橡固有値セミナー2_棚橋改.PDF
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1 1 II
2 2... Arnoldi. Lanczos. Jacobi-Davidson
3 . 3
4 4 Ax = x A A Ax = Mx M: M
5 5 Householder ln ln-1 0 l3 0 l2 l1
6 6 Lanczos Lanczos, 1950 Arnoldi Arnoldi, 1951 Hessenberg Jacobi-Davidson Sleijpen & van der Vorst, 1996
7 7. Ritz Ritz
8 8 N Ax = x S m = span {v 1, v 2,, v m } {v i } m << N Ax = x x S m <w, Ax x > = 0 for w S m Ax x S m 0 x S m
9 9 Ritz Ritz V m = [v 1, v 2,, v m ] x = V m s <w, Ax x > = 0 for w S m V mt (AV m s V m s ) = 0 (V mt AV m ) s = s k (V mt AV m ) s = s x Ritz s Ritz
10 10 v 1 V 1 = [v 1 ] DO i = 2, m u u V i 1 v i V i = [V i 1 v i ] END DO (V mt AV m ) s = s s x = V m s A Lanczos Arnoldi JD
11 11 S m (1) v 1 Implicit Restart Arnoldi Thick Restart Lanczos (2) u Jacobi-Davidson
12 12. Arnoldi Arnoldi Arnoldi Implicit Restart Arnoldi
13 13 Arnoldi (1) v 1 S m = K m (A, v 1 ) = span {v 1, Av 1, A 2 v 1,, A m 1 v 1 } Krylov Krylov m
14 Arnoldi (2) 14 j v j A v 1, v 2,, v j v j+1 Arnoldi v j+1 Av j v 1, v 2,, v j+1 V m = [v 1, v 2,, v m ] AV m = V m H m + h m+1, m v m+1 e m t H m m m Hessenberg h m+1, m e m m m A m-step Arnoldi 0 Hessenberg
15 15 v 1 v 1 := v 1 / v 1 2 V 0 = DO j = 1, m V j = [V j 1 v j ] u= Av j A DO i = 1, j h ij = v it u u:= u h ij v i Modified Gram-Schmidt END DO h j+1, j = u 2 v j+1 = u / h j+1, j END DO H m = ( h ij ) m i, j = 1 H m s = s s x = V m s A
16 16 H m s = s x= V m s Ax x = AV m s V m H m s = h m+1, m v m+1 e mt s = h m+1, m e mt s E = h m+1, m (e mt s) v m+1 x t x ( A + E ) x = x A E h m+1, m e mt s
17 17 Arnoldi i v 1, v 2,, v i i i m O(m), O(m 2 )
18 18 m v m v 1 Arnoldi v 1, v 2,, v m 1
19 19 Implicit Restart Arnoldi (1) v m S m k k < m k v 1, v 2,, v m k m
20 Implicit Restart Arnoldi (2) 20 m-step Arnoldi AV m = V m H m + h m+1, m v m+1 e t m 1, 2,, m k H m m k QR V m+ = V m Q H m+ = Q t H m Q AV m+ = V m+ H m+ + h m+1, m v m+1 e mt Q Q = Q 1 Q 2 Q m 1 QR k k-step Arnoldi AV k+ = V k+ H k+ + h k+1, k v k+1+ e t k Arnoldi k
21 Implicit Restart Arnoldi (3) 21 QR h m+1, m v m+1 A V m + = V m + H m + + e m t Q m k+1 k m k H m + h m+1, m v m+1 e mt Q A V m + = V m + + k-step Arnoldi H k + h k+1, k v k+1+ e k t A V k + = V k + + k k
22 Implicit Restart Arnoldi (4) 22 QR 1, 2,, m k QR = (H m+ 1 ) (H m+ 2 ) (H m+ m k ) R = R m k R 2 R 1 R i QR V + m 1 v + 1 v 1 + = V m Q e 1 = V m (H m+ 1 ) (H m+ 2 ) (H m+ m k ) R 1 e 1 = r 11 1 (A 1 ) (A 2 ) (A m k ) v 1 implicit restart v 1 (A 1 ) (A 2 ) (A m k ) v 1
23 23 Implicit Restart Arnoldi (5) v 1
24 24 Arnoldi AV m = V m H m + h m+1, m v m+1 e t m DO l = 1, 2, 3, 1, 2,, m k H m m k QR Q = Q 1 Q 2 Q m 1 V m+ = V m Q H m+ = Q t H m Q k = (H m+ ) k+1, k k = Q m, k v k+1+ = v k+1 k + h m+1, m v m+1 k h k+1, k = v k+1+ 2 v k+1+ := v k+1+ / h k+1, k V + m k V k+ H + m k k H k+ Arnoldi AV k+ = V k+ H k+ + h k+1, k v k+1+ e t k Arnoldi m k END DO
25 (1) 25 Ax = Mx M M I M = LL t Cholesky (L 1 AL t )(L t x) = (L t x) fill-in L M M
26 (2) 26 II Ax = Mx (A M) x = ( ) Mx (A M) 1 M x = x = 1 / ( ) M (A M) 1
27 (3) 27 III M- Arnoldi v t u 2 M- v t Mu M- u t Ax = Mx M- II
28 ( ) Arnoldi 28 SR8000(F1) 2 2 Arnoldi vs. IRAM E( r) + k0εre( r) = log( ε (ARPACK) Ex E( r) = exp( iβz) E y 2µm 12µm [ ) E( r) ] r
29 ( ) Arnoldi 29 Ax x / < N = 13112, NE = 30, 100 IRAM Arnoldi 339G 26.1G IRAM 40.6G G Krylov N=13112 N=51482 NE=30 Arnoldi 242G 382G IRAM 77.0G N=13112 N=51482 NE=100
30 ( ) Arnoldi (1) 30 n=51482 θ 1, θ 30 IRAM 1.0E E E E E E E q 1 Step=100 q 30 Arnoldi IRAM Step=60 (Gflop) 1.0E-14
31 ( ) Arnoldi 31 N=51482 θ 100 IRAM Arnoldi 1.0E E E E E E E q 100 IRAM Step=160 Arnoldi Step=500 (Gflop) 1.0E-14
32 32. Lanczos Lanczos Lanczos Lanczos Thick Restart Lanczos
33 Lanczos (1) 33 Arnoldi S m Krylov S m = K m (A, v 1 ) = span {v 1, Av 1, A 2 v 1,, A m 1 v 1 } j v j A v 1, v 2,, v j v j+1 j < m 1 (Av m ) t v j = v mt (Av j ) = v mt i=1 j+1 c i v i = i=1 j+1 c i v mt v i = 0 Av m v m 1 v m
34 Lanczos (2) 34 Lanczos Lanczos Arnoldi Arnoldi AV m = V m H m + m v m+1 e t m V mt V mt AV m = H m H m Hessenberg T m AV m = V m T m + m v m+1 e m t A m-step Lanczos
35 v 1 0 = 0, v 0 = 0, v 1 := v 1 / v 1 2 V 0 = DO j = 1, m V j = [V j 1 v j ] u= Av j j 1 v i 1 A j = v jt u u:= u j v j j = u 2 v j+1 = u / j END DO 35 T m = m-1 m-1 m T m s = s s x = V m s A
36 36 T m s = s x= V m s Ax x = AV m s V m T m s = m+1 v m+1 e mt s = m+1 e mt s E = m+1 (e mt s) v m+1 x t x ( A + E ) x = x A E i < E 2 A i m+1 e mt s
37 37 A 1 > 2 > > N v 1 {x i } N i=1 v 1 = i=1n c i x i m Lanczos 1 (m) (m) 1 ( 1 N )( i=2n c i 2 ) / c ( (1+ ) + ) 4(m 1) = ( 1 2 ) / ( 1 N )
38 Lanczos I 38 Av j v j 1 v j v j 2 Lanczos
39 39 Selective orthogonalization (Parlett & Scott, 1979) Ritz Av j Ritz Partial orthogonalization (Simon, 1984) ij = v it v j ij
40 Lanczos II 40 Krylov v 1 = i=1n c i x i A j 1 v 1 = i=1n j 1 i c i x i i = i x i x i c i c i m
41 41 Lanczos (Cullum and Donath, 1974) p p Lanczos p p p (Underwood, 1975)
42 Lanczos V 1 0 = 0, V 0 = 0 V 0 = p DO J = 1, M p p V J = [V J 1 V J ] U = AV J V J 1 J 1 A J = V Jt U U := U V J J V J+1 J = U U QR END DO T M = m-1 m-1 m T M s = s s x = V M s A
43 43 Thick Restart Lanczos (1) Lanczos m m m Ritz k k < m Lanczos k m
44 Thick Restart Lanczos (2) 44 m-step Lanczos AV m = V m T m + m v m+1 e t m T m k Y AV m Y = V m T m Y + m v m+1 e mt Y AV m Y = V m Y T k+ + m v m+1 e mt Y AV k+ = V k+ T k+ + m v k+1+ s t --- V m+ = V m Y s= Y t e m T m+ : Y k k k-step Lanczos e mt Lanczos v k+2 v k+3, k
45 45 Thick Restart Lanczos (3) v k+2 Av k+1 v 1, v 2,, v k+1 + k+1 v k+2 = ( I V k+1 V k+1t ) Av k+1 = ( I v k+1 v k+1t V k V kt ) Av k+1 = ( I v k+1 v k+1t ) Av k+1 V k m s V k Av k+1 = m s
46 Thick Restart Lanczos (4) 46 v k+i+1 i 2 Av k+i v 1, v 2,, v k+i k+i v k+i+1 = ( I V k+i V k+it ) Av k+i = ( I v k+i 1 v k+i 1t v k+i v k+it V k+i 2 V k+i 2t ) Av k+i = ( I v k+i 1 v k+i 1t v k+i v k+i t ) Av k+i V k+i 2 (AV k+i 2 ) t v k+i = ( I v k+i 1 v k+i 1t v k+i v k+i t ) Av k+i AV k+i 2 v 1, v 2,, v k+i 1 v k+i Av k+i v k+i 1, v k+i Lanczos
47 47 Thick Restart Lanczos (5) Thick Restart Lanczos j k+1 k+1 AV j = V j T j + j v j+1 e j t k+1 T j j j T j k-step Lanczos m T m
48 ( ) Lanczos 48 EP8000 /690Turbo (Power4 1.3GHz) Matrix Market (Harwell-Boeing) Lanczos vs. TR-Lanczos (bcsstk21,bcsstk39) Matrix n nz bcsstk bcsstk bcsstk bcsstk bcsstk bcsstk bcsstk bcsstk bcsstk bcsstk bcsstk crystk Gap Gap=(λ 1 -λ 100 )/λ 1
49 ( ) Lanczos 49 Ax x / <10-10 NE=30,100 TR: simple TR bcsstk16 bcsstk17 bcsstk18 bcsstk21 bcsstk23 bcsstk24 bcsstk25 bcsstk35 bcsstk36 bcsstk37 bcsstk39 crystk02 bcsstk16 bcsstk17 bcsstk18 bcsstk21 bcsstk23 bcsstk24 bcsstk25 bcsstk35 bcsstk36 bcsstk37 bcsstk39 crystk02 NE=30 NE=100
50 50 ( ) Lanczos NE=30 NE=100 TR (EP8000) bcsstk16 bcsstk17 bcsstk18 bcsstk21 bcsstk23 bcsstk24 bcsstk25 bcsstk35 bcsstk36 bcsstk37 bcsstk39 crystk02 [Mflop] [sec] simple TR simple TR bcsstk16 bcsstk17 bcsstk18 bcsstk21 bcsstk23 bcsstk24 bcsstk25 bcsstk35 bcsstk36 bcsstk37 bcsstk39 crystk02 [Mflop] [sec] simple TR simple TR
51 ( ) Lanczos (1) bcsstk39 28,29, E E E E E E E E E E Conv. check Simple: 100 TR : 90 TR simple θ 30 θ 29 θ 28
52 1.0E E E E E E E E E-16 ( ) Lanczos (2) bcsstk21 TR-Lanczos Conv. check Simple: 20 TR : 70 TR simple θ 30 θ 29 θ 28 52
53 53. Jacobi-Davidson Jacobi-Davidson Jacobi-Davidson
54 Jacobi-Davidson 54 Lanczos Arnoldi (A M) 1
55 Jacobi-Davidson (1) 55 S m {v 1, v 2,, v m } V m = [v 1, v 2,, v m ] S m A Ritz Ritz ( j, u j = V m s j ) ( j = 1,, m) u j u j t A ( u j + t ) = ( u j + t ) (A I ) t = (A I ) u j Jacobi-Davidson u j t S m
56 Jacobi-Davidson (2) 56 (A I ) u j ( I u j u jt )(A I )( I u j u jt ) t ( I u j u jt )(A I )( I u j u jt ) t = (A I ) u j j ( I u j u jt )(A j I )( I u j u jt ) t = (A j I ) u j JD t {v 1, v 2,, v m } v m+1
57 57 Jacobi-Davidson (3) u j u j t 0 = 0 GMRES CGS Krylov u j GMRES K u j
58 58 t = v 0 DO m = 1, 2, DO i = 1, 2,, m 1 t:= t (t t v i ) v i Modified Gram-Schmidt END DO v m = t / t 2, v ma = Av m DO i = 1, 2,, m M i,m = v it v A m END DO m m M s u= Vs V = [v 1, v 2,, v m ] u A = V A s V A = A[v 1, v 2,, v m ] r= u A u r 2 = x= u ( I u u t )(A I )( I u u t ) t = r t END DO
59 59 JD JD S m Krylov Ritz Ritz
60 60 x 1,, x l A I ( I X l X lt )(A I )( I X l X lt ) X l = [x 1,, x l ] X l
61 JD N=1000, A ii =i, A i-1,i =0.5, A 1000,1 =0.5 Ax x < Arnoldi JD with GMRES(5) Residual norm Residual norm Iteration number Iteration number cf) Sleijpen, Jacobi-Davidson algorithms for various eigenproblems(1999),p.15 Fig.4-1
62 62 /5 Z. Bai, J. Demmel, J. Dongarra, A.Ruhe and H. Van der Vorst (eds.): Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, F. Chatlin: Valeurs Propres de Matrices, Masson, Paris, W. Ledermann Eigenvalues of Matrices, John-Wiley and Sons, Chichester, J. W. Demmel: Numerical Linear Algebra, SIAM, Philadelphia, G. H. Golub and C. F. van Loan: Matrix Computations, 3rd edition, The Johns Hopkins University Press, B. N. Parlett: The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, Y. Saad: Numerical Methods for Large Eigenvalue Problems, Halsted Press, New York, J. H. Wilkinson: The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965.
63 Lanczos (1) (2/5) J. Cullum and W. E. Donath: A Block Lanczos Algorithm for Computing the q Algebraically Largest Eigenvalues and a Corresponding Eigenspace of Large Sparse Real Symmetric Matrices, Proc. of the 1974 IEEE Conference on Decision and Control, Phoenix, Arizona, pp (1974). J. Cullum and R. A. Willoughby: Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume 1, Theory, Birkhauser, Boston, R. G. Grimes, J. G. Lewis and H. D. Simon: A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems, SIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 1, pp (1994). C. Lanczos: An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, J. Research of the National Bureau of Standards, Vol. 45, No. 4, pp (1950).,, :,,
64 Lanczos (2) (3/5) B. N. Parlett and D. Scott: The Lanczos Algorithm with Selective Orthogonalization, Mathematics of Computation, Vol. 33, pp (1979). H. D. Simon: The Lanczos Algorithm with Partial Reorthogonalization, Mathematics of Computation, Vol. 42, pp (1984). R. Underwood: An Iterative Block Lanczos Method for the Solution of Large Sparse Symmetric Eigenproblems, Report STAN-CS , Department of Computer Science, Stanford University, Stanford, California (1975). K. Wu and H. Simon: Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems, SIAM Journal on Matrix Analysis and Applications, Vol. 22, No. 2, pp (2000). 64
65 65 Arnoldi (4/5) W. E. Arnoldi: The Principle of Minimized Iterations in the Solution of the Matrix Eigenvalue Problem, Quart. J. Applied Mathematics, Vol. 9, pp (1951). R. B. Lehoucq and D. C. Sorensen: Deflation Techniques for an Implicitly Restarted Arnoldi Iteration, SIAM Journal on Matrix Analysis and Applications, Vol. 17, No. 4, pp (1996). R. B. Lehoucq, K.J. Maschhoff.: implementation of an implicitly restarted block Arnoldi method, Preprint MCS-P , Argonne National Laboratory,IL,1997. R. B. Lehoucq, D. C. Sorensen and C. Yang: ARPACK User s Guide, SIAM, Philadelphia, D. C. Sorensen: Implicit Application of Polynomial Filters in a k-step Arnoldi Method, SIAM Journal on Matrix Analysis and Applications, Vol. 13, No. 1, pp (1992).
66 66 Jacobi-Davidson (5/5) D. R. Fokkema, G. L. G. Sleijpen and H. A. van der Vorst: Jacobi-Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils, SIAM Journal on Scientific Computing, Vol. 20, pp (1999). M.Genseberger, G. L. G. Sleijpen: Alternative correction equations in the Jacobi- Davidson method, Preprint No. 1073,1999. M. Nool and A. van der Ploeg: A Parallel Jacobi-Davidson-Type Method for Solving Large Generalized Eigenvalue Problems in Magnetohydrodynamics, SIAM Journal on Scientific Computing, Vol. 22, No. 1, pp (2000). Y. Saad and M. H. Schultz: GMRES: A Generalized Minimum Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM Journal on Scientific and Statistical Computing, Vol. 7, pp (1986). G. L. G. Sleijpen and H. A. van der Vorst: A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems, SIAM Journal on Matrix Analysis and Applications, Vol. 17, pp (1996). G. L.G. Sleijpen, H. A. Van der Vorst, Z. Bai : Jacobi-Davidson algorithms for various Eigenproblems A working document-, Preprint nr.1114,1999.
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