Multilevel Krylov Methods Deflation Deflation, DD, MG Reinhard - PowerPoint PPT Presentation

Multilevel Krylov Methods Reinhard Nabben Yogi A. Erlangga Multilevel Krylov Methods Deflation Deflation, DD, MG Reinhard Nabben Multilevel Krylov methods Numerical Yogi A. Erlangga examples MK methods for Helmholtz equation AMK methods Conclusion supported by Deutsche Forschungsgemeinschaft (DFG) 12.09.2008

Multilevel Krylov Outline Methods Reinhard Nabben Yogi A. Erlangga Deflation Deflation Deflation, DD, MG Deflation, DD, MG Multilevel Krylov methods Numerical Multilevel Krylov methods examples MK methods for Helmholtz Numerical examples equation AMK methods Conclusion MK methods for Helmholtz equation AMK methods Conclusion

Multilevel Krylov Methods Deflated CG Reinhard Nabben Yogi A. Erlangga Nicolaides 1987, Mansfield 1988, 1990, Kolotilina 1998, Deflation Vuik, Segal, and Meijerink 1999, Morgan 1995, Saad, Deflation, DD, MG Yeung, Erhel, and Guyomarch 2000, Frank and Vuik Multilevel Krylov 2001, Blaheta 2006 methods Numerical examples MK methods for Helmholtz Deflation and restarted GMRES equation AMK methods Morgan 1995, Erhel, Burrage, and Pohl 1996, Chapman Conclusion and Saad 1997, Eiermann, Ernst, and Schneider 2000, Morgan 2002 Clemens et al. 2003,2004, de Sturler et al. 2006, Aksoylu, H. Klie, and M.F . Wheeler 2007

Multilevel Krylov Deflation with general vectors Methods Reinhard Nabben A symmetric positive definite Yogi A. Erlangga Deflation Deflation, DD, MG Multilevel Krylov E = Z T AZ Z = [ z 1 , . . . , z r ] rankZ = r methods Numerical examples MK methods for Helmholtz equation P D = I − AZE − 1 Z T , Z ∈ R n × r , AMK methods Conclusion P D AZ = 0 spectrum ( P D A ) = { 0 , . . . , 0 , µ r + 1 , . . . µ n }

Multilevel Krylov Deflation for linear systems Methods Reinhard Nabben Z ∈ R n × r Yogi A. Erlangga Z = [ z 1 , . . . , z r ] rankZ = r Deflation Deflation, DD, MG P D = I − AZE − 1 Z T Ax = b Multilevel Krylov methods Numerical x = ( I − P T D ) x + P T We have: D x Compute both! examples MK methods for Helmholtz equation 1. ( I − P T D ) x = Z ( Z T AZ ) − 1 Z T b AMK methods preconditioner M − 1 : 2. Solve P D A ˜ x = P D b Conclusion M − 1 P D A ˜ x = M − 1 P D b 3. Build P T x = P T D ˜ D x

Multilevel Krylov Deflation Methods M − 1 preconditioner, ILU Z appro. eigenvectors Reinhard Nabben ZE − 1 Z T Yogi A. Erlangga Deflation Deflation, DD, MG Domain decomposition Multilevel Krylov M − 1 add. Schwarz Z grid transfer operator methods ZE − 1 Z T coarse grid correction Numerical examples MK methods for Helmholtz equation Multigrid AMK methods M − 1 smoother Z grid transfer operator Conclusion ZE − 1 Z T coarse grid correction

Multilevel Krylov Methods Reinhard Nabben Name Method Operator Yogi A. Erlangga M − 1 PREC Traditional Preconditioned CG Deflation M − 1 + Q AD Additive Coarse Grid Correc. Deflation, DD, MG M − 1 P D DEF1 Deflation Variant 1 Multilevel Krylov methods P T D M − 1 DEF2 Deflation Variant 2 Numerical M − 1 P D + Q A-DEF1 Adapted Deflation Variant 1 examples D M − 1 + Q P T A-DEF2 Adapted Deflation Variant 2 MK methods for Helmholtz P T D M − 1 P D + Q BNN Abstract Balancing equation P T D M − 1 P D AMK methods R-BNN1 Reduced Balancing Variant 1 P T D M − 1 Conclusion R-BNN2 Reduced Balancing Variant 2 Q = ZE − 1 Z T = Z ( Z T AZ ) − 1 Z T

Multilevel Krylov Methods Reinhard Nabben Name Method Operator Yogi A. Erlangga M − 1 PREC Traditional Preconditioned CG Deflation M − 1 + Q AD Additive Coarse Grid Correc. Deflation, DD, MG M − 1 P D DEF1 Deflation Variant 1 Multilevel Krylov methods P T D M − 1 DEF2 Deflation Variant 2 Numerical M − 1 P D + Q A-DEF1 Adapted Deflation Variant 1 examples D M − 1 + Q P T A-DEF2 Adapted Deflation Variant 2 MK methods for Helmholtz P T D M − 1 P D + Q BNN Abstract Balancing equation P T D M − 1 P D AMK methods R-BNN1 Reduced Balancing Variant 1 P T D M − 1 Conclusion R-BNN2 Reduced Balancing Variant 2 Q = ZE − 1 Z T = Z ( Z T AZ ) − 1 Z T Nabben, Vuik 04 , Nabben, Vuik 06, Nabben Vuik 08 Tang, Nabben, Vuik, Erlangga 07 Tang, MacLachlan, Nabben, Vuik 08

Multilevel Krylov Non-symmetric Problems Methods Reinhard Nabben ◮ Erlangga, Nabben 06: Yogi A. Erlangga Z T → Y T E → Y T AZ Deflation Deflation, DD, MG P D = I − AZE − 1 Y T Multilevel Krylov methods Numerical P T D → Q D = I − ZE − 1 Y T A examples MK methods for Helmholtz equation P BNN = Q D M − 1 P D + ZE − 1 Y T AMK methods Conclusion � M − 1 ( b − Au k , D ) � 2 ≤ � M − 1 ( b − Au k , BNN ) � 2 .

Multilevel Krylov Multilevel Krylov methods (MK methods) Methods Reinhard Nabben Yogi A. Erlangga A ∈ C N × N , u , b ∈ C N . Au = b , Deflation Deflation, DD, MG A is in general nonsymmetric, sparse and large Multilevel Krylov methods Problems: Numerical ◮ Diffusion problem (symmetric) examples MK methods for ◮ Convection-diffusion equation (nonsymmetric) Helmholtz equation ◮ Helmholtz equation (symmetric, indefinite) AMK methods Conclusion Preconditioned system: M − 1 1 AM − 1 u = M − 1 2 � � 1 b , u = M 2 u , M 1 , M 2 nonsingular . Here, � u = � � � A := M − 1 A , b := M − 1 b . A � � b , u := u ,

Multilevel Krylov Consider Methods Reinhard Nabben Yogi A. Erlangga E = Y T � P N = P D + λ N Z � E − 1 Y T , � AZ , Deflation where Deflation, DD, MG Multilevel Krylov methods P D = I − � AZ � E − 1 Y T , ( Deflation ) Numerical examples and solve the system MK methods for Helmholtz equation P N � u = P N � A � b . AMK methods Conclusion ◮ Z , Y ∈ R n × r are full rank ◮ � E : Galerkin product ◮ λ N Approximation of largest eigenvalue of � A .

Properties of P N � Multilevel Krylov A Methods Reinhard Nabben Spectral relation between P D � A and P N � A . Yogi A. Erlangga Theorem Deflation Z , Y are arbitrary rectangular matrices with rank r. Deflation, DD, MG σ ( P D � Multilevel Krylov A ) = { 0 , . . . , 0 , µ r + 1 , . . . , µ N } methods σ ( P N � = ⇒ A ) = { λ N , . . . , λ N , µ r + 1 , . . . , µ N } . Numerical examples MK methods for Helmholtz equation ◮ σ ( P N � A ) is similar to σ ( P D � A ) AMK methods Conclusion

Properties of P N � Multilevel Krylov A Methods Reinhard Nabben Deflation: Yogi A. Erlangga ◮ P 2 D = P D (Projection) Deflation ◮ P D � A = � AQ D Deflation, DD, MG ◮ If � A is symmetric, then P D � A is also symmetric Multilevel Krylov methods In contrast: Numerical examples ◮ P 2 N � = P N MK methods for ◮ P N � A � = � AQ N . However, σ ( P N � A ) = σ ( � Helmholtz AQ N ) equation ◮ P N � A is not symmetric even if � AMK methods A is symmetric. Conclusion

Multilevel Krylov Multilevel Krylov method Methods Reinhard Nabben Yogi A. Erlangga E = Y T � P N = P D + λ N Z � � Deflation E − 1 Y T , AZ , Deflation, DD, MG Need to solve the coarse system with � A H := � Multilevel Krylov E . methods Numerical examples ◮ P N is stable w.r.t. inexact solves. MK methods for Helmholtz ◮ Applying P N at the “second” level, i.e. use P N , H equation AMK methods � A H � instead of x H = b H Conclusion P N , H � A H � x H = P N , H b H solve: using a Krylov method ◮ With inner Krylov iterations, P N is i.g. not constant Use flexible Krylov subspace method (FGMRES, FQMR, . . . )

Multilevel Krylov Multilevel Krylov method Methods Reinhard Nabben ◮ The choice of Z and Y Yogi A. Erlangga Sparsity of Z and Y ; Deflation May be the same as prolongation and restriction Deflation, DD, MG matrices in multigrid Multilevel Krylov (piece-wise constant, bi-linear interpolation, etc.); methods Numerical But not eigenvectors; examples Y = Z ; MK methods for Helmholtz equation ◮ About λ N AMK methods Expensive to compute, but an approximate is Conclusion sufficient: → by Gerschgorin’s theorem.

Multilevel Krylov Numerical example: 2D Poisson equation Methods Reinhard Nabben The 2D Poisson equation: Yogi A. Erlangga in Ω ∈ ( 0 , 1 ) 2 , −∇ · ∇ u = g , Deflation u = 0 , on Γ = ∂ Ω . Deflation, DD, MG Multilevel Krylov methods Discretization: finite differences. Numerical examples Ω with index set I = { i | u i ∈ Ω } . MK methods for Helmholtz Ω is partitioned into non-overlapping subdomain Ω j , equation AMK methods j = 1 , . . . , l , with respective index I j = { i ∈ I| u i ∈ Ω j } . Conclusion Then, Z = [ z ij ] : � 1 , i ∈ I j , z ij = 0 , i / ∈ I j . Y = Z

Multilevel Krylov Numerical example: 2D Poisson equation Methods Reinhard Nabben Convergence results: relative residual ≤ 10 − 6 Yogi A. Erlangga Gerschgorin estimate for λ N Deflation Deflation, DD, MG Multilevel Krylov N MK(2,2,2) MK(4,2,2) MK(6,2,2) MK(4,3,3) MG methods 32 2 15 14 14 14 11 Numerical 64 2 16 14 14 14 11 examples 128 2 MK methods for 16 14 14 14 11 Helmholtz 256 2 16 14 14 14 11 equation ◮ MK(4,2,2,): Multilevel Projection with 4,2,2 AMK methods Conclusion FGMRES iterations at level no. 2,3 and 4. etc. ◮ MG: Multi Grid (here, V-cycle, one pre- and post RB-GS smoothing, bilinear interpolation) Observation: ◮ h -independent convergence ◮ Convergence of MK is comparable with MG.