211 9 12, GMRES,.,., Look-Back.,, Ax = b, A C n n, x, b C n (1),., Krylov., GMRES [5],.,., Look-Back [3]., 2 Krylov,. 3, Look-Back, 4. 5. 1
Algorith 1 The GMRES ethod 1: Choose the initial guess x and copute r = b Ax 2: Set β = r 2, v 1 = r /β 3: For j = 1, 2,..., k, Do: 4: Copute w j = Av j 5: For i = 1, 2,..., j, Do: 6: h i,j = (w j, v i ) 7: w j = w j h i,j v i 8: End For 9: h j+1,j = w j 2. If h j+1,j = then Set k := j and go to 12 1: v j+1 = w j /h j+1,j 11: End For 12: Define the (k + 1) k Hessenberg atrix H k = {h i,j } 1 i k+1,1 j k 13: x k = x + V k s k, where s k = arg in s C k βe 1 H k s 2 (1) Krylov, A x C n r := b Ax Krylov.,, Krylov. [4, 6]. GMRES Arnoldi Krylov, Algorith 1. GMRES, Arnoldi, 1.,,, GMRES., 3. Part 1. 1 x (1). Part 2. l, Ax = b x (l) GMRES ( ), x (l). Part 3. x (l), (l + 1) x (l+1), x (l+1) := x (l). Part 1 3 Algorith 2. 2
Algorith 2 The ethod 1: Choose the restart frequency and the initial guess x (1) 2: For l = 1, 2,..., until convergence Do: 3: Solve (approxiately) Ax = b by iterations of GMRES with the initial guess x (l), and get the approxiate solution x (l) 4: Update the initial guess x (l+1) := x (l) 5: End For Look-Back, 1 GMRES,., Part 1 Part 2., Part 2, Part 1., Part 3 [2]., x (l+1) x (l+1) := x (l) x (l), y (l+1) C n, x (l+1) := x (l) + y (l+1). [2].,, Look-Back, Look-Back [3]., Look-Back y (l+1) y (l+1) := µ (l) x (l), x (l) := { x (l) x (l k 2 ) x (l) k 1 (l 2 ) (k : ) x (k : )., k (k 2, k N), µ (l) C µ (l) = arg in µ C r(l) µa x (l) 2 = (r(l), A x (l) ) (A x (l), A x (l) )., Look-Back. Look-Back Algorith 3., Look-Back 1 Table 1., Mat-Vec, AXPY Inner-Product, 3
Algorith 3 A Look-Back ethod 1: Choose the restart frequency, the paraeter k 2 and the initial guess x (1) 2: For l = 1, 2,..., until convergence Do: 3: Solve (approxiately) Ax = b by iterations of GMRES with the initial guess x (l), and get the approxiate solution x (l) 4: Copute the vector y (l+1) as follows: If l = 1 then y (l+1) = If l 2 then If (l = k = 2) or (k : even, l k k 1 2 ) or (k : odd, l 2 ) then x (l) := x (l) x (1) Else x (l) := End If x (l) x (l k 2 ) x (l) k 1 (l 2 ) (k : even) x (k : odd) y (l+1) = µ (l) x (l), µ (l) = arg in µ C r (l) µa x (l) 2 End If 5: Update the initial guess x (l+1) := x (l) + y (l+1) 6: End For Table 1 The nuber of operations per restart cycle and storage requireents of and Look-Back. Method Look-Back k : even Mat-Vec + 1 ( + 1) + 1 k : odd AXPY ( 2 + 5 + 4)/2 ( 2 + 5 + 4)/2 + 3 Inner-Product ( 2 + 3 + 2)/2 ( 2 + 3 + 2)/2 + 2 Storage ( + 2)n ( + 2)n + k k+1 2 n ( + 2)n + 2 n 1,., Strage.,. + 1 [5], Look-Back + 2.,. 4
, The University of Florida Sparse Matrix Collection [1] 6, Look-Back (Algorith 3),. AMD Pheno II X4 94 (3.GHz) FORTRAN 77. Table 2 Characteristics of the coefficient atrices of the test probles for and Look- Back. Matrix (Type) n Nnz Ave.Nnz Application area CAVITY1 (R) 2597 76367 29.41 Coputational fluid dynaics KIM1 (C) 38415 933195 24.29 2D/3D proble LIGHT IN TISSUE (C) 29282 4684 13.87 Electroagnetics proble RAJAT3 (R) 762 32653 4.3 Circuit siulation WAVEGUIDE3D (C) 2136 33468 14.43 Electroagnetics proble XENON2 (R) 157464 3866688 24.56 Materials proble Table 2., (R) (C),,., n, Nnz, Ave.Nnz,, 1 (1 ). = 3, Look-Back k = 3., b = [1, 1,..., 1] T x (1) = [,,..., ] T, r k 2 / b 2 1 1. [ ] Table 3, Fig. 1., Table (1 ).,, 1 ( ) 3.,. (Iter), Look-Back,,., CAVITY1 RAJAT3,, Look-Back (Table 3 TRR )., Look-Back., Fig. 1 Look-Back. KIM1 WAVEGUIDE3D Look-Back, CAVITY1, LIGHT IN TISSUE, RAJAT3 5
Table 3 Convergence results (Iter : nuber of iterations, t Total : total coputation tie, t Restart : coputation tie per restart cycle, TRR : log 1 of explicitly coputed relative residual 2-nor) of and Look-Back for = 3. Matrix Method Iter Tie[sec.] TRR t Total t Restart CAVITY1 8.48 1 3-7.3 LB- 24811 7.28 1 8.8 1 3-1. KIM1 2824 3.72 1 1 3.96 1 1-1. LB- 3971 5.35 1 1 4.5 1 1-1. LIGHT IN TISSUE 2964 2.6 1 1 2.62 1 1-1. LB- 938 8.29 1 2.66 1 1-1. RAJAT3 1.43 1 2 -.55 LB- 14881 8.24 1 1.65 1 2-1.1 WAVEGUIDE3D 35268 2.36 1 2 2.3 1 1-1. LB- 29745 2.2 1 2 2.3 1 1-1. XENON2 15677 3.92 1 2 7.5 1 1-1. LB- 2371 6.5 1 1 7.65 1 1-1. XENON2, Look-Back, Look-Back., 1 ( ) (t Restart ). Look-Back, 1 1%., Look-Back 1 1 AXPY (Table 1 ). (t Total ). Look-Back,, (Iter),, 1 (t Restart ),, KIM1., Look-Back,., Look-Back 1, 6
log 1 of relative residual 2-nor (d) CAVITY1 Look-Back -1 2 4 6 8 1 Nuber of Iterations log 1 of relative residual 2-nor (a) KIM1 Look-Back -1 1 2 3 4 Nuber of Iterations log 1 of relative residual 2-nor (b) LIGHT IN TISSUE Look-Back -1 5 1 15 2 25 3 Nuber of Iterations log 1 of relative residual 2-nor (d) RAJAT3-1 Look-Back 2 4 6 8 1 Nuber of Iterations log 1 of relative residual 2-nor (d) WAVEGUIDE3D Look-Back -1 2 4 Nuber of Iterations log 1 of relative residual 2-nor (f) XENON2 Look-Back -1 5 1 15 2 Nuber of Iterations Fig.1 The relative residual 2-nor history of and Look-Back of = 3 without preconditioners for KIM1, LIGHT IN TISSUE, NS3DA, RAJAT3, RDB5 and XENON2.., Look-Back,,.,., AXPY,.. 7
[1] Davis, T. A., The University of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/research/sparse/atrices/. [2],,,,, 19(29), 551 564. [3],,, Look-Back, 21, (21), 25 26. [4] Saad, Y., Iterative ethods for sparse linear systes. 2nd edition, SIAM, Philadelphia, PA, 23. [5] Saad, Y. and Schultz, M. H., GMRES: A generalized inial residual algorith for solving nonsyetric linear systes, SIAM J. Sci. Stat. Coput., 7(1986), 856 869. [6],,,,, 29. 8