Krylov A04 October 8, 2010 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
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1 Krylov A04 October 8, 2010 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
2 Krylov QCD, RSDFT, Shell model Block Krylov MATLAB Scilab T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
3 Krylov Krylov n A n v K k (A, v) span(v, Av, A 2 v,..., A k 1 v). Krylov Krylov Alexei Krylov (Wikipedia ) Krylov T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
4 Krylov Krylov 1 Ax = b x 0 2 r 0 = b Ax 0 3 k = 1, 2,... x k x 0 K k (A, r 0 ) x k ρ (k) j x k = x 0 + k 1 j=0 ρ (k) j A j r 0 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
5 Krylov x k r k = b Ax k r k = I k 1 ρ (k) j A j+1 j=0 r 0 = R k (A)r 0 K k+1 (A, r 0 ) R k (A) A k r k Krylov r k x k T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
6 Krylov Krylov 4 1 Ritz-Galerkin b Ax k K k (A, r 0 ) x k 2 b Ax k 2 K k (A, r 0 ) x k 3 Petrov-Galerkin b Ax k k x k 4 x k x 0 2 x k A T K k (A T, r 0 ) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
7 (CG) Krylov 1952 Hestenes, Stiefel x e = x x f(x) f(x) = 1 2 (e, Ae) = 1 2 (x, Ax) (x, b) (x, b) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
8 (CG) p k k + 1 x k+1 x k+1 = x k + α k p k α k f(x k+1 ) x k+1 r k+1 r k+1 = r k α k Ap k. p k+1 p k+1 = r k+1 + β k p k p k+1 (p i, Ap j ) = 0, i j β k+1 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
9 Krylov A A + σi Krylov K k (A + σi, v) = K k (A, v). A Krylov (A + σi)x σ = b x σ T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
10 CG r 0 = b Ax 0 r k = R k (A)r 0 K k+1 (A, r 0 ) R k (z) Lanczos Lanczos R k+1 (z) R 0 (z) = 1, P 0 (z) = 1 R k+1 (z) = R k (z) α k zp k (z) P k+1 (z) = R k+1 (z) + β k P k (z) P k (z) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
11 CG Ax = b x 0 = 0 r 0 = b r k = R k (A)b K k+1 (A, b). (A + σi)x σ = b x σ 0 = 0 rσ 0 = b r σ k = Rσ k (A + σi)b K k+1(a + σi, b). Krylov K k (A, b) = K k (A + σi, b) R σ k (z + σ) = ξσ k R k(z). T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
12 CG Lanczos R σ k (z + σ) = ξσ k R k(z) ξ σ k+1 = ξ σ k ξσ k 1 α k 1 (ξ σ k 1 ξσ k )α kβ k 1 + ξ σ k 1 α k 1(1 + σα k ) α σ k = ξσ k+1 α ξk σ k β σ k = ( ξ σ k+1 ξ σ k ) 2 β k r σ k = ξσ k r k T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
13 SS(Sakurai-Sugiura) N ( ) zj γ k+1 (z j I A) 1 ˆF k = 1 N j=1 ρ span{ŝ 0, ŝ 0,, ŝ m 1 } Sakurai, et.al. (2003) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
14 ŝ k = 1 N N j=1 ( ) zj γ k w j (z j I A) 1 v ρ Krylov N (z 1 I A)y 1 = v (z 2 I A)y 2 = v. (z N I A)y N = v 1/N T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
15 A R n n A λ 1,..., λ n x 1,..., x n A n A = λ j P j, j=1 P j = x j x T j. P j λ j T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
16 (zi A) 1 re(a) (zi A) 1, z re(a) A A n A = λ j P j j=1 n (zi A) 1 P j = z λ j j=1 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
17 Γ α Jordan f(z) Γ f(α) = 1 f(z) 2πi Γ z α dz. r(z) Γ α 1, α 2,..., α n Γ r(z)dz = 2πi n Res z=αj r(z) j=1 Res z=αj r(z) z = α j r(z) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
18 α 1,..., α n R ν 1,..., ν n R 0 1 2πi Γ r(z)dz = r(z) = n j=1 n j=1 1 2πi ν j z α j Γ ν j z α j dz = n j=1 ν j 1 2πi Γ z k r(z)dz = n j=1 1 2πi Γ ν j z k z α j dz = n ν j α k j. j=1 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
19 Γ A Jordan 1 (zi A) 1 dz = 2πi Γ n P j. j=1 f(z) f(a) = 1 2πi Γ f(z)(zi A) 1 dz. v R n A k v = 1 z k (zi A) 1 dz v. 2πi Γ T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
20 Γ Jordan λ 1,..., λ m Γ (zi A) 1 P Γ (A) = 1 (zi A) 1 dz 2πi Γ m P Γ (A) = j=1 P j P Γ (A) x 1,..., x m T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
21 P Γ (A) v s = P Γ (A)v s = 1 (zi A) 1 dz v = 2πi Γ m P j v. j=1 v F (x) = 1 1 2πi z x dz Γ T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
22 Γ γ ρ z j = γ + ρ e 2πi N (j+1/2), j = 0, 1,..., N 1 ŝ = 1 N N 1 j=0 e 2πi N (j+1/2) (z j I A) 1 v ˆF (x) = 1 N N 1 j=0 e 2πi N (j+1/2) z j x T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
23 QCD QCD all-to-all propagator A = γ 5 M Low mode part: (λ j, v j ) High mode part: A 1 High = 1 m A 1 Low = n j=1 1 m φ j 1 j=1 λ j v j v j n v j v j η j j=1 A 1 = A 1 Low + A 1 High T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
24 QCD O(a) κ = (497, , 380, 864) 20 shifted CG (Ohno, et.al. accepted) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
25 QCD T2K-Tsukuba 1 node SS ( ) SS ( ) PARPACK [sec] [sec] ( ) ( ) Arnoldi SS QCD Implicitly restarted Arnoldi T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
26 48 Cr in the model space consisting of single-particle orbits f 7/2, p 3/2, f 5/2, p 1/2. M-scheme dimension for M = 0 : : double-lanczos : SS with Shifted COCG (Mizusaki, et.al. 2010) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
27 Arnoldi SS Jordan Γ µ Γ = 1 2πi Γ tr((zb A) 1 B)dz. trace tr((zb A) 1 ) 1 s s v T j (z jb A) 1 v j, j=1 v j 1 1 n v j T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
28 (RSDFT) Si ,248, 2000 (Futamura, et.al. (submitted)) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
29 T2K cores time[sec] speedup Linear Speedup NVIDIA Tesla C1060( ) GPUs time[sec] speedup GPU GPU 8 GPU Speedup T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
30 Matrix Market: olm100 ( et.al (2010)) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
31 Block Krylov B n L AX = B Block BiCG O Leary(1980) Block GMRES Vital(1990) Block QMR Freund(1997) Block BiCGSTAB Guennouni(2003) 1 1 Block Krylov K k (A, R 0 ) = (R 0, AR 0, A 2 R 0,..., A k 1 R 0 ), R 0 = B AX 0 X 0 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
32 Block Krylov 1 R k X k B AX k R k T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
33 BiCGGR Tadano, et.al (2009) Block Krylov T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
34 Block Krylov QCD , 572, , 216, 064 Block BiCGSTAB Block BiCGGR T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
35 (NEPs) A 0, A 1,..., A p C n n, A p O p T (z) = z i A i, z C p i=0 T (z) T (λ)x = 0 λ x 0 p = 1 A 1 = I T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
36 (NEPs) T (z) Γ T (z) T (λ)x = 0 λ x 0 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
37 : T (z) = zi + A 0 + A 1 e τ z E. Jarlebring, 2008 : T (z) = K z 2 M + i l j=1 z 2 σ 2 j W j Acoustic field computations: Rich Lee, et. al., 2008 T (z) = K(z) + zd + z 2 M, K(z) J. Mehrmann, 2008 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
38 : f(x, y, z) = 0, g(x, y, z) = 0, h(x, y, z) = 0. Dixon δ 1 f(x, y, z) g(x, y, z) h(x, y, z) δ = f(α, y, z) g(α, y, z) h(α, y, z) (x α)(y β) f(α, β, z) g(α, β, z) h(α, β, z) 1 x = [1, α,, α n 1 β n 1 ] T (z). x n 1 y n 1 T (z)u = 0 Sakurai, et. al, 2008 T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
39 m n n m 3 T (z) = A 0 + za 1 + z 2 A 2 + z 3 A 3 C T = O O A 0 I O A 1 O I A 2 T (z) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
40 : T (z) = K z 2 M + i tx j=1 q z 2 σ 2 j W j, t = 1, σ 1 = 0. : 1,100,242 1,100,242 : T2K-Tsukuba System Quad-core AMD Opteron 2.3GHz (Barcelona) 4 sockets, 32GB of memory, 10,368 cores (95 TFlops), Infiniband Fat-tree Network Compiler : Intel C Library : Intel MKL PARDISO in MKL (shared memory) SuperLU DIST(distributed memory) Rich Lee, Stanford Linear Accelerator Center T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
41 Bottom Level PARDISO PARDISO 1 sec. Solution time in seconds sec Number of cores SS Top Level, Second Level MPI T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
42 4 (Γ 1 Γ 4 ) PARDISO 8 ( = 1024 ) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
43 (Linear Solver) z shift-invert T (z) (SuperLU DIST) LU LU factorization (sec) T2K Franklin #procs T2K 1 (16 ) Franklin T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
44 (GEP) (GEP) 40 SS Lanczos (TRLan) SS sec Franklin(LBNL) TRLan 100 SS #procs 128 (TRLan) (SS method) T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
45 (NEP) (NEP) SS 40 sec 5000 T2K Franklin(LBNL) T2K 100 Franklin #procs Franklin 1 2 SS T2K Franklin T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
46 I Krylov,,,, Shifted Linear Systems Krylov,, Vol. 14, No. 3, pp (2004)., (2010), ISBN T. Mizusaki, K. Kaneko, M. Honma, and T. Sakurai, Filter diagonalization of shell-model calculations, Phys. Rev. C 82, (2010). [10 pages] H. Ohno, Y. Kuramashi, H. Tadano, T. Sakurai, A quadrature-based eigensolver with a Krylov subspace method for shifted linear systems for Hermitian eigenproblems in lattice QCD, JSIAM Letters (accepted). Y. Futamura, H. Tadano, T. Sakurai, Parallel stochastic estimation method of eigenvalue distribution, JSIAM Letters, submitted. Block Krylov H. Tadano, T. Sakurai, On single precision preconditioners for Krylov subspace iterative methods, Lecture Notes in Computer Science, No. 4818, pp (2007). H. Tadano, T. Sakurai and Y. Kuramashi, Block BiCGGR: A new block Krylov subspace method for computing high accuracy solutions, JSIAM Letters, Vol.1, pp (2009). T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
47 II,, Block Krylov,, Vol. 2 No. 2, pp (2009). T. Sakurai, H. Tadano and Y. Kuramashi, Application of block Krylov subspace algorithms to the Wilson-Dirac equation with multiple right-hand sides in lattice QCD, Comput. Phys. Commun., Vol. 181, pp (2010). H. Tadano, Y. Kuramashi and T. Sakurai, Application of preconditioned block BiCGGR to the Wilson-Dirac equation with multiple right-hand sides in lattice QCD, Comput. Phys. Commun., Vol. 181, pp (2010). SS T. Sakurai and H. Sugiura, A projection method for generalized eigenvalue problems, J. Comput. Appl. Math. Vol. 159, pp (2003). J. Asakura, T. Sakurai, H. Tadano, T. Ikegami and K. Kimura, A numerical method for nonlinear eigenvalue problems using contour integrals, JSIAM Letters, Vol.1, pp (2009). T. Sakurai, J. Asakura, H. Tadano and T. Ikegami, Error analysis for a matrix pencil of Hankel matrices with perturbed complex moments, JSIAM Letters, Vol. 1, pp (2009). T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
48 III T. Sakurai, J. Asakura, H. Tadano, T. Ikegami and K. Kimura, A method for finding zeros of polynomial equations using a contour integral based eigensolver, Proc. Symbolic Numeric Computations 2009, Kyoto, pp (2009). T. Ikegami, T. Sakurai and U. Nagashima, A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method, J. Comput, Appl. Math., Vol. 233, (2010). J. Asakura, T. Sakurai, H. Tadano, T. Ikegami and K. Kimura, A numerical method for polynomial eigenvalue problems using contour integral, Japan J. Indust. Appl. Math., Vol. 27, pp (2010). I. Yamazaki, M. Okada, H. Tadano, T. Sakurai and K. Teranishi, A block sparse approximate inverse with cutoff preconditioner for semi-sparse linear systems derived from Molecular Orbital calculations, JSIAM Letters, Vol. 2, pp (2010).,,,,, Cutoff 2,, Vol. 18, No. 4, pp (2008).,,,,, Vol. 17, No. 3, pp (2007). T. Sakurai (Univ. Tsukuba) Krylov October 8, / 48
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