domain Decomposition Algorithms

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1 Adaptive Coarse Spaces and Multiple Search Directions: Tools for Robust Domain Decomposition Algorithms Nicole Spillane Center for Mathematical Modelling at the Universidad de Chile in Santiago. July 9th, 2015 DD23 Jeju Island, Korea

2 Nicole Spillane (U. de Chile) 2 / 33 Acknowledgements Collaborators Pierre Gosselet (CNRS, ENS Cachan) Frédéric Nataf (CNRS, Université Pierre et Marie Curie) Daniel J. Rixen (University of Munich) François-Xavier Roux (ONERA, Université Pierre et Marie Curie) Funding Financial support by CONICYT through project Fondecyt

K ΓΓ K ΓI K 1 II K IΓ, b := f Γ K ΓI K 1 II f I.

3 icole Spillane (U. de Chile) 3 / 33 Balancing Domain Decomposition (BDD): Ku = f (K spd) BDD reduces the problem to the interface Γ : Ω Γ Ω s Γ s Au,Γ = b, where A := K ΓΓ K ΓI K 1 II K IΓ, b := f Γ K ΓI K 1 II f I. The operator A is a sum of local contributions : A = N R s S s R s, s=1 S s := K s Γ s Γ s K s Γ s I s (K s I s I s) 1 K s I s Γ s, R s : Γ Γ s. J. Mandel. Balancing domain decomposition. Comm. Numer. Methods Engrg., 9(3): , The preconditioner H also: H := N R s D s S s D s R s, with s=1 The coarse space is range(u) := N R s D s Ker(S s ). s=1 N R s D s R s = I. s=1

4 Nicole Spillane (U. de Chile) 4 / 33 Illustration of the Problem: Heterogeneous Elasticity N = 81 subdomains, ν = 0.4, E 1 = 10 7 and E 2 = Young s modulus E Error (log scale) Metis Partition Iterations Problem: We want to design new DD methods with three objectives: Reliability: robustness and scalability. Efficiency: adapt automatically to difficulty. Simplicity: non invasive implementation.

5 Nicole Spillane (U. de Chile) 4 / 33 Illustration of the Problem: Heterogeneous Elasticity N = 81 subdomains, ν = 0.4, E 1 = 10 7 and E 2 = Young s modulus E Metis Partition Error (log scale) Global Local Simultaneous PPCG GenEO Iterations Problem: We want to design new DD methods with three objectives: Reliability: robustness and scalability. Efficiency: adapt automatically to difficulty. Simplicity: non invasive implementation.

6 Nicole Spillane (U. de Chile) 5 / 33 Contents Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG

7 Nicole Spillane (U. de Chile) 6 / 33 Projected PCG [Nicolaides, 1987 Dostál, 1988] for Ax = b preconditioned by H and projection Π Assume that A, H R n n are spd and U R n n0 is full rank, Define Π := I U(U AU) 1 U A. 1 x 0 = U(U AU) 1 U b; Initial Guess 2 r 0 = b Ax 0 ; Initial residual 3 z 0 = Hr 0; 4 p 0 =Πz 0 ; Initial search direction 5 for i = 0, 1,..., convergence do 6 q i = Ap i ; 7 α i = (q i p i ) 1 (p i r i ); 8 x i+1 = x i + α i p i ; Update approximate solution 9 r i+1 = r i α i q i ; Update residual 10 z i+1 = Hr i+1 ; Precondition 11 β i = (q i p i ) 1 (q i z i+1 ); 12 p i+1 = Πz i+1 β i p i ; Project and orthogonalize 13 end 14 Return x i+1 ; Convergence [Kaniel, 66 Meinardus, 63] [ ] i x x i A κ 1 2 x x 0 A κ + 1 κ = λ max λ min, λ max and λ min : extreme eigenvalues of HAΠ excluding 0. GenEO is a choice of range(u) that guarantees fast convergence.

8 Nicole Spillane (U. de Chile) 7 / 33 Bibliography (1/2) (coarse spaces based on generalized eigenvalue problems) Multigrid M. Brezina, C. Heberton, J. Mandel, and P. Vaněk. Technical Report 140, University of Colorado Denver, April Additive Schwarz J. Galvis and Y. Efendiev. Multiscale Model. Simul., Y. Efendiev, J. Galvis, R. Lazarov, and J. Willems. ESAIM Math. Model. Numer. Anal., F. Nataf, H. Xiang, V. Dolean, and N. S. SIAM J. Sci. Comput., V. Dolean, F. Nataf, R. Scheichl, and N. S. Comput. Methods Appl. Math., N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl. Numer. Math., Optimized Schwarz Methods S. Loisel, H. Nguyen, and R. Scheichl. Technical report, submitted, R. Haferssas, P. Jolivet, and F. Nataf. Technical report, Submitted, hal , 2015.

9 Bibliography (2/2) (coarse spaces based on generalized eigenvalue problems) Substructuring Methods P.E. Bjørstad, J. Koster and P. Krzyżanowski. Applied Parallel Computing, J. Mandel and B. Sousedík. Comput. Methods Appl. Mech. Engrg., B. Sousedík, J. Šístek, and J. Mandel. Computing, 2013 N. S. and D. J. Rixen. Int. J. Numer. Meth. Engng., H. H. Kim and E. T. Chung, Model. Simul., A. Klawonn, P. Radtke, and O. Rheinbach. DD22 Proceedings, A. Klawonn, P. Radtke, and O. Rheinbach. SIAM J. Numer. Anal., And many other talks at this conference: Minisymposium on monday: Olof B. Widlund, Clark R. Dohrmann (with Clemens Pechstein), Pierre Jolivet (HPDDM Frédéric Nataf s plenary talk tomorrow! Session on friday morning (CT 7)! Nicole Spillane (U. de Chile) 8 / 33

10 Nicole Spillane (U. de Chile) 9 / 33 Adaptive Coarse Space: Strategy (1/5) Relative error bounded w.r.t C/c Prove (2) Prove (1) Problem dependant Use trace theorems, Poincaré inequalities, Korn inequalities... (1) holds (2) holds DD method dependant FETI BDD Additive Schwarz Abstract Framework ([Toselli, Widlund (2005)]) (1) λ min > c if stable decomposition (2) λ max < C if local solver is stable [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014]

11 Nicole Spillane (U. de Chile) 10 / 33 Adaptive Coarse Space: Strategy (2/5) Relative error bounded w.r.t C/c Prove (2) Prove (1) Problem dependant Use trace theorems, Poincaré inequalities, Korn inequalities... (1) holds (2) holds DD method dependant FETI BDD Additive Schwarz Abstract Framework ([Toselli, Widlund (2005)]) (1) λ min > c if stable decomposition (2) λ max < C if local solver is stable [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014]

12 Nicole Spillane (U. de Chile) 11 / 33 Adaptive Coarse Space: Strategy (3/5) Relative error bounded w.r.t C/c In Problem dependant Prove (2) Prove (1) Make (1) or (2) local: x s M s x s τ x s N s x s for all x s. Use trace theorems, Poincaré inequalities, (Bottleneck Korn inequalities... Estimate) (1) holds (2) holds DD method dependant FETI BDD Additive Schwarz Abstract Framework ([Toselli, Widlund (2005)]) (1) λ min > c if stable decomposition (2) λ max < C if local solver is stable [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014]

13 Nicole Spillane (U. de Chile) 12 / 33 Adaptive Coarse Space: Strategy (4/5) Relative error bounded w.r.t C/c In Problem dependant Solve Prove M(2) s x s k = λs k Ns xprove s k. (1) bottleneck estimate x s M s x s τ x s N s x s holds Use intrace span{x theorems, s Poincaré inequalities, Korn λs k τ}, k ; inequalities... does not hold in span{x s k ; λs k > τ}. (1) holds (2) holds DD method dependant FETI BDD Additive Schwarz Abstract Framework ([Toselli, Widlund (2005)]) (1) λ min > c if stable decomposition (2) λ max < C if local solver is stable [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014]

14 Nicole Spillane (U. de Chile) 13 / 33 Adaptive Coarse Space: Strategy (5/5) Π: A-orthogonal projection onto the space where the bottleneck estimate holds Ax = b is equivalent to : In (I Π) Ax = (I Π) b } Problem {{ dependant } Direct Solver and Π Ax = Π b. }{{} DD method Relative error bounded w.r.t τ and number of neighbours N Solve Prove M(2) s x s k = λs k Ns xprove s k. (1) bottleneck estimate x s M s x s τ x s N s x s holds Use intrace span{x theorems, s Poincaré inequalities, Korn λs k τ}, k ; inequalities... does not hold in span{x s k ; λs k > τ}. (1) holds (2) holds DD method dependant FETI BDD Additive Schwarz Abstract Framework ([Toselli, Widlund (2005)]) (1) λ min > c if stable decomposition (2) λ max < C if local solver is stable [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014]

15 Nicole Spillane (U. de Chile) 14 / 33 GenEO Coarse Space for BDD λ min 1. the bottleneck estimate for λ max is x s R s AR s x s (1/τ) x s D s 1 SD s 1 x s. So we solve in each subdomain : D s 1 S s D s 1 x s k = λs k Rs AR s x s k, and define the coarse space as range(u) = span{r s x s k ; s = 1,..., N and λs k τ}. The the effective condition number is bounded by : κ(haπ) N τ ; N : number of neighbours of a subdomain.

16 icole Spillane (U. de Chile) 15 / 33 Numerical Illustration: Heterogeneous Elasticity N = 81 subdomains, ν = 0.4, E 1 = 10 7 and E 2 = 10 12, τ = 0.1 Young s modulus E Metis Partition Error (log scale) BDD BDD + GenEO Iterations Size of the coarse space: n 0 = 349 including 212 rigid body modes.

17 Conclusion for GenEO and introduction for MPCG Convergence guaranteed in few iterations, This is achieved with local contributions to the coarse space. Only drawback could be the cost of the eigensolves. Instead of precomputing a coarse space, take advantage of the local components already being computed: H := N } R s D s {{ S s D s R } s. :=H s s=1 R. Bridson and C. Greif. SIAM J. Matrix Anal. Appl., 27(4): , C. Greif, T. Rees, and D. Szyld. DD 21 proceedings, D. J. Rixen. PhD thesis, Université de Liège, Belgium, Collection des Publications de la Faculté des Sciences appliquées, n.175, P. Gosselet, D. J. Rixen, F.-X. Roux, and N. S.. International Journal for Numerical Methods in Engineering, Currently Published Online, Nicole Spillane (U. de Chile) 16 / 33

18 Multi Preconditioned CG for Ax = b prec. by {H s } s=1,...,n and Π A, H R n n spd U R n n0 full rank, H = N s=1 Hs, where H s spsd. MPCG 1 x 0 = U(U AU) 1 U b; Initial Guess 2 r 0 = b Ax 0; 3 Z 0 = [ H 1 r 0... H N r i+1 ] ; 4 P 0 = ΠZ 0; Initial search directions 5 for i = 0, 1,..., convergence do Remark r i, x i R n Z i, P i, Q i R n N β i,j R N N α i R N 6 Q i = AP i ; 7 α i = (Q i P i ) (P i r i ); 8 x i+1 = x i + P i α i ; Update approximate solution 9 r i+1 = r i Q i α i ; Update residual 10 Z i+1 = [ H 1 r i+1... H N r i+1 ] ; Multi Precondition 11 β i,j = (Q j P j ) (Q j Z i+1 ), j = 0,..., i; 12 P i+1 = ΠZ i+1 i j=0 P jβ i,j ; Project and orthog. 13 end 14 Return x i+1 ; n: size of problem, N: nb of precs. Properties 1. P i AP j = 0 (i j). 2. r i P j = 0 (j < i). 3. no short recurrence. 4. x x i+1 A = min{ x x A ; x x i + range(p i )}. icole Spillane (U. de Chile) 17 / 33

19 Nicole Spillane (U. de Chile) 17 / 33 Multi Preconditioned CG for Ax = b prec. by {H s } s=1,...,n and Π A, H R n n spd U R n n0 full rank, H = N s=1 Hs, where H s spsd. MPCG PPCG 1 x 0 = U(U AU) 1 U b; Initial Guess 2 r 0 = b Ax 0; 3 Z 0 = [ H 1 r 0... H N r i+1 ] ; 4 P 0 = ΠZ 0; Initial search directions 5 for i = 0, 1,..., convergence do 6 Q i = AP i ; 7 α i = (Q i P i ) (P i r i ); 8 x i+1 = x i + P i α i ; Update approximate solution 9 r i+1 = r i Q i α i ; Update residual 10 Z i+1 = [ H 1 r i+1... H N r i+1 ] ; Multi Precondition 11 β i,j = (Q j P j ) (Q j Z i+1 ), j = 0,..., i; 12 P i+1 = ΠZ i+1 i j=0 P jβ i,j ; Project and orthog. 13 end 14 Return x i+1 ; x 0 = U(U AU) 1 U b; r 0 = b Ax 0 ; z 0 = Hr 0; p 0 =Πz 0; for i = 0, 1,..., conv. do q i = Ap i ; α i = (q i p i ) 1 (p i r i ); x i+1 = x i + α i p i ; r i+1 = r i α i q i ; z i+1 = Hr i+1 ; β i = (q i p i ) 1 (q i z i+1 ); p i+1 = Πz i+1 β i p i ; end Return x i+1 ;

20 Numerical Illustration (Heterogeneous Elasticity E 2 /E 1 = 10 5 ) Error (log scale) Simultaneous (MPCG) PPCG GenEO Iterations Dimension of the minimization space 1,500 1, Simultaneous (MPCG) PPCG GenEO Number of local solves Simultaneous (MPCG) PPCG Nicole Spillane (U. de Chile) Iterations Iterations 18 / 33

Simultaneous FETI: Tests with CPU time (F.-X. Roux) 100 subdomains, 17 10 6 dofs, ν = 0.45 or 0.4999, 1 E 10 5. 2.6 GHz 8-core Xeon processors, Intel fortran, MKL-pardiso.

21 Simultaneous FETI: Tests with CPU time (F.-X. Roux) 100 subdomains, dofs, ν = 0.45 or , 1 E GHz 8-core Xeon processors, Intel fortran, MKL-pardiso. Checker Slices 1 Slices 2 Decomp. ν Solver #it dim Max solves Time (s) Slices FETI > 800 > 800 > 1600 > 7300 S-FETI Slices FETI S-FETI Checker FETI S-FETI Slices FETI > 800 > 800 > 1600 > 7300 S-FETI Slices FETI > 800 > 800 > 1600 > 7300 S-FETI Nicole Spillane (U. de Chile) 19 / 33

22 icole Spillane (U. de Chile) 20 / 33 Good convergence but two possible limitations Local contributions H s r i form a good minimization space. Cost of inverting P i AP i R N N at each iteration in α i = (P i AP i ) (P i r i ) and β i,j = (P j AP j ) (Q j Z i+1 ). Simultaneous BDD may work too hard on easy problems. Error (log scale) Simultaneous (MPCG) PPCG GenEO Number of local solves 4,000 3,000 2,000 1,000 Simultaneous (MPCG) PPCG Iterations Introduce adaptativity into multipreconditioned CG. From the GenEO section we know that: Iterations a few local vectors should suffice to accelerate convergence. they are the low frequency eigenvectors of: D s 1 S s D s 1 x s k = λs k Rs AR s x s k.

23 icole Spillane (U. de Chile) 21 / 33 Adaptive Multi Preconditioned CG for Ax = b preconditioned by N s=1 Hs and projection Π. (τ R + is chosen by the user) 1 x 0 = U(U AU) 1 U b; 2 r 0 = b Ax 0; Z 0 = Hr 0; P 0 = ΠZ 0; 3 for i = 0, 1,..., convergence do 4 Q i = AP i ; 5 α i = (Qi P i ) (P i r i ); 6 x i+1 = x i + P i α i ; 7 r i+1 = r i Q i α i ; 8 t i = (P iα i ) A(P i α i ) r i+1 Hr ; i+1 9 if t i < τ then τ-test 10 Z i+1 = [ H 1 r i+1... H N ] r i+1 11 else 12 Z i+1 = Hr i+1 ; 13 end 14 β i,j = (Qj P j ) (Qj Z i+1 ), j = 0,..., i; 15 P i+1 = ΠZ i+1 i P j β i,j ; j=0 16 end 17 Return x i+1 ; Remark x i, r i R n, Z i, P i, Q i R n N or R n, α i R N or R, β i,j R N N, R N, R 1 N or R, P i α i R n. n: size of problem, N: nb of precs.

24 icole Spillane (U. de Chile) 22 / 33 Theoretical Result (1/3): PPCG like Properties x 0 = U(U AU) 1 U b; r 0 = b Ax 0; Z 0 = Hr 0; P 0 = ΠZ 0; for i = 0, 1,..., convergence do Q i = AP i ; α i = (Q i P i ) (P i ri ); x i+1 = x i + P i α i ; r i+1 = r i Q i α i ; t i = (P iα i ) A(P i α i ) r i+1 Hr ; i+1 if t i < τ then ] Z i+1 = [H 1 r i+1... H N r i+1 else end Z i+1 = Hr i+1 ; β i,j = (Q j P j ) (Q j Z i+1 ); P i+1 = ΠZ i+1 i P j β i,j ; j=0 end Return x i+1 ; Remark No short recurrence property as soon as the minimization space has been augmented. Theorem x n n0 = x. Blocs of search directions are pairwise A-orthogonal: P j AP i = 0 (i j). Residuals are pairwise H-orthogonal: Hr j, r i = 0 (i j). x { x i A = min x x A ; x range(u) + } i 1 j=0 range(p j).

25 icole Spillane (U. de Chile) 23 / 33 Theoretical Result (2/3): Two types of iterations x 0 = U(U AU) 1 U b; r 0 = b Ax 0; Z 0 = Hr 0; P 0 = ΠZ 0; for i = 0, 1,..., convergence do Q i = AP i ; α i = (Q i P i ) (P i ri ); x i+1 = x i + P i α i ; r i+1 = r i Q i α i ; t i = (P iα i ) A(P i α i ) r i+1 Hr ; i+1 if t i < τ then ] Z i+1 = [H 1 r i+1... H N r i+1 else end Z i+1 = Hr i+1 ; β i,j = (Q j P j ) (Q j Z i+1 ); P i+1 = ΠZ i+1 i P j β i,j ; j=0 end Return x i+1 ; Theorem If the τ test returns t i 1 τ then ( ) x x i A 1 1/2, x x i 1 A 1 + λ min τ λ min = 1 for BDD.

26 Theoretical Result (2/3): Two types of iterations x 0 = U(U AU) 1 U b; r 0 = b Ax 0; Z 0 = Hr 0; P 0 = ΠZ 0; for i = 0, 1,..., convergence do Q i = AP i ; α i = (Q i P i ) (P i ri ); x i+1 = x i + P i α i ; r i+1 = r i Q i α i ; t i = (P iα i ) A(P i α i ) r i+1 Hr ; i+1 if t i < τ then ] Z i+1 = [H 1 r i+1... H N r i+1 else end Z i+1 = Hr i+1 ; β i,j = (Q j P j ) (Q j Z i+1 ); P i+1 = ΠZ i+1 i P j β i,j ; j=0 end Return x i+1 ; Theorem If the τ test returns t i 1 τ then ( ) x x i A 1 1/2, x x i 1 A 1 + λ min τ λ min = 1 for BDD. Proof x = x 0 + n n 0 n n 0 P i α i = x i + i=0 j=i n n 0 x x i 2 A = P j α j 2 A j=i P j α j x x i 1 2 A = x x i 2 A + P i 1α i 1 2 A. (Similar to [Axelsson and Kaporin, 2001].) x x i 1 2 A x x i 2 = 1 + r i 2 H P i 1 α i 1 2 A P i 1 α i 1 2 A A d i 2 A r i λ min H r i λ min τ. H }{{}}{{} = d i 2 AHA / d i 2 =t i 1 A icole Spillane (U. de Chile) 23 / 33

27 icole Spillane (U. de Chile) 24 / 33 Theoretical Result (3/3): No extra work for easy pbs. x 0 = U(U AU) 1 U b; r 0 = b Ax 0; Z 0 = Hr 0; P 0 = ΠZ 0; for i = 0, 1,..., convergence do Q i = AP i ; α i = (Q i P i ) (P i ri ); x i+1 = x i + P i α i ; r i+1 = r i Q i α i ; t i = (P iα i ) A(P i α i ) r i+1 Hr ; i+1 if t i < τ then ] Z i+1 = [H 1 r i+1... H N r i+1 else end Z i+1 = Hr i+1 ; β i,j = (Q j P j ) (Q j Z i+1 ); P i+1 = ΠZ i+1 i P j β i,j ; j=0 end Return x i+1 ; Theorem If λ max (HA) 1/τ then the algorithm is the usual PPCG.

28 icole Spillane (U. de Chile) 24 / 33 Theoretical Result (3/3): No extra work for easy pbs. x 0 = U(U AU) 1 U b; r 0 = b Ax 0; Z 0 = Hr 0; P 0 = ΠZ 0; for i = 0, 1,..., convergence do Q i = AP i ; α i = (Q i P i ) (P i ri ); x i+1 = x i + P i α i ; r i+1 = r i Q i α i ; t i = (P iα i ) A(P i α i ) r i+1 Hr ; i+1 if t i < τ then ] Z i+1 = [H 1 r i+1... H N r i+1 else end Z i+1 = Hr i+1 ; β i,j = (Q j P j ) (Q j Z i+1 ); P i+1 = ΠZ i+1 i P j β i,j ; j=0 end Return x i+1 ; Theorem If λ max (HA) 1/τ then the algorithm is the usual PPCG. Proof We begin with r i = r i 1 Q i 1 α i 1 and take the inner product by Hr i : Hr i, r i = Hr i, r i 1 Hr i, AP i 1 α i 1 or equivalently: r i 1/2 HAH P i 1α i 1 1/2 A (C.S.), Hr i, r i Hr i, AHr i P i 1α i 1, AP i 1 α i 1. Hr i, r i Finally: τ 1 Hr i, r i λ max Hr i, AHr i P i 1α i 1, AP i 1 α i 1 = t i 1. Hr i, r i

29 icole Spillane (U. de Chile) 25 / 33 Local Adaptive MPCG for N s=1 As x = b preconditioned by N s=1 Hs and projection Π. (τ R + is chosen by the user) 1 x 0 = U(U AU) 1 U b; r 0 = b Ax 0; Z 0 = Hr 0; P 0 = ΠZ 0; 2 for i = 0, 1,..., convergence do 3 Q i = AP i ; 4 α i = (Q i P i ) (P i r i ); 5 x i+1 = x i + P i α i ; 6 r i+1 = r i Q i α i ; 7 Z i+1 = Hr i+1 ; initialize Z i+1 8 for s = 1,..., N do 9 ti s = P iα i, A s P i α i r ; i+1 Hs r i+1 10 if ti s < τ then local τ-test 11 Z i+1 = [Z i+1 H s r i+1 ]; 12 end 13 end 14 β i,j = (Qj P j ) (Qj Z i+1 ), j = 0,..., i; 15 P i+1 = ΠZ i+1 i P j β i,j ; j=0 16 end 17 Return x i+1 ; A = N s=1 Rs }{{ S s R } s. :=A s Between 1 and N search directions per iteration. Reduces the cost of inverting P i AP i (one limitation of MPCG) Theorem If the τ test returns t s i 1 τ for all s = 1,..., N then ( ) x x i A 1 1/2. x x i 1 A 1 + λ min τ

30 Implementation for BDD: Saving local solves 1 x 0 = U(U AU) 1 U b; r 0 = b Ax 0; Z 0 = Hr 0; P 0 = Z 0; 2 for i = 0, 1,..., convergence do 3 Q i = N s=1 Qs i where Qi s = A s Z i A s U(U AU) 1 (AU) Z i i 1 Qj sβ i 1,j ; j=0 4 α i = (Qi P i ) (P i r i ); x i+1 = x i + Π P i α i ; r i+1 = r i Q i α i ; 5 Z i+1 = Hr i+1 ; 6 for s = 1,..., N do 7 ti s = Π P i α i, Qi sα i r ; i+1 Hs r i+1 8 if ti s < τ then 9 Z i+1 = [Z i+1 H s r i+1 ]; 10 Subtract H s r i+1 from the first column in Z i+1 ; 11 end 12 end 13 β i,j = (Qi P i ) (Q j Z i+1 ) (j = 0,..., i); P i+1 = Z i+1 i P j β i,j ; j=0 14 end 15 Return x i+1 ; Same trick as in : P. Gosselet, D. J. Rixen, F.-X. Roux, and N. S. Simultaneous FETI and block FETI: Robust domain decomposition with multiple search directions. International Journal for Numerical Methods in Engineering, Nicole Spillane (U. de Chile) 26 / 33

31 Nicole Spillane (U. de Chile) 27 / 33 Numerical Illustration (1/4): Homogenous subdomains Error (log scale) Iterations Global Local Simultaneous PPCG GenEO Dimension of the minimization space Iterations Global Local Simultaneous PPCG GenEO Number of local solves 4,000 3,000 2,000 1,000 0 Global Local Simultaneous PPCG Iterations

32 Numerical Illustration (2/4): Metis subdomains Error (log scale) Global Local Simultaneous PPCG GenEO Iterations Dimension of the minimization space 1,500 1, Global Local Simultaneous PPCG GenEO Number of local solves Global Local Simultaneous PPCG Nicole Spillane (U. de Chile) Iterations Iterations 28 / 33

33 Numerical Illustration (2/4): Zoom on new methods Error (log scale) Iterations Global Local Simultaneous GenEO Nicole Spillane (U. de Chile) 29 / 33

34 Numerical Illustration (3/4): Channels Error (log scale) Global Local Simultaneous PPCG GenEO Iterations Dimension of the minimization space 2,500 2,000 1,500 1, Global Local Simultaneous PPCG GenEO Nicole Spillane (U. de Chile) Iterations Iterations 30 / 33 Number of local solves Global Local Simultaneous PPCG

35 Numerical Illustration (3/4): Zoom on new methods Error (log scale) Iterations Global Local Simultaneous GenEO Nicole Spillane (U. de Chile) 31 / 33

36 Nicole Spillane (U. de Chile) 32 / 33 Numerical Illustration (4/4) Variable heterogeneity for N = 81 (k-scaling) : Global τ test Local τ test Simultaneous usual BDD GenEO E 2 /E 1 it solves it solves it solves it solves it dim < 554 dim < 423 dim < 1428 dim < 353 dim < 372 Variable number of subdomains for E 2 /E 1 = 10 5 (k scaling) : Global τ test Local τ test Simultaneous usual BDD GenEO N it solves it solves it solves it solves it dim < 693 dim < 379 dim < 1193 dim < 320 dim < 327

37 Nicole Spillane (U. de Chile) 32 / 33 Numerical Illustration (4/4) Variable heterogeneity for N = 81 (k-scaling) : Global τ test Local τ test Simultaneous usual BDD GenEO E 2 /E 1 it solves it solves it solves it solves it dim < 554 dim < 423 dim < 1428 dim < 353 dim < 372 Variable number of subdomains for E 2 /E 1 = 10 5 (k scaling) : Global τ test Local τ test Simultaneous usual BDD GenEO N it solves it solves it solves it solves it dim < 693 dim < 379 dim < 1193 dim < 320 dim < 327

38 Nicole Spillane (U. de Chile) 33 / 33 Conclusion Reliability, Efficiency and Simplicity through adaptive coarse spaces and adaptive multiple search directions. Perspectives Test on industrial test cases and measure CPU times. Use the τ test to recycle part of the minimization spaces. Simultaneous BDD: Theoretical Analysis at the PDE level. Adaptive Multi PCG for other DD methods: when λ max is known or neither λ min or λ max. N. S., D. J. Rixen, Automatic spectral coarse spaces for robust FETI and BDD algorithms. IJNME, P. Gosselet, D. J. Rixen, F.-X. Roux, and N. S. Simultaneous FETI and block FETI: Robust domain decomposition with multiple search directions. International Journal for Numerical Methods in Engineering, N. S. Adaptive Multi Preconditioned Conjugate Gradient: Algorithm, Theory and an Application to Domain Decomposition Submitted, Preprint available online <hal >, Downloadable at

Multipréconditionnement adaptatif pour les méthodes de décomposition de domaine. Nicole Spillane (CNRS, CMAP, École Polytechnique)

Multipréconditionnement adaptatif pour les méthodes de décomposition de domaine Nicole Spillane (CNRS, CMAP, École Polytechnique) C. Bovet (ONERA), P. Gosselet (ENS Cachan), A. Parret Fréaud (SafranTech),