T.F. Chan. Fourier analysis of relaxed incomplete factorization preconditioners. SIAM J. Sci. Stat. Comput., 12:668--680, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a data flow front farer and farer away from the location of this variable. An estimate for the rate of convergence of this limit process is also given. This analysis is related to the theory of continued fractions [21] and may also serve as a justification for the infinite domain analysis used in [8] and [14] The results are stated in pure algebraic manner, avoiding the use of the grids, orders and norms defined in [22] In the case of [27] this convergence means that when the overlapping in Figure 3 is large enough, the approximate values in the lower right domain almost reach their exact ....

....Note that the implementation of Figure 1(a) requires more communication steps than that of Figure 1(b) List of examples 1. The Poisson equation with Dirichlet boundary conditions. We have used central second order difference scheme and the parameter ff = 1 Gamma 8=128 suggested in [8]. An interesting phenomenon is the inferiority of the domain decomposition of Figure 3 to that of Figure 4. This may be explained by the less strict diagonal dominance in the former relatively to the latter (see Lemma 5) 2. The anisotropic diffusion equation 0:01u xx u yy = f with Dirichlet ....

Chan T.F., "Fourier Analysis of Relaxed Incomplete Factorization Preconditioners", SIAM J. Sci. Stat. Comput. 12 (1991), 668-680.

....the minor, and consequently the factorisation, an M matrix and therefore positive de nite. 2.6.2 Accuracy of MILU factorisations Often the ll is multiplied by a parameter less than 1 before being moved to the diagonal. This is refered to as a relaxed modi ed incomplete factorisation ; 7 see [7, 11, 20, 44]. Also, some methods perturb the diagonal by adding elements of order h 2 to the diagonal; see [2, 8] Such (perturbed) modi ed incomplete factorisation algorithms can be proved to give (M 1 A) O(h 1 ) see [26, 36, 10] There is of course an equivalence of sorts between the relaxed and ....

Tony F. Chan. Fourier analysis of relaxed incomplete factorization preconditioners. SIAM J. Sci. Stat. Comput., 12:668-680, 1991.

....a data flow front farer and farer away from the location of this variable. An estimate for the rate of convergence of this limit process is also given. This analysis is related to the theory of continued fractions [19] and may also serve as a justification for the infinite domain analysis used in [7] and [13] The results are stated in pure algebraic manner, avoiding the use of the grids, orders and norms defined in [20] In the case of [24] this convergence means that when the overlapping in Figure 3 is large enough, the approximate values in the lower right domain almost reach their exact ....

....a diffusion equation with discontinuous coefficients and (5) an anisotropic diffusion equation with discontinuous coefficients. The 5 coefficient discretization schemes of [1] and [6] are used on a 128 Theta 128 grid. The MILU parameter is 0. 95 (except of Example 1, where the optimal parameter of [7] is used) The amount of overlapping is 8 grid points in each spatial direction. The Transpose Free Quasi Minimal Residual method (Algorithm 5.2 in [10] is used for acceleration. The numbers of iterations required for reducing the l 2 norm of the residual by 6 orders of magnitude are displayed in ....

Chan T.F., "Fourier Analysis of Relaxed Incomplete Factorization Preconditioners", SIAM J. Sci. Stat. Comput. 12 (1991), 668-680.

....recursion. In other words, the solution is taken only in the subdomain under consideration and not in its extension toward its neighbor. The present analysis is related to the theory of continued fractions ( 22] 24] It may also be interpreted to justify the infinite domain analysis used in [8] and [16] It uses pure algebraic terminology, avoiding the use of the grids, orders and norms used in [25] In the case of [31] the present convergence theory implies that, when the overlapping area in Figure 3 gets larger, the solution of the truncated recursion at the lower right triangle ....

....Note that the implementation of Figure 4(a) requires more communication steps than that of Figure 4(b) List of examples 1. The Poisson equation with Dirichlet boundary conditions. We have used central second order difference scheme and the parameter ff = 1 Gamma 8=128 2 recommended in [8]. An interesting phenomenon is the inferiority of the domain decomposition of Figure 3 to that of Figure 5. This may be explained by the more strict diagonal dominance of the triangular matrices in the latter (see Lemma 5) which yields more accurate truncated recursion in the forward elimination ....

Chan T.F., "Fourier Analysis of Relaxed Incomplete Factorization Preconditioners", SIAM J. Sci. Stat. Comput. 12 (1991), 668-680.

....a data flow front farer and farer away from the location of this variable. An estimate for the rate of convergence of this limit process is also given. This analysis is related to the theory of continued fractions [21] and may also serve as a justification for the infinite domain analysis used in [8] and [14] The results are stated in pure algebraic manner, avoiding the use of the grids, orders and norms defined in [22] In the case of [27] this convergence means that when the overlapping in Figure 3 is large enough, the approximate values in the lower right domain almost reach their exact ....

....Note that the implementation of Figure 1(a) requires more communication steps than that of Figure 1(b) List of examples 1. The Poisson equation with Dirichlet boundary conditions. We have used central second order difference scheme and the parameter ff = 1 Gamma 8=128 2 suggested in [8]. An interesting phenomenon is the inferiority of the domain decomposition of Figure 3 to that of Figure 4. This may be explained by the less strict diagonal dominance in the former relatively to the latter (see Lemma 5) 2. The anisotropic diffusion equation 0:01u xx u yy = f with Dirichlet ....

Chan T.F., "Fourier Analysis of Relaxed Incomplete Factorization Preconditioners", SIAM J. Sci. Stat. Comput. 12 (1991), 668-680.

....17, 18] 2. it is robust with respect to discontinuities and anisotropy [4, 34] Two major inconveniences are that the optimal value of the relaxation parameter strongly varies from a problem to another and the behavior could be very sensitive to variations of around the observed opt [12, 39]. In [34] a new variant of RILU has been proposed. There, the relaxation parameter is variable and dynamically computed during the incomplete factorization phase. Like its precursor RILU, it is robust with respect to both existence and performance. In addition, its performance does not critically ....

....robustness problems [17, 22, 25, 26, 34, 40] Optimal performances are achieved with 0 1. The trouble is that opt strongly depends on the problem (size) Most severe is the fact that performances could be highly sensitive to the variation of around opt which is very hard to estimate [4, 12, 39]. In the case of 6 M.M. MAGOLU AND Y. NOTAY uniform grid of mesh size h in all directions, it is advocated in [4] to use = 1 Gamma ffi h with ffi = O(1) 3.7) ffi opt 2:4 for some 2D problems [4] in which case one has (see Subsection 3.3) that max (B Gamma1 A) 2 ffi h Gamma1 ....

T.F. Chan, Fourier analysis of relaxed incomplete factorization preconditioners, SIAM J. Sci. Stat. Comput., 12 (1991), pp. 668--680.

.... suggesting that the modified incomplete LU preconditioning has reduced the condition number from O(h Gamma2 ) to O(h Gamma1 ) although this behaviour can only be proved to occur theoretically if the diagonal elements are perturbed to increase the 8 positive definiteness (see for example [1]) Unpreconditioned RGMRES is somewhat slower than the other two iterative methods, and fails to converge within 1000 iterations on the finer meshes, although this situation might be improved by an alternative choice of the subspace dimension m. On the finest mesh PCGS is the best iterative method ....

....below. These values can therefore be thought of as the number of false starts of F11DAF required. The CPU times presented do not include the time spent in false starts; this is generally a fairly small proportion of the total time. N 200 400 800 1600 NNZ 1001 2008 4005 8011 PRGMRES 10 (0. 16) [1] 10 (0.46) 2] 20 (0.90) 10 (1.55) 1] PCGS 6 (0.16) 1] 3 (0.38) 2] 7 (0.68) 5 (1.44) 1] PBICGSTAB 8 (0.20) 1] 4 (0.43) 2] 8 (0.78) 8 (1.87) 1] RGMRES CGS BICGSTAB F01BRF F04AXF (0.22) 1.08) 6.09) 46.23) F11DCF(Direct) 0.42) 2.46) 16.75) 113.39) MA48 (0.16) 0.62) 3.59) 22.02) ....

[Article contains additional citation context not shown here]

T F Chan (1991) Fourier analysis of relaxed incomplete factorization preconditioners SIAM J. Sci. Statist. Comput. 12 668--680.

....T A. Preconditioning techniques that accelerate the convergence of these methods have received extensive attention in the literature. Some examples of the literature of preconditioning symmetric positive definite linear systems and least squares problems are [25] 2] 39] 59] 21] 4] 61] 3] [16]. They can be summarized as follows: ffl Column scaling. C = diag(d i ) where d i are norms of columns of A. ffl SSOR preconditioning [9] C = I L T , where A has been normalized so that A T A = L I L T with L strictly lower triangular, and is a scalar parameter. ffl Incomplete ....

T. F. Chan. Fourier analysis of relaxed incomplete factorization preconditioners. SIAM Journal of Scientific and Statistical Computing, 12(3):668--680, May 1991.

....a i;i in the innermost loop is replaced by: a i;i : a i;i Gamma ea r;j ; where 0 1 is a user specified relaxation parameter. Obviously, 0 and 1 correspond to ILU and MILU respectively. It was observed empirically by van der Vorst (1990a) and verified using the Fourier analysis method (Chan 1991), that a value of = 1 Gamma ch 2 gives the best results for some classes of matrices coming from elliptic problems. The optimal value of c can be estimated and is related to the optimal value of c in the DKR method of Dupont et al. 1968) Notay (1994) gave strategies for choosing = i;j ....

Chan, T. F. (1991), `Fourier analysis of relaxed incomplete factorization procedures', SIAM J. Scientific and Statistical Computing 12, 668--680.

....for a certain variable converges rapidly when the initial conditions are given on a data front farer and farer away from the location of this variable. This analysis is related to the theory of continued fractions [19] and may also serve as a justification for the infinite domain analysis used in [7] and [13] The results are stated in pure algebraic manner, avoiding the use of the grids, orders and norms defined in [20] In the case of [24] this convergence means that when the overlapping in Figure 3 is large enough, the approximate values in the lower right domain approach their exact ....

....a diffusion equation with discontinuous coefficients and (5) an anisotropic diffusion equation with discontinuous coefficients. The 5 coefficient discretization schemes of [1] and [6] are used on a 128 Theta 128 grid. The MILU parameter is 0. 95 (except of Example 1, where the optimal parameter of [7] is used) The amount of overlapping is 8 grid points in each spatial direction. The Transpose Free Quasi Minimal Residual method (Algorithm 5.2 in [10] is used for acceleration. The numbers of iterations required for reducing the l 2 norm of the residual by 6 orders of magnitude are displayed in ....

Chan T.F., "Fourier Analysis of Relaxed Incomplete Factorization Preconditioners", SIAM J. Sci. Stat. Comput. 12 (1991), 668-680.

....to a complete Cholesky factorization. The same parameter DROPTL is also used to perturb the diagonal elements of the matrix, scaling them by the factor (1 DROPTL) prior to the factorization. This can have the effect of accelerating convergence for some model problems involving M matrices [1], and can be used to ensure that the preconditioning matrix is positive definite for more general positive definite matrices [3] 4 Given a symmetric positive definite matrix A, a complete Cholesky factorization will always produce positive pivots. However, an incomplete factorization will not ....

....Cholesky preconditioning used by F11JCF and F04MAF is worthwhile and reduces the order of the condition number of the preconditioned system from O(h Gamma2 ) to O(h Gamma1 ) although theoretically this should only happen for non zero DSCALE. This behaviour has been noted elsewhere [1]. On the finest mesh ICCG is roughly 13 times faster than the direct method, and 5 times faster than unpreconditioned CG. For this simple model problem F01MAF F04MAF is slightly more efficient than F11JAF F11JCF. This is principally because F01MAF F04MAF scales the diagonal matrix elements before ....

[Article contains additional citation context not shown here]

Chan T F (1991) Fourier analysis of relaxed incomplete factorization preconditioners SIAM J. Sci. Stat. Comput. 12 (3) 668--680.

....elliptic problems. It turns out the stencils of the incomplete LU factors in this case tend to a limiting constant coefficient one as one moves away from the boundary of the domain. Therefore, a Von Neumann type analysis can be applied to the preconditioned system. The reader is referred to [30, 49, 29] for more details. 3. Variants. Many variants of the basic ILU and MILU methods have been proposed in the literature. These variants are designed to either improve the performance or reduce the drawbacks of the basic methods. To describe all of these variants is beyond the scope of this chapter; ....

....of a i;i in the inner most loop is replaced by: a i;i : a i;i Gamma ea r;j ; 9 where 0 1 is a user specified relaxation parameter. Obviously, 0 and 1 correspond to ILU and MILU respectively. It has been observed empirically in [94] and verified using the Fourier analysis method [29], that a value of = 1 Gamma ch gives the best results. The optimal value of c can be estimated and is related to the optimal value of c in the DKR method in [53] see Remark 1 in Sec. 2.3) Van der Vorst [98] suggested to use the simple rule of = 95 in practice. Van der Vorst [93] ....

T.F. Chan. Fourier analysis of relaxed incomplete factorization preconditioners. SIAM J. Sci. Stat. Comput., 12:668--680, 1991.

....elliptic problems. It turns out the stencils of the incomplete LU factors in this case tend to a limiting constant coefficient one as one moves away from the boundary of the domain. Therefore, a Von Neumann type analysis can be applied to the preconditioned system. The reader is referred to [30, 49, 29] for more details. 3. Variants. Many variants of the basic ILU and MILU methods have been proposed in the literature. These variants are designed to either improve the performance or reduce the drawbacks of the basic methods. To describe all of these variants is beyond the scope of this chapter; ....

....update of a i;i in the inner most loop is replaced by: a i;i : a i;i Gamma ea r;j ; where 0 1 is a user specified relaxation parameter. Obviously, 0 and 1 correspond to ILU and MILU respectively. It has been observed empirically in [94] and verified using the Fourier analysis method [29], that a value of = 1 Gamma ch 2 gives the best results. The optimal value of c can be estimated and is related to the optimal value of c in the DKR method in [53] see Remark 1 in Sec. 2.3) Van der Vorst [98] suggested to use the simple rule of = 95 in practice. Van der Vorst [93] ....

T.F. Chan. Fourier analysis of relaxed incomplete factorization preconditioners. SIAM J. Sci. Stat. Comput., 12:668--680, 1991.

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T. F. Chan. Fourier analysis of relaxed incomplete factorization preconditioners. SIAM Journal of Scientific and Statistical Computing, 12(3):668 680, May 1991. 197

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