情報処理学会研究報告 IPSJ SIG Technical Report Vol.2014-HPC-144 No /5/ CRS 2 CRS Performance evaluation of exclusive version of preconditioned ite
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1 1 2 3 CRS 2 CRS Performance evaluation of exclusive version of preconditioned iterative method for dense matrix Abstract: As well known, only nonzero entries of a sparse matrix are stored in memory in CRS format. In this case, nonzero entries are refered using indirect access, because it requires a small amount of memory and short time of computation. CRS format, however, is inefficient in case of dense matrix because direct access can be done using two-dimensional array. In this article, we implement an exclusive version of iterative methods with two-dimensional array for dense matrix. Through numerical experiments, we examine speedup of computation time of the proposed version. Keywords: Dense matrix, Iterative method, CRS format 1. Ax = b A x b A CRS(Compressed Row Storage) [3] 1 Graduate School of Information Science and Electrical Engineering, Kyushu University 2 Research Institute for Information Technology, Kyushu University 3 Faculty of Science and Engineering, Doshisha University 2 [3] [4] 2 ILU(0) 1 2 Eisenstat SSOR [11] (exclusive version) 2 Eisenstat SSOR 2 Eisensatat SSOR 3 1
2 Eisenstat SSOR M M 1 Ax = M 1 b ILU(0) 2 Eisenstat SSOR ( E-SSOR )[11] 1 2 E-SSOR ILU(0) [7] E-SSOR 2.1 SSOR SSOR [2] SSOR A = L + D + U K K = (L + D/ω)(D/ω) 1 (U + D/ω) (1) ω à x b à = (L + D/ω) 1 A(U + D/ω) 1 (D/ω), (2) x = (D/ω) 1 (U + D/ω)x, (3) b = (L + D/ω) 1 b (4) k r k := b Ax k r k r n = b à x n = (L + D/ω) 1 b (L + D/ω) 1 A (U + D/ω) 1 (D/ω)(D/ω) 1 (U + D/ω)x n = (L + D/ω)(b Ax n ) = (L + D/ω) 1 r n (5) 2.2 Eisenstat trick SSOR Eisenstat trick [9] (2) à = ((U + D/ω) 1 +(L + D/ω) 1 (I + (1 2/ω)D(U + D/ω) 1 ))(D/ω) Ãv [5][8] 1. y = (U + D/ω) 1 (D/ω)v, 2. z = (D/ω)v + (1 2/ω)Dy, 3. w = (L + D/ω) 1 z, 4. Ãv = y + w. (6) CRS(Compressed Row Storage) [3] 1 CRS n pivot lval L lrowptr L lcolind L omega w 1: CRS 1. do i = 1, n 2. tmp = w(i) 3. do j = lrowptr(i), lrowptr(i + 1) 1 4. tmp = tmp lval(j) w(lcolind(j)) 5. end do 6. w(i) = omega tmp pivot(i) 2 CRS uval U urowptr U ucolind U 2: CRS 1. do i = n, 1, 1 2. tmp = w(i) 3. do j = urowptr(i), urowptr(i + 1) 1 4. tmp = tmp uval(j) w(ucolind(j)) 5. end do 6. u(i) = omega tmp pivot(i) 1 2 CRS 4 w(ucolind(j)) ucolind(j) w w w [6] [3] mat 2 2
3 プログラム例 3: 2 次元配列における前進代入演算 1. do i = 1, n 2. tmp = w(i) 3. do j = 1, i 1 4. tmp = tmp mat(j, i) w(j) 5. end do 6. w(i) = omega tmp pivot(i) プログラム例 4: 2 次元配列における後退代入演算 1. do i = n, 1, 1 2. tmp = w(i) 3. do j = i + 1, n 4. 図 1 行列 boxshield の解析モデルとメッシュ図と磁束密度分布 Fig. 1 Analytic model, mesh and distribution of density tmp = tmp mat(j, i) w(j) 5. end do 6. w(i) = omega tmp pivot(i) in case of matrix boxshield. プログラム例 3 同例 4 のように 行列要素を 2 次元配列 で格納した場合 4 行目の配列 w の値を直接参照で求めら れるため 密行列に対して反復法の高速化が期待できるが どの程度改善できるかを知りたい 4. 数値実験 4.1 電磁界解析で現れたテスト行列 同志社大学理工学部 藤原 高橋研究室より提供された 電磁界解析モデルのテスト行列に対する密行列専用版反 復法の収束性を調査する 使用した行列は 2 個で 共に 密行列である 行列 boxshield (次元数 4,902) は 2 個の コイルに挟まれた箱の解析モデル 行列 problem13 (次元 図 2 行列 problem13 の解析モデルとメッシュ図と磁束密度分布 Fig. 2 Analytic model, mesh and distribution of density in case of matrix problem13. 数 16,800) は TEAM (Testing Electromagnetic Analysis Method) Workshop により公開されている検証用の解析モ 復回数は 回とした 解法は GPBiCG 法 [14] GP- デルである 図 1 に行列 boxshield 図 2 に行列 problem13 BiCG variant4 法 (以下 GPBiCG v4 と表記)[1] BiCGSafe の解析モデルとメッシュ図と磁束密度分布を示す ここ 法 [10] GMRES 法 [12] GBiCGSTAB(s,L)[13] 法の 5 種 で メッシュ図は対称性から図 1 では全体の 1/8 図 2 で 類を使用した は 1/4 のみ表示した 実験結果 計算機環境と計算条件 表 1 表 2 に行列 boxshield に対する E-SSOR 前処理つ 計算機は Dell PowerEdge R210 戦略名 Sandy Bridge, き反復法の収束性を示す 表中の GMRES 法のリスタート CPU Intel Xeon E クロック周波数 3.1GHz メ 係数は 500 加速係数は 0.1 から 1.0 まで 0.1 刻みで実験を モリ 8.2Gbytes L3 キャッシュメモリ 8.2Mbytes, OS 行った ここで 0.6 から 0.9 までの実験結果については全 Red Hat Linux Enterprise を使用した プログラムは For- ての解法で最短で収束するケースが無かったので 省略し tran90 を用いて実装し コンパイラは Intel Fortran Com- た GBiCGStab(s,L) 法では s を 4,8,10,12,14 L を 1,2 と piler ver を使用した 最適化オプションは -O3 を 変化させた TRR は True Relative Residual の略で常用 使用した 右辺項は物理的条件から得られる値を用いた 収 対数 log10 ( b Axk+1 2 / b Ax0 2 ) の値を意味する 8 束判定条件は相対残差の 2 ノルム r k+1 2 / r acc-c. は accelerated coefficient の略で加速係数を示す とした 初期近似解 x0 はすべて 0 とした 前処理は対 また pre-t. は前処理にかかった時間 itr-t. は反復計 角スケーリングと E-SSOR 前処理を使用した 最大反 算にかかった時間 tot-t. は前処理と反復計算の合計時 2014 Information Processing Society of Japan 3
4 1 boxshield E-SSOR Table 1 Convergence rate of four E-SSOR preconditioned iterative methods for matrix boxshield. method acc-c. itr. pre-t. itr-t. tot-t. ratio TRR GPBiCG GPBiCG v BiCGSafe GMRES ratio GPBiCG , 2 ( 1 ) GMRES ( 2 ) GPBiCGStab (s,l)=(10,2) , 4 problem13 E-SSOR 3 4 ( 1 ) GMRES ( 2 ) BiCGSafe , ,870,864( ) 20, ,811,536( ) Dell PowerEdge R210 Sandy Bridge, CPU Intel Xeon E boxshield E-SSOR GBiCGStab(s,L) Table 2 Convergence rate of E-SSOR preconditioned GBiCGStab(s,L) method for matrix boxshield. (a) in case of s = max , , , , , , , GHz 8.2Gbytes L3 8.2Mbytes OS Red Hat Linux Enterprise Fortran90 Intel Fortran Compiler ver CX400 CPU Intel Xeon E GHz 128Gbytes 20Gbytes OS Red Hat Linux Enterprise Fortran90 Fujitsu Fortran Compiler -O3 2 r k+1 2 / r x 0 0 E-SSOR 4
5 (b) in case of s = 12 and , , problem13 E-SSOR Table 3 Convergence rate of four E-SSOR preconditioned iterative methods for matrix problem13. method acc-c. itr. pre-t. itr-t. tot-t. ratio TRR GPBiCG GPBiCG v BiCGSafe GMRES Table 4 problem13 E-SSOR GBiCGStab(s,L) Convergence rate of E-SSOR preconditioned GBiCGStab(s,L) method for matrix problem13. (a) in case of s = max max , , , max , , , E-SSOR GMRES(k) E-SSOR GMRES(k) 18%. 6 E-SSOR ratio
6 (b) in case of s=12 and , , , , E-SSOR GMRES(k) Table 5 Convergence rate of exclusive version of E-SSOR preconditioned GMRES(k) method for small size of model. store itr. itr-t. tot-t. ratio TRR format [sec.] [sec.] CRS format 2, Exclusive ver. 2, Table 6 E-SSOR Convergence rate of exclusive version of four E-SSOR preconditioned iterative methods for middle size of model. method acc-c. itr. pre-t. itr-t. tot-t. ratio TRR GPBiCG (CRS) GPBiCG (exclu.) GPBiCG v (CRS) GPBiCG v (exclu.) BiCGSafe (CRS) BiCGSafe (exclu.) GMRES (CRS) GMRES (exclu.) Fig. 3 3 Comparison of the conventional CRS version and exclusive version of iterative methods in view of the ratio of total computation time ( 1 ) ( 2 ) GMRES 19% ( 3 ) GPBiCG 7.9%
![Fig. 4 4 Comparison of the conventional CRS version and exclusive version of iterative methods in view of total computation time. Sci. Stat. Compute., Vol.2, pp.1-4, 1981. [10] Fujino, S.](/docs-images/103/161546470/images/7-0.jpg ", Fujiwara, M., Yoshida, M.: A proposal of preconditioned BiCGSafe method with safe convergence, Proc. of The 17th IMACSWorld Congress on Scientific Computation, Appl. Math. Simul.")
7 Fig. 4 4 Comparison of the conventional CRS version and exclusive version of iterative methods in view of total computation time. Sci. Stat. Compute., Vol.2, pp.1-4, [10] Fujino, S., Fujiwara, M., Yoshida, M.: A proposal of preconditioned BiCGSafe method with safe convergence, Proc. of The 17th IMACSWorld Congress on Scientific Computation, Appl. Math. Simul., CD-ROM, Paris, France, [11] : Eisenstat IDR(s)-R [12] Saad, Y. : Iterative Methods for Sparse Linear Systems 2nd edition, SIAM, Philadelphia, [13] : GBiCGSTAB(s,L) 1733 pp , [14] Zhang, S.-L. : GPBi-CG: Generalized product-type methods preconditionings based on Bi-CG for solving nonsymmetric linear systems, SIAM J. Sci. Comput., pp , E-SSOR E-SSOR GMRES GMRES 19% GPBiCG 7.9% GMRES [1] Abe, K., Sleijpen, Gerard L.G.: Solving linear equations with a stabilized GPBiCG method, Appl. Numer. Math., doi: /j.apnum , [2] Axelsson, O. : A generalized SSOR method, BIT, Vol.12, pp , [3] Barrett, R., Berry, M., et al.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Templates 1996 [4] Benzi, M.: Preconditioning Techniques for Large Linear Systems: A Survey, Journal of computational Physics, Vol.182, pp , [5] 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 , [6] Chen, K: Matrix Preconditioning Techniques and Applications, Cambridge Monographs on Applied and Computational Mathematics, [7] Chen, K.: On a class of preconditioning methods for dense linear systems from boundary elements, SIAM J. Sci. Comput., Vol.20, No.2, pp , [8] 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 , [9] Eisenstat, S. C.: Efficient implementation of a class of preconditioned conjugate gradient methods, SIAM J. 7
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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