IDRstab(s, L) GBiCGSTAB(s, L) 2. AC-GBiCGSTAB(s, L) Ax = b (1) A R n n x R n b R n 2.1 IDR s L r k+1 r k+1 = b Ax k+1 IDR(s) r k+1 = (I ω k A)(r k dr
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1 1 2 IDR(s) GBiCGSTAB(s, L) IDR(s) IDRstab(s, L) GBiCGSTAB(s, L) Verification of effectiveness of Auto-Correction technique applied to preconditioned iterative methods Keiichi Murakami 1 Seiji Fujino 2 Abstract: Auto-Correction technique has been proposed by Sakurai et al. in order to avoid spurious convergence. Tukada et al. expanded the AC technique, and applied it to their GBiCGSTAB(s, L) method. In general, iterative methods are used together with preconditioning. Therefore, in this paper, we apply the AC technique to preconditioned iterative methods, and verify effectiveness of the AC technique. Keywords: Auto-Cerrection technique, IDR(s) method, IDRstab(s, L) method, GBiCGSTAB(s, L) method, preconditioning Sonneveld IDR(s) Induced Dimension Reduction [15] Sleijpen IDR(s) BiCGstab(l) [12] IDRstab(s, L) [13][14] IDRstab(s, L) GBiCGSTAB(s, L) [16] IDR(s) Bi-CGSTAB IDR(s) 1 Graduate School of Information Science and Electrical Engineering, Kyushu University 2 Research Institute for Information Technology, Kyushu University r k = b Ax k [11] IDR(s) GBiCGSTAB(s, L) [17] ILU(0) SSOR [2][3][4] IDR(s) 1
2 IDRstab(s, L) GBiCGSTAB(s, L) 2. AC-GBiCGSTAB(s, L) Ax = b (1) A R n n x R n b R n 2.1 IDR s L r k+1 r k+1 = b Ax k+1 IDR(s) r k+1 = (I ω k A)(r k dr k c). (2) ω k 0 c s dr k dr k = r k+1 r k dr k := (dr k 1,, dr k s ) dr k dr k = Adx k IDR I k = dr k b Range(c). (3) [11]3 Range(c) Range(c) = max 1 i s c(i) min 1 j s c(j). (4) I k I th dr k IDR(s) Set I k if I k < I th then Compute dr k recurcively else Compute dr k = Adx k end if r k+1 = r k + dr k GBiCGSTAB(s, L) I k [17] GBiCGSTAB(s, L) r k+1 = r k L k=1 γ k+1 U j k,1 U (j) k,1 αj k [rl k,1,, r L k,l]γ k+1 (5) γ k+1 = argmin γ r 0 [r 1,, r L ]γ, U j k,1 = AU j k (6) GBiCGSTAB(s, L) I k I k = r k r 0 max ( j Range(αj ) ) Range(γ) (7) 2.2 Eisenstat trick SSOR ILU SSOR [2] SSOR Eisenstat trick [6] [3][4][8][9] SSOR A = L A + D + U A K K = (L A + D/ω)(D/ω) 1 (U A + D/ω) (8) L A D U A ω SSOR Eisenstat trick [6] Ã = ((U A + D/ω) 1 + (L A + D/ω) 1 (I + (1 2/ω)D(U A + D/ω) 1 ))(D/ω) (9) Ãv [3][4] 1. y = (U A + D/ω) 1 (D/ω)v 2. z = (D/ω)v + (1 2/ω)Dy 3. w = (L A + D/ω) 1 z 4. Ãv = y + w Eisenstat trick SSOR 2
3 E-SSOR E-SSOR AC-GBiCGSTAB(s, L) E-SSOR AC-GBiCGSTAB(s, L) 1. Let x 0 be an initial guess, r 0 = b Ax 0, set R 0 R n s 2. Compute x 0 = (D/ω) 1 (U A + D/ω)x 0, r 0 = (L A + D/ω) 1 r 0 3. Set U 0, Compute U p, (p = 1) 4. Solve Mα = m 5. r 0 = r 0 U 1 α, x 0 = x 0 + U 0 α 6. while r k / r 0 ϵ do 7. {BiCG PART} 8. for j = 1... L do 9. if (k = 0) (j = 1) then 10. M 0 = R T 0 U 1, m 0 = R T 0 r Go to line end if 13. for i = 1... s do 14. if i = 1 then 15. Solve Mβ = m 16. U p e 1 = r p U p β, (p = 0, 1,, j 1) 17. else 18. Solve [ m, M[1 : i 2], M[i : s] ] β = Me i U p e i = U p+1 e i 1 [ r p, U p+1 [1 : i 2], U p [i : s] ] β 20. end if (p = 0, 1,, j 1) 21. y = (U A + D/ω) 1 (D/ω)e i 22. z = (D/ω)Ue i + (1 2 ω )Dy 23. w = (L A + D/ω) 1 z 24. Ue 1 = y + w 25. U p e i = U pe i, (p = 0, 1,, j) U j e i 26. Me i = R T 0 Ue i 27. end do 28. Solve Mα = m 29. r p = r p U p+1 α, (p = 0, 1,, j 1) 30. y = (U A + D/ω) 1 (D/ω) r j z = (D/ω) r j 1 + (1 2 ω )Dy 32. w = (L A + D/ω) 1 z 33. r j = y + w 34. end do 35. {MR PART} 36. γ = argmin γ r 0 [ r 1,, r L ]γ 37. I k = r k r 0 max ( j Range(αj ) ) Range(γ) 38. dx k = L U k α j + [ r 0,, r L 1 ]γ, x k+1 = x k + dx k j=1 39. if I k > θ then 40. y = (U A + D/ω) 1 (D/ω)dx k 41. z = (D/ω)dx k + (1 2 ω )Dy 42. w = (L A + D/ω) 1 z 43. r k+1 = r k (y + w) 44. else 45. r k+1 = r k [ r 1,, r L ]γ 46. end if 47. U k+1 = U k [U 1,, U L ]γ 48. M k+1 = γm k, m k = R T 0 r k, k = k Compute r k = (L A + D/ω) r k 50. end while 51. Compute x k = (U A + D/ω) 1 (D/ω) x k ( 1 ) ( 2 ) Dell PowerEdge R210II CPU: Intel Xeon E : 3.1GHz : 8Gbytes OS: Scientific Linux 6.0 ( 3 ) Intel Fortran Compiler version fast ( 4 ) cputime ( 1 ) 2 : r k+1 2 / r ( 2 ) x 0 0 ( 3 ) 1.0 ( 4 ) ( 5 ) s L ( 6 ) SSOR ILU(0) ω [5] 6. 2 dc3 TRRTrue Relative Residual b Ax k+1 / b NaN 3
4 1 Table 1 Characteristics of test matrices. air-cfl5 1,536,000 19,435, airfoil 2d 14, , poisson3db 85,623 2,374, sherman5 3,312 20, raefsky2 3, , watt 2 1,856 11, OK01 54,903 3,990, comsol 1,500 97, sme3dc 42,930 3,148, add20 2,395 17, dc3 116, , memplus 17, , epb3 84, , k3plates 11, , TRR -8 2 dc3 : (a) IDRstab(s, L) Table 2 Convergence rate of preconditioned iterative methods for matrix dc3: (a) non-preconditioned IDRstab(s, L) method , , , , , , , , , , , , , , , , , , , , , , , , , , , (b) ILU(0) IDRstab(s, L) (b) ILU(0) preconditioned IDRstab(s, L) method , , TRR 3 IDRstab(s, L) ( 1 ) AC TRR AC 7 ( 2 ) AC max ( 3 ) SSOR AC TRR -8 1 ( 4 ) AC TRR ILU(0) 136 SSOR 118 GBiCGSTAB(s, L) ( 1 ) AC TRR AC 31 ( 2 ) max AC AC ( 3 ) SSOR AC TRR -8 0 ( 4 ) AC TRR ILU(0) 86 SSOR 88 4
5 (c) E-SSOR IDRstab(s, L) (c) E-SSOR preconditioned IDRstab(s, L) method , , (d) GBiCGSTAB(s, L) (d) non-preconditioned GBiCGSTAB(s, L) method. [s] [s] 1 1 5, , , , , , , , , , , , , , , , , , , , , IDRstab(s, L) GBiCGSTAB(s, L) ( 1 ) AC IDRstab(s, L) TRR GBiCGSTAB(s, L) 103 ( 2 ) AC IDRstab(s, L) max 35 GBiCGSTAB(s, L) 63 ( 3 ) AC IDRstab(s, L) TRR GBiCGSTAB(s, L) 188 ( 4 ) AC SSOR IDRstab(s, L) TRR GBiCGSTAB(s, L) TRR 1 GBiCGSTAB(s, L) IDRstab(s, L) AC GBiCGSTAB(s, L) IDRstab(s, L) TRR IDRstab(s, L) GBiCGSTAB(s, L) (a) IDRstab(s, L) (b) GBiCGSTAB(s, L) 1 TRR Fig. 1 TRR distribution of preconditioned iterative method
6 (e) ILU(0) GBiCGSTAB(s, L) (e) ILU(0) preconditioned GBiCGSTAB(s, L) method (f) E-SSOR GBiCGSTAB(s, L) (f) E-SSOR preconditioned GBiCGSTAB(s, L) method , [1],, : IDRstab ( ),,, No.1791, pp (2012). [2] Axelsson, O.: A generalized SSOR method, BIT, Vol.12, pp (1972). [3] Chan, T. F., van der Vorst, H. A.: Approximate and Incomplete Factorizations, Parallel Numerical Algorithms, ICASE/LaRC Interdisciplinary Series in Sci. and Eng., Kluwer Academic, Vol.4, pp (1997). [4] Chen, X., Toh, K. C., Phoon, K. K.: A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations, Int. J. Numer. Meth. Engng, Vol.65, pp (2006). [5] Davis, T.:University of Florida Spares Matrix Collection: research/sparse/matrices/index.html [6] Eisenstat, S. C.: Efficient implementation of a class of preconditioned conjugate gradient methods, SIAM J. Sci. Stat. Compute., Vol.2, pp.1-4 (1981). [7], Sonneveld, P.,, van Gijzen, M.B.: IDR(s)-SOR,, Vol.20, No.4, pp (2010). [8],, : Eisenstat GS MRTR,, No.2011, (2011). [9] : Eisenstat, Table 3 3 Comparison of the accuracy between the preconditioned iterative methods. (a) IDRstab(s, L) AC TRR max (46%) (13) (12) (8) (12) (10) ILU(0) (46) (13) (12) (8) (12) (10) SSOR (34) (7) (9) (14) (29) (7) (2) (3) (9) (14) (61) (11) ILU(0) (5) (3) (8) (12) (67) (5) SSOR (0) (2) (7) (13) (69) (9), Vol.3, No.2, pp (2011). [10] Saad, Y., van der Vorst, H.A.: Iterative solution of linear systems in the 20th century, J. of Compute. Appl. Math., Vol.123, pp.1-33 (2000). [11] : AC-IDR(s) HPCS2009 pp.81-88, (2009) [12] Sleijpen, G.L.G., Fokkema, D.R.: BiCGstab(l) for equations involving unsymmetric matrices with complex spectrum, Electronic Transactions on Numerical Analysis, 6
7 (b) GBiCGSTAB(s, L) AC TRR max (32%) (5) (8) (7) (29) (18) ILU(0) (25) (8) (13) (8) (39) (9) SSOR (25) (6) (13) (8) (39) (9) (9) (3) (11) (6) (54) (18) ILU(0) (5) (2) (5) (13) (74) (2) SSOR (0) (1) (7) (13) (72) (7) Vol.1, pp (1993). [13] Sleijpen, G.L.G., Sonneveld, P., van Gijzen, M.B.: Bi- CGSTAB as an induced dimension reduction method, Appl. Numer. Math., Vol.60, pp (2010). [14] Sleijpen, G.L.G., van Gijzen, M.B.: Exploiting BiCGstab(l) strategies to induce dimension reduction, SIAM J. Sci. Comput., Vol.35, No.5, pp (2010). [15] Sonneveld, P., van Gijzen, M.B.: IDR(s): a family of simple and fast algorithms for solving large nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., Vol.31, No.2, pp (2008). [16] Tanio,M., Sugihara, M.: Bi-CGSTAB(s, L)(= IDR(s, L),,, No.1638, pp (2009). [17],,, : GBiCGSTAB(s, L) ( ),,, No.1733, pp (2011). [18] : IDR, (2010). [19] van der Vorst, H.A.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, Vol.12, pp (1992). [20] Wesseling, P., Sonneveld, P.: Numerical Experiments with a Multiple Grid-and a Preconditioned Lanczos Type Methods, Lecture Notes in Math., Springer, No.771, pp (1980). 7
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