On convergence of Inverse-based IC decomposition

Transcription

1 On convergence of Inverse-based IC decomposition Seiji FUJINO and Aira SHIODE (Kyushu University) On convergence of Inverse-basedIC decomposition p.1/26 Outline of my tal 1. Difference between standard Incomplete Cholesy and Inverse-based IC decompositions 2. Extension of Crout version of ILU decomposition to Inverse-based IC decomposition for solving a linear systems with symmetric positive definite maytrices 3. Numerical experiments 4. Concluding remars On convergence of Inverse-basedIC decomposition p.2/26

2 Bacground and Objectives ILUC(Crout) decomposition for solving a linear system of equations with nonsymmetric matrix has been proposed by Li et al. in ILUC decomposition enables efficient implementation of rigorous dropping strategy based on estimating the norms of the inverse factor L 1. We extend ILUC decomposition to IC decomposition for solving linear systems with symmetric positive definite matrix. On convergence of Inverse-basedIC decomposition p.3/26 preconditioned equation Solved equation (1) Ax = b preconditioned equation (2) (U T AU 1 )(Ux) =U T b U: upper triangular matrix, T : transpose The error due to the inverse of the factors will be more significant than that of the factors themselves. On convergence of Inverse-basedIC decomposition p.4/26

3 Error due to inverse of factor 1. When A = Ū T Ū and A U T U, using U Ū + X (U T = Ū T + X T ) with error matix X, we can gain the following inverses (3) U 1 = Ū 1 + X, U T = Ū T + X T. 2. However, the preconditioned matrix is given by (4) U T AU 1 = I + ŪX + XT Ū T + X T AX, Here we cannot find the error X. 3. Then we estimate the error X of U 1 instead of the error matrix X of U measured in the usual IC decomposition. On convergence of Inverse-basedIC decomposition p.5/26 Inverse-based dropping 1. We use representation of L = U T for convenience. 2. Entry a i,i is corresponded to the diagonal entry of the ith row of A, and updated in the decomposition process. (5) l 1,1 0 0 l 2, l,.... L =... a +1, l n,1 l n, 0 0 a n,n On convergence of Inverse-basedIC decomposition p.6/26

4 Norm estimation of L 1 1. We suppose that fill-in l j, (j>) is dropped when we update the column of matirx L. 2. Lower matrix L is presented as below. e i (1 i n) is the unit vector with the ith row of 1. (6) L = L l j, e j e T. 3. For j> from L e j = a j,j e j, we can gain (7) L = L l j, e j e T = L (I l j, e j e T /a j,j) On convergence of Inverse-basedIC decomposition p.7/26 Norm estimation of L 1 (contd.) 1. As a result, we can represent L 1 as (8) L 1 = (I l j, e j e T /a j,j) = + l j, e j e T L 1 1 L 1 1 L /a j,j. 2. In dropping strategy, it is preferable for us to utilize (9) l j, e j e T 1 L /a j,j = l j, /a j,j e j e T 1 L 3. This leads to rough norm estimation for the th row of L 1. On convergence of Inverse-basedIC decomposition p.8/26

5 Maximum norm of A 1. We define maximum norm of A as (10) A =max x 0 Ax x =max i n a i,j j=1 2. Moreover we can approximate e j e T certain vector b ( 0) 1 L with a (11) e j e T L 1 = max b 0 e j e T L 1 b b. On convergence of Inverse-basedIC decomposition p.9/26 Decision of vector b 1. For estimating of e j e T L 1, it is sufficient that we decide a vector b( 0) which satisfies equation (11). 2. It is nown that, at the definition of maximum norm, when at i = m (12) n a i,j j=1 is maximum, the jth component of vector b which gives the maximum value, is written as (13) b j =sign(a m,j ). (j =1, 2,...,n) 3. Here sign means the sign function as (14) sign(t) = +1 (t 0), sign(t) = 1 (t<0) On convergence of Inverse-basedIC decomposition p.10/26

6 Decision of vector b (contd.) 1. To compute directly L 1 is too expensive. 2. Then we search approximately for a vector b so as to mae e j e T L 1 b as large as possible in the vector b as (15) b = ( b 1, b 2,..., b n ) T, (b 1,b 2,...,b n = ±1) On convergence of Inverse-basedIC decomposition p.11/26 Decision of vector b (contd.) 1. The following estimation holds approximately using the vector b which maes e j e T L 1 b as large as possible. (16) e j e T L 1 e je T L 1 b b. 2. R.H.S. of this equation can be estimated the th component of the solution Lξ = b. 3. The ξ component of vector ξ is gained with the th component of vector b of R.H.S. of the above equation ξ = (ξ 1,ξ 2,...,ξ, 0,...,0) T, b =(b 1,b 2,...,b, 0,...,0) T (17) ξ =(b e T L 1 ξ 1 )/l,. On convergence of Inverse-basedIC decomposition p.12/26

7 Estimation of L 1 1. Finally, we can show the algorithm which estimates the norm of e j e T L ξ correponds to e j e T L 1. Also ν correponds to e T L 1 ξ 1. set ξ 1 =1/l 1,1 ; ν i =0(i =1,...n) for =2,n temp + =1 ν,temp = 1 ν if temp + > temp then ξ = temp + /l, else ξ = temp /l, end if for j = +1,n(l j, 0) ν j = ν j + ξ l j, end for end for. On convergence of Inverse-basedIC decomposition p.13/26 Inverse-based dropping 1. In inverse-based dropping, when the following equation holds, fill-in l j, is dropped in the decomposition process. (18) l j, /a j,j e j e T L 1 = l j, ξ / a j,j τ On convergence of Inverse-basedIC decomposition p.14/26

8 Numerical Experiments 1. A linear system of equations was solved by the preconditioned CG method. 2. We adopt the IC with standard dropping and that with Inverse-based dropping. 3. The right-hand side of vector b was imposed as all The stopping criterion for convergence is less than 10 8 of the relative residual 2-norm r n+1 2 / r In all cases the iteration was started with x 0 = The maximum iteration is set as same as the dimension of each matrix. On convergence of Inverse-basedIC decomposition p.15/26 Analytic conditions 1. Tolerance values τ were varied from up to 0.15 with the interval of In total 30 cases were examined for dropping strategies and matrices, respectively. 3. The shifted parameter α of linear systems A + αi was fixed as 0.10, where I denotes the unit matrix. On convergence of Inverse-basedIC decomposition p.16/26

9 Test matrices 1. Four test matrices were derived from Matrix-Maret database and matrix T_ was stemed from a realistic analysis. 2. Table shows description of four test matrices. 3. In Table total nnz means the total number of nonzero entries of each matrix. Similarly ave. means average number of nonzero entries per one row of each matrix. matrix n total nnz ave. analysis S3DKQ4M2 90,449 2,455, cylindrical shells S3DKT3M2 90,449 1,921, cylindrical shells CT20STIF 52,329 1,375, engine bloc ENGINE 143,571 2,424, engine head T_ ,524 6,053, structural analysis On convergence of Inverse-basedIC decomposition p.17/26 Results for S3DKQ4M2 Vertical axis means the total time preconditioning and CG iteration time, and horizontal axis means tolerance. Color bars in red and blue mean times of ICCG method with Inverse-based and standard dropping, respectively. (a)matrix S3DKQ4M2 On convergence of Inverse-basedIC decomposition p.18/26

10 Results for S3DKQ4M2 (contd.) For larger tolerance of ICCG method with the standard dropping could not converge. On the other hand, ICCG method with Inverse-based dropping converged over the whole range of tolerance. The much computation times of the former require sometimes as tolerances of 0.050, and On convergence of Inverse-basedIC decomposition p.19/26 Results for S3DKT3M2 At tolerances τ of.125,.13,.145 and.15, ICCG with standard dropping diverged. In contrast, ICCG with Inverse-based dropping converged. The former requires much computation times irregularly at tolerances of.095,.12,.135 and.14. In contrast, the latter converged smoothly over the whole range. On convergence of Inverse-basedIC decomposition p.20/26

11 History of residual Figure (a) shows the results of CG method with usual IC decomposition at tolerances of.09(red),.095(green) and.1(blue). Similarly Figure (b) depicts the results of CG method with ib_ic decomposition at the above same tolerances. 0-1 Shift_IC(0.090) Shift_IC(0.095) Shift_IC(0.100) 0-1 Shift_ib_IC(0.090) Shift_ib_IC(0.095) Shift_ib_IC(0.100) Relative Residual Relative Residual Iterations (a)ic decomposition Iterations (b)ib_ic decomposition On convergence of Inverse-basedIC decomposition p.21/26 History of residual (contd.) 0-1 Shift_IC(0.090) Shift_IC(0.095) Shift_IC(0.100) 0-1 Shift_ib_IC(0.090) Shift_ib_IC(0.095) Shift_ib_IC(0.100) Relative Residual Relative Residual Iterations (a)ic decomposition Iterations (b)ib_ic decomposition In case of usual IC decomposition, much times are required at tolerance of only compared with those of tolerance of.09 and.1. This means that error matrix X of U measured in the usual IC has possibility of misleading estimation. In case of ib_ic decomposition, times for three tolerances are approximately as same as each other. On convergence of Inverse-basedIC decomposition p.22/26

12 Results fors4dkt3m2 Figure (a) illustrates that at tolerances τ of.09 and.095 ICCG method with the standard dropping diverged. Contrastively general behaviour of ICCG method with Inverse-based dropping converges nicely. On convergence of Inverse-basedIC decomposition p.23/26 Results for CT20STIF Figure (b) illustrates that the range of Inverse-based IC is larger than that of usual IC. On convergence of Inverse-basedIC decomposition p.24/26

13 Results for T_ Figures (a),(b) show times vs. tol. and history of residual of CG method with _IC and ib_ic decompositions at tol. =.09 for T_ Obviously we notice that ib_ic decomposition yields better result than the usual IC decomposition. 0-1 ShiftIC Shift_ib_IC -2 Relative Residual (a)cpu times vs. tol Iterations (b)history of residual (tol.=.09) On convergence of Inverse-basedIC decomposition p.25/26 Concluding remars We concluded that Inverse-based dropping is superior to the standard dropping from the viewpoint of robustness for tolerance value and stability of convergence. It may be an alternative promising option if ICCG method with the standard dropping fails, though it is unliely to be the fastest choice of convergence in general cases. On convergence of Inverse-basedIC decomposition p.26/26