New Adaptive Projection Technique for Krylov Subspace Method

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1 1 2 New Adaptive Projection Technique for Krylov Subspace Method Akinori Kumagai 1 and Takashi Nodera 2 Generally projection technique in the numerical operation is one of the preconditioning commonly used when the linear system is solved. In this paper, we describe the method that the value of some eigenvalues to the coefficient matrix by using this preconditioning is shifted. Moreover, a new projection technique that uses this projection at the multi-level is derived, and the technique for applying to the Krylov subspace method is given. 1. Krylov Au = b, A R n n (1) A A SPD( ) (1) x 1 Graduate School of Science and Technology, Keio University 2 Faculty of Science and Technology, Keio University κ(a) = λmax(a) λ min(a) λ max(a) λ min(a) A A Krylov 0 A SPD κ(m 1 A) 1 M M 1 A I(I: ) LU Cholesky A 0 Nicolaides 4) CG Krylov Morgan Augmented GMRES 5) Krylov 6) P N = P D + λ N ZE 1 Y T, E = Y T AZ P D = I AZE 1 Y T Z, Y R n r Z, Y ; rank (r) Z A SPD Z = Y A P N P D σ(p N A) = {λ n,..., λ n, λ r+1,..., λ n}. σ(p DA) = {0,..., 0, λ r+1,..., λ n}. A r λ n 0 r(z ) (2) Z Krylov E 1 Z Krylov E 1 E 1 E = Y T AZ Galerkin Galerkin Galerkin (2) 1 c 2009 Information Processing Society of Japan

2 1) Galerkin Krylov ( Multi-level ) Multi-level projection Krylov method Krylov GMRES 2. Krylov n A n b Krylov K n K n = span{b, Ab, A 2 b,..., A n 1 b} 1 b, Ab A A 2 b Krylov Krylov Lanczos Arnoldi Krylov Arnoldi Lanczos GMRES generalized minimum residual BiCGSTAB stabilized biconjugate gradient Multilevel projection Krylov method GMRES 2.1 GMRES GMRES r n = Ax n b x n K n Ax = b Krylov K n = span{b, Ab, A 2 b,..., A n 1 b} Arnoldi K n {q 1, q 2,..., q n } x n K n y n R n x n = Q ny n Q n q 1, q 2,..., q n m n Arnoldi AQ n = Q n+1h n (n + 1) n Hessenberg H n Q n e 1 = (1, 0, 0,..., 0) R n+1 Ax n b = H ny n e 1 x n r n = H ny n e 1 GMRES ( 1 ) 1 ( 2 ) r n y n ( 3 ) x n=q ny n ( 4 ) 3. GMRES (1) Âû = ˆb Â (3) Â = M 1 A, û = u, ˆb = M 1 b 1 Â σ(â) = {λ1,..., λn}, λi λi+1, i N 3.1 ˆ A 1,γ = Â γ1z1yt, y T z 1 = 1, γ R. 3 y T i Aˆ 1,γ = y T i (Â γ1z1yt ) = λ iz 1y T i 2 c 2009 Information Processing Society of Japan

3 Â 1,γy i = (Â γ1z1yt )y i = Ây1 γ1z1yt y 1 = λ iy 1 Â 1,γz 1 = (Â γ1z1yt )z 1 = Âz1 γ1z1yt z 1 = (λ 1 γ 1)z 1 3 σ(â1,γ) = {λ1 γ1, λ2,..., λn} γ 1 = λ 1 σ(â1,γ) = {0, λ2,..., λn} γ 1 = λ 1 λ n σ(â1,γ) = {λn, λ2,..., λn} γ 1z 1y T λ 1 γ 1 r λ n 0 Â r = Â ZΓrY T, Y T Z = 1 Z = [z 1,..., z r] Γ = diag(γ 1,..., γ r) Â rz i = (Â ZΓrY T )z i = Âzi [z1,..., zr]diag(γ1,..., γr)[y1,..., yr]t z i = λ iz i γ iz i = (λ i γ i)z i. (3) i = 1,..., r Ârzi = (λi γi)zi i = r + 1,..., n Ârzi = λizi (γ1,..., γr) = (λ1,..., λr) Γ r = E = diag(λ 1,..., λ r) ÂZ = ZÊ Aˆ T Y = Y Ê Â r = Â ZÊY T = Â ZÊ ˆ E 1ÊY T = Â ÂZ ˆ E 1 Y T Â (4) Â r = (I ÂZ E ˆ 1 Y T )Â := P ˆDÂ (5) Â r = Â(I ÂZE ˆ 1 Y T ) = ÂQ ˆD (6) (γ 1,..., γ r) = (λ 1 λ n,..., λ r λ n) Γ r = E = diag(λ 1 λ n,..., λ r λ n) Â r,γ = (I ÂZ E ˆ 1 Y T + λ nze ˆ 1 Y T )Â = P ˆN Â (7) Â r,γ = Â(I ÂZE ˆ 1 Y T + λ nze ˆ 1 Y T ) = ÂQ ˆN (8) P (e.g., P ˆD) Q(e.g., Q ˆD) 3.2 (5) (6) (7) (8) σ(âr) = σ(p ˆDÂ) = σ(âq ˆD) σ( ˆ A r,γ) = σ(p ˆN Â) = σ(âq ˆN ) 3 c 2009 Information Processing Society of Japan

4 σ(p ˆDÂ) = σ(âq ˆD) = {0,..., 0, λ r+1,..., λ n} σ(p ˆN Â) = σ(âq ˆN ) = {λ n,..., λ n, λ r+1,..., λ n} A A λ 1 λ n P ˆDÂ P ˆN Â λ r+1 λ n κ(a) = λ n/λ 1 λ n/λ r+1 = κ(âr,γ) Krylov (GMRES ) 4. GMRES GMRES 4.1 P ˆN P ˆD Q ˆD P ˆN P ˆD Q ˆD r 0 Galerkin Galerkin P ˆD Q ˆD < ɛ λ 1 P ˆD Q ˆD P ˆN λ n P ˆN 4.2 P ˆN P ˆN P ˆN ( ) E 1 :λ n,est n λ n λ n,est E E E E E E + 4 Q ˆN 4.3 λ n Q ˆN λ n λ n λ n ρ(â) max aˆ i.j i N j N ρ(â) Â λ n max a i.j = λ n,est. i N j N Poisson λ n (3.24) λ n,est 1 1 λ n,est λ n λ n,est 4.4 GMRES ÂQ ˆN û = ˆb, u = Q ˆN û Q ˆN = I ZE 1 Y T Â + ωλ N,estZE 1 Y T Ê = Y T ÂZ ω λ N = ωλ N,est GMRES 1 4 x j := Q ˆN v j Q ˆN Q ˆN 4 c 2009 Information Processing Society of Japan

5 1. Choose u 0, ω, and λ n,est.compute Q N. 2. Compute r 0 = b Au 0,β = r 0 2,and v 1 = r 0 β. 3. For j = 1,..., k 4. x j := Q ˆN v j; 5. w := AM 1 x j. 6. For i = 1,..., j 7. h i,j = (w, v j); 8. w := w h i,jv i. 9. Endfor 10. h j+1,j := w 2 and v j+1 = w h j+1,j 11. Endfor 12. Set X k := [x 1... x k ] and Ĥk = {h i,j} 1 i j+1;1 j k 13. Compute y k = argmin y βe 1 Ĥky 2 and u k = u 0 + M 1 X k y k 1 GMRES x j = v j ZE 1 Y T Âv j + ωλ N,estZE 1 Y T v j = v j ZE 1 Y T (AM 1 ωλ N,estI)v j s = AM 1 v j ˆv = Y T (s ωλ n,estv j) x j x j = v j ZE 1ˆv Galerkin E 1ˆv ˆv = Êṽ ṽ ṽ t := Zṽ x j := s t Z : R r R n Y T : R n R r Z prolongation operater Y T restrictio operater) 2 1. Choose u 0, ω, and λ n,est. 2. Compute r 0 = b Au 0,β = r 0 2,and v 1 = r 0 β. 3. For j = 1,..., k 4. s := AM 1 v j. 5.Restriction:ˆv = Y T (s ωλ n,estv j) 6.Solve for ṽ:êṽ = ˆv 7.Prolongation:t := Zṽ 8.x j := v j t. 9. w := AM 1 x j. 10. For i = 1,..., j 11. h i,j = (w, v j); 12. w := w h i,jv i. 13. Endfor 14. h j+1,j := w 2 and v j+1 = w h j+1,j 15. Endfor 16. Set X k := [x 1... x k ] and Ĥk = {h i,j} 1 i j+1;1 j k 17. Compute y k = argmin y βe 1 Ĥky 2 and u k = u 0 + M 1 X k y k 2 GMRES 5. multilevel projection Krylov method 3 Galerkin ˆv = Êṽ A (2) := E = Y T A (1) Z Q (1) ˆN = I (1) Z (1,2) A (2) 1 Y (1,2)T A (1) + ω 1 λ (1) N,estZ (1,2) A (2) 1 Y (1,2)T level Galerkin A (2) Q N (2) p (2) = ˆv (2), ˆv (2) = Q N (2) p (2) 5 c 2009 Information Processing Society of Japan

6 Galerkin Krylov Galerkin 5.2 Multi-level Galerkin Multi-level Krylov (4,3) Galerkin Galerkin A (l+1) = Y (l,l+1)t A (l) Z (l,l+1) Q N Q (l) ˆN = I (l) Z (l,l+1) A (l+1) 1 Y (l,l+1)t A (l) + ω (l) λ (l) N,estZ (l,l+1) A (l+1) 1 Y (l,l+1)t l = m Galerkin A (m) Galerkin GMRES Galerkin Galerkin Galerkin Galerkin (Z Y ) ( ) Swopp09 1. l = 1. With an initial guess u 0 (1) = 0, Solve A (1) u (1) = b (1) with FGMRES by computing 2. s (1) := A (1) M (1) 1 v j (1) ; 3. restriction:ˆv (2) = Y (1,2)T (s (1) ω (1) λ (1) n,estv j (1) ); 4. if l + 1 = m 5. solve exactly Â(2) 1 ˆv (2) = ṽ (2) ; 6. else 7. l=l+1. With ṽ (2) 0 = 0, solve Â(2) ṽ (2) = ˆv (2) with FGMRES by computing 8. s (2) := Â(2) v (2) ; 9. restriction: ˆv (3) := Y (2,3)T (s (2) ω (2) λ (2) n,estv (2) ); 10. if l + 1 = m 11. solve exactly Â(3) 1 ˆv (3) = ṽ (3) ; 12. else 13. l = l + 1; endif 16. prolongation: t (2) := Z (2,3) ṽ (3) ; 17. x (2) = v (2) t (2) ; 18. w (2) = Â(2) x (2) ; 19. endif 20. prolongation: t (1) := Z (1,2) ṽ (2) ; 21. x (1) = v (1) t (1) ; 22. w (2) = A (1) M (1) 1 x (2) ; 3 multilevel projection Krylov method 6 c 2009 Information Processing Society of Japan

7 1) Y. A. Erlangga and R. Nabben, Multilevel projection-based nested Krylov iteration for boundary value problems, SIAM J.Sci.Comput.,pp , ) D. Day, An Efficient Implementation Of The Nonsymmetric Lanczos Algorithm, SIAM J.Matrix Anal.Appl.,18, pp , ) Y. Saad and M. H. Schultz, GMRES: A Generalized Minimal Residual Algorithm For Solving Nonsymmetric Linear Systems, SIAM J.Sci.Comput.,7,pp , ) R. A. Nicolaides, Deflation of conjugate gradients with applications to boundary value problems, SIAM J.Numer. Anal.,24,pp , ) R. B. Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J.Matrix Anal.Appl.,16,pp , ) J. Frank and C. Vuik, On the construction of deflation-based preconditioners, SIAM J.Sci.Comput.,23,pp , c 2009 Information Processing Society of Japan

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