O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numerische Mathematik, 48 (1986), pp. 479--498.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of problems. Among all these preconditioners the Incomplete LU factorizations [69, 21] are the most popular ones, and attempts have been made to improve them, for instance by including more fill [70] or by modifying the diagonal of the ILU factorization in order to force rowsum constraints [58, 6, 5, 73, 95, 34], or by changing the ordering of the matrix [96, 97] A collection of experiments with respect to the effects of ordering is contained in [30] More recently, it was discovered that a multigrid inspired ordering can be very effective for discretized diffusion convection equations, leading in some ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48:479--498, 1986.

....MILU does not always perform better than ILU. Part of the reason is the favourable eigenvalue distribution for ILU and part of it has to do with the higher sensitivity of MILU to round off errors [94] This provides motivation for an interpolated version between ILU and MILU, first proposed in [10, 1]. Namely, in the MILU algorithm, the update 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 ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning matrices. Numer. Math., 48:479--498, 1986.

....ICCG(1) or ICCG(1,2) method. The incomplete factorization can be modified in various way: adding the deleted entries to the diagonal entries in the same row, or modifying the pivot entry by adding a positive (small) number to it. For further study, see Gustafsson [18, 17] Axelsson and Lindskog [4] and Axelsson [7] The ICCG method needs the solution of lower and upper triangular matrices, which is not readily vectorizable or parallelizable. Johnson and Paul [30] and van der Vorst [54] have proposed some variants for vector machines. These variants have ample data parallelism, however ....

.... 1] and a[i 1] j] In this case, the number of the inner points is O(n) and the number of the boundary 72 for (i = 1; i N 1; i ) for (j = 1; j M 1; j ) b[i] j] dd[i] j] 0] a[i 1] j] dd[i] j] 1] a[i] j 1] dd[i] j] 2] a[i] j] dd[i] j] 3] a[i] j 1] dd[i] j][4] a[i 1] j] Figure 9.1: Fragment of pentadiagonal matrix vector multiplication compute inner points send data receive data compute edge points Node ID 0 network latency compute inner points send data receive data compute edge points Node ID 1 Figure 9.2: Nodal code for overlapping ....

Axelsson, O. and G. Lindskog, "On the eigenvalue distribution of a class of preconditioning methods," Numer. Math., vol. 48, pp. 479--498, 1986.

....(see e.g. 1, 6, 11, 18, 19] for solving system (1. 2) Relaxed incomplete factorizations (RILU) are precondition ing techniques that interpolate, through a relaxation parameter, between the popular incomplete LU factorization ILU and its modified variant MILU that preserves row sums of A [3, 13]. As opposed to ILU and MILU, the two main advantages of RILU are following: t Research supported by the Commission of the European Communities HCM Contract No. ERB CHBG CT93 0420, at Utrecht University, Mathematical Institute, The Netherlands. Current ad dress: Universit Libre de Bruxelles, ....

....Nuclaire (CP 165) 50 av. F.D. Roosevelt, B 1050 Brussels, Belgium. ynotay ulb.ac.be. Research supported by the Fonds National de la Recherche Scientifique , Belgium. 1. it does not suffer a lot from existence problem [13, 16, 17] 2. it is robust with respect to discontinuities and anisotropy [3, 34]. Two major inconveniences are that the primal value of the relaxation parameter 0: strongly varies from a problem to another and the behavior could be very sensitive to variations of 0: around the observed :opt [12, 38] In [34] a new variant of RILU has been proposed. There, the relaxation ....

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O. AXELSSON AND G. LINDSKOG, On the eigenvalue distribution of class of preconditioning methods, Numer. Math., 48 (1986), pp. 479 498.

....(sometimes known as IC(0) or ICCG(0) 20] and drop tolerance incomplete factorization. When performing incomplete factorizations, the code can modify the diagonal to ensure that the row sums of LL T are equal to those of A [14] or it can use a relaxed modification, which is more robust [2, 4, 5, 27]. The performance of this code is similar to the performance of other drop tolerance incomplete Cholesky codes, but it is slower than the multifrontal code unless L remains very sparse. The iterative solver that we use is preconditioned conjugate gradients (see, for example, 3, 13] 3.3. Test ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning matrices, Numerische Mathematik, 48 (1986), pp. 479--498.

....It is impossible in general to adequately estimate the number of iterations a preconditioned nonsymmetric iterative solver will take. Although upper bounds have been established for the number of conjugate gradient iterations needed for some simple problems with known eigenvalue distributions [AL86] no realistic estimates for practical problems are available. Three additional complicating factors also occur. First, the targeted systems are nonsymmetric. In this case, even a complete a priori knowledge of the eigenvalues of the system does not allow estimating the number of iterations. ....

Axelsson O. and Lindskog G. (1986) On the eigenvalue distribution of a class of preconditioning methods. Numer. Math. 48: 479--498.

....a Cartesian grid of 80 Theta 80 cells is used. The subdomain problems are solved using GMRES with RILU(ff) preconditioning [50, 48] and a relative stopping criterion. For ff = 0 we get the standard ILU preconditioner [36] and for ff = 1 we get the Modified ILU preconditioner [27] RILU(ff) [1] lies in between these two. With RILUD(ff) we mean RILU(ff) restricted to the diagonal. The momentum equations are solved using a RILUD(0:95) preconditioner and the pressure equations using a RILU(0:975) As a short hand, we will use RILU(ff) to mean RILUD(0:95) whenever the momentum, and ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numerische Mathematik, 48:479--498, 1986.

.... strategy in common use is to take K = LL T to be an incomplete Cholesky factorization of A [16] For discretizations of second order PDEs in two dimensions, defined on a grid with spacing h, we have with incomplete Cholesky factorization, h Gamma2 ; with a modified IC factorization[9, 1], h Gamma1 ; and with a multigrid cycle, 1. Preconditioners such as multigrid and some domain decomposition methods, for which the condition number of the preconditioned system is independent of the grid size, are termed optimal. Another preconditioning strategy that has proven successful ....

....by a random vector gives the same qualitative behavior reported here. 6. Numerical experiments 15 Convergence is declared when, in the Jth iteration, kr J k tol Delta kr 0 k, for tol = 10 Gamma6 . The preconditioner used on the blocks is the relaxed incomplete LU (RILU) factorization of [1], with relaxation parameter = 0:975. We choose this preconditioner because it is simple to implement (for a five point stencil, modifications only occur on the diagonal) and is reasonably effective. Certainly, more advanced preconditioners could be designed based on the blocks of A . 6.1 ....

O. Axelsson and G. Linskog. On the eigenvalue distribution of a class of preconditioning methods. Numerische Mathematik, 48:479--498, 1986.

....entry a i;j of the original system matrix. In such a situation, the dropped value may be added to the diagonal entries a i;i and a j;j . This technique, which is known as diagonal compensation [1] preserves the rowsum of the system matrix. In the case of the relaxed variants of IC introduced in [2], all the dropped values, possibly weighted, will be automatically added to the diagonal entries. Figs. 3 and 4 show two examples where the physical domain is partitioned into 1 Theta 8 (stripes) and 2 Theta 4 subdomains, respectively. Definition 1 A standard IC( combined with the above ....

....strategy, are accepted [25,26] It would be interesting to choose the fill level proportional to the number of subdomains, for a fixed mesh size problem. This needs further investigations. Another point which deserves to be explored is the usage of other variants of the basic IC (see, e.g. [1,2,4,18,23]) 16 M. Magolu monga Made and H.A. van der Vorst Table 3. Problem 2. h Gamma1 = 480; n = 230880. Extremal eigenvalues (min , 2 , and max ) effective spectral condition number ( 2) number of pcg iterations (iter) overall elapsed time (time) in seconds. Precond and Part denote the ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math, 48, 479--498, 1986.

....row sums, D ii = A ii Gamma X j i X k j A ij D Gamma1 jj A jk (2) 2.1. 1 Momentum equations For the momentum equations we use a row by row combination of 0:05 times the D from ILUD and 0:95 times the D taken from MILUD 2 , as described in [15] this is similar to RILUD as described in [1]. To describe the MILUD 2 preconditioner we use the following splitting u = u (1) u (2) 3) 3 where u (1) contains a vector of the first momentum vector component in all cells and u (2) contains a vector of the second momentum vector component in all cells. The corresponding ....

....A ij = P j M ij i.e. D ii X j i X k j L ij D Gamma1 jj U jk X j i L ij X i j U ij = X j A ij : 16) In the ISNaS solver we use RILU(0.975) which is a row by row combination of the two diagonal matrices, 0.025 times the ILU version and 0. 975 times the MILU version [14] [1]. 4 2.1.3 A parallel version On a parallel machine ILU is not very efficient, but in [2] it is shown that we can use the fact that we have a 2D rectangular domain to obtain some parallelism as follows: make vertical strips, distribute the strips over the processors and overlap calculation of ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48:479--498, 1986.

....experiments in this paper are only performed for the standard modified incomplete factorizations (MIC) 8] and for two dimensional problems. Since the premodification gives diagonally dominant M matrices in general, the technique is applicable in connection with other variants of modification, [3], 4] 6] 15] and for three dimensional problems as well. An important feature of the idea is the preservation of the nullspace of the element matrices. In this respect we have so far only considered the case when the nullspace (for internal elements) is spanned by the constant vector. We give ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48, 479-498, 1986.

.... Since ILU and IC factorizations were the most popular preconditioners, at least in a sequential environment, many attempts have been made to improve them, for instance by including more fill [118] by 15 modifying the diagonal of the ILU factorization in order to force rowsum constraints [87, 7, 6, 122, 171, 61], or by changing the ordering of the matrix [173, 174] A set of experiments with respect to the effects of ordering is contained in [56] Saad [144] proposed a few variants on the incomplete LU approach for the matrix A, one of which is in fact an incomplete LQ decomposition. In this approach it ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48:479--498, 1986.

....finite element problems, MILU(s,1) gives a preconditioner which is exact on constant functions. Although it can be shown to have better order of accuracy on model problems, on practical problems it generally gives a worse preconditioner; for an explanation of this phenemonen, see the results in [5]. ILUT is Yousef Saad s algorithm (and implementation) of thresholding as a dropping criterion. This allows dropping of elements during the incomplete factorization based on their relative size instead of their positions within the matrix. See the documentation in the routine precond ilut.f for ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48 (1986), pp. 479--498.

.... required in the general algorithm given in [19] We give the algorithm for the DRIC method [18, 21] which we chosed for the numerical tests in the next section; it is easily converted in an algorithm for the IC method [15] the MIC method (without perturbations) see e.g. 8] or the RIC method [3] by using respectively i j 0, i j 1 or i j for some 0 1. ff is a parameter such that 0 ff 1 (see Section 5) and oe an auxiliary vector. FOR ALL p : ffl initialize, for all i i = a (p) ii oe i = X j i a (p) ij 8 ffl set Gamma Sigma f ( oe Gamma Sigma f ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48 (1986), pp. 479--498.

....The preconditioning matrix B is selected as the generalized relaxed incomplete LPL t factorization described in Fig. 2. The set D specifies where fill in entries have to be ignored, while the ae j are the relaxation parameters: ae j = ae, Gamma1 ae 1. This corresponds to the relaxed method [5], which includes the standard incomplete Cholesky factorization (ae = 0) 31,32] as well as the classical modified variant (ae = 1) for which Be = Ae [18,24] The variables ae j encompass dynamically relaxed methods [7,35,28] Two basic strategies for accepting or discarding fill in have been ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math. 48 (1986) 479--498.

....the system matrix itself. Next, a symbolic factorization is performed to determine the structure of the triangular factors, i.e. fix the set of locations in which fill in will be permitted. Finally, the numerical factorization is performed according to either the IC, the (unperturbed) MIC, the RIC [1], the DMIC [3] or the DRIC [7] algorithm. This will be further described in Section 3. ITSOL is much embedded to the SPARSKIT toolkit by Y. Saad [9] Most routines for general sparse matrix computation have been retrieved from that module. Some of them have been slightly modified to better match ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48 (1986), pp. 479--498.

....elements of L and U are calculated by the following rule: LU) ij = a ij for all i Gamma j 2 P n fill and j Gamma i 2 P n fill nf0g. For problems where the solution is a smoothly varying function, it is a good idea to use the Modified ILU preconditioner [11] or the Relaxed ILU preconditioner [2] instead of the classic ILU preconditioner [12] The RILU fill(ff) preconditioner is an average of the ILU fill and MILU fill preconditioner. In the MILU fill preconditioner the following rules are used: 10 Table 1: The methods used in the comparison Not. Description Problem 1 2 3 4 5 6 D ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Num. Math., 48:479--498, 1986.

.... is based on the divide and conquer method for linear systems [21] there is freedom in the choice of the modification, which can be used to improve the eigenvalue distribution and the condition number for the preconditioned matrix which is again advantageous for the convergence of the CG method [1, 2, 24]. We will present some numerical examples run on a parallel computer to demonstrate that the CG method with one of the modified Jacobi preconditioners is always faster than the standard one and the CG method with another of the modified Jacobi preconditioners is faster than the standard one under ....

....the interval (0; 1] with a cluster of eigenvalues at 1. In contrast all eigenvalues of the standard block Jacobi preconditioner for block tridiagonal matrices are in the interval (0; 2] For this case a detailed analysis on the convergence rate of the CG method was given by Axelsson and Lindskog [1, 2]. They showed that if the spectrum of S Gamma1 A satisfies spec(S Gamma1 A) ae [a; b] f1g; 9) where 0 a b 1, then the number of iterations of the CG method needed to obtain kx Gamma x (k) k A 1=2 fflkx Gamma x (0) k A 1=2 is at most k = int[ 1 2 r b a ln (2=ffl) 2] ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48:479--498, 1986.

....METHOD GU GUIDING Numerical Analysis Group, Trinity College, Dublin 2, Ireland Abstract: When the Conjugate Gradient Squared (CGS) method is used to solve a linear system, an efficient preconditioning method is expected to accelerate its speed. The relaxed incomplete LU factorisation (RILU) [9] is such a preconditioner. This report presents the conditions for the existence of the RILU, and analyses the condition in [8] for the stability of the computations involving the triangular factors in the preconditioning operation in the iterative process of the preconditioned CGS. A choice of ....

....quickly by solving (1.2) rather than (1.1) The problem is how to choose a better C to make a smaller (C Gamma1 A) Incomplete LU factorisation ( ILU(k) 2] 5] and modified incomplete LU factorisation (MILU(k) 3] are two efficient preconditioning methods. After both methods, Axelsson [9] presented the relaxed incomplete LU factorisation (RILU( k) which contains a parameter (0 1) For = 0 the method reduces to the ILU(k) while for = 1 the method reduces to the MILU(k) Moreover, it is said in [3] that the MILU(k) is more efficient, i.e. x can be obtained more quickly, ....

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O. Axelsson and G. Lindskog, (1986). "On the eigenvalue distribution of a class of preconditioning method," Numer.Math. v. 48, pp.479--498.

....solvers. Since ILU and IC factorizations were the most popular preconditioners, at least in a sequential environment, many attempts have been made to improve them, for instance by including more fill [115] by modifying the diagonal of the ILU factorization in order to force rowsum constraints [82, 7, 6, 118, 164, 57], or by changing the ordering of the matrix [165, 166] A set of experiments with respect to the effects of ordering is contained in [52] Saad [140] proposed a few variants on the incomplete LU approach for the matrix A, one of which is in fact an incomplete LQ decomposition. In this approach it ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48:479--498, 1986.

....this produces the same incomplete factor R as IMGS, and therefore inherits the robustness of IMGS. The approach of relaxed IMGS is based on CIMGS extended by a relaxation parameter 2 [0; 1] The idea is similar with RILU, Relaxed Incomplete LU factorization proposed by Axelsson and Lindskog in [1, 2]. For a fixed sparse pattern, the parameter value = 0 in relaxed IMGS reproduces the corresponding CIMGS factorization. Algorithmically, let B = A T A. When A is a real matrix with full rank, B is symmetric positive definite. Given a drop set P Pn , Relaxed IMGS generates the upper ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numerische Mathematik, 48:479--498, 1986.

....a bound on the largest eigenvalue of the preconditioned system; see [4, Ch. 10] for a review. Experimental Study of ILU 13 These methods apply to elliptic problems in one variable, where they lower the order of the spectral condition number of the preconditioned matrix. Relaxed ILU (RILU) [2, 3] parameterizes the fraction of the modification to perform, giving it the same effect as the perturbation. Negative relaxation factors in RILU have a stabilizing effect for M matrices. They were used for multigrid smoothing by Wittum [42] in his ILU fi method. The diagonal is augmented with fi ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Num. Math., 48:479--498, 1986.

....e.g. 1, 2, 6, 11, 19, 20, 35] for solving system (1. 2) Relaxed incomplete factorizations (RILU) are powerful preconditioning techniques that interpolate, through a relaxation parameter, between the popular incomplete LU factorization ILU and its modified variant MILU that preserves rowsums of A [4, 13]. As opposed to ILU and MILU, the two main advantages of RILU are following : y Research supported by the Commission of the European Communities HCM Contract No. ERBCHBG CT93 0420, at Utrecht University, Mathematical Institute, The Netherlands. Current address : Universit e Libre de Bruxelles, ....

....Roosevelt, B 1050 Brussels, Belgium. ynotay ulb.ac.be. Research supported by the Fonds National de la Recherche Scientifique , Belgium. 2 M.M. MAGOLU AND Y. NOTAY 1. it does not suffer a lot from existence problem [13, 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 ....

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O. Axelsson and G. Lindskog, On the eigenvalue distribution of class of preconditioning methods, Numer. Math., 48 (1986), pp. 479--498.

....a given environment that emit light, i.e. few patches i with E i 6= 0. To use this starting guess we replace line 2 of the above algorithm by: B (0) i = E i ae i Ambient The speed of convergence of the CG method depends on the overall distribution of eigenvalues of the matrix G. Upper bounds [1] for the number of iterations are often based on the condition number of G. Since G is symmetric and positive definite, its condition number is = max (G) min (G) The number k of iterations needed to reduce the residual by a factor of ffl in the G norm is then bounded by k 0:5 p ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Tech. Rep. Report 3, Goteborg Numerical Analysis, 1989.

....conjugate gradient solution of large sparse linear systems arising from the discretization of second order elliptic PDEs. Incomplete factorization preconditionings are quite popular in this context, and several recent works focused on the design of efficient techniques, most of them (see e.g. [3, 6, 8, 16, 21, 24]) making some subtle compromise between the standard or unmodified Incomplete Cholesky (IC) method, and the modified (unperturbed) version (MIC) where the diagonal part of the triangular factors is computed so that the preconditioner satisfies a row sum criterion. Assuming a lexicographic ....

....system; the numerical tests presented in [24] effectively display the poor behaviour of that family of methods under such circumstancies. This gave rise to the development of a further technique [24] referred to as DRIC, and which uses a dynamic version of AxelssonLindskog relaxation [6] to reproduce the behaviour of the former methods in isotropic regions, while bringing a pronounced improvement in anisotropic ones by taking benefit of the small size of the fill entries. We now recall for completeness the algorithm associated to DRIC. We use a pseudo programming language to ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48 (1986), pp. 479--498.

....by the results obtained in [8] In that paper, GMRES was used as an approximate solver, with tolerances varying from 10 Gamma4 to 10 Gamma1 ; also a blockwise application of the RILUD preconditioner was used. The RILUD preconditioner is a diagonal version of the method introduced in [1]. This factorization is an average of an ILUD preconditioner [27] and an MILUD preconditioner [15] with a weight7 ing parameter , assigned a value of 0.95 in our experiments. See also [43] for useful results with RILU factorizations applied to Navier Stokes equations. The use of incomplete ....

O. Axelsson and G. Linskog. On the eigenvalue distribution of a class of preconditioning methods. Numerische Mathematik, 48:479--498, 1986.

....finite element problems, MILU(s,1) gives a preconditioner which is exact on constant functions. Although it can be shown to have better order of accuracy on model problems, on practical problems it generally gives a worse preconditioner; for an explanation of this phenemonen, see the results in [5]. ILUT is Yousef Saad s algorithm (and implementation) of thresholding as a dropping criterion. This allows dropping of elements during the incomplete factorization based on their relative size instead of their positions within the matrix. See the documentation in the routine precond ilut.f for ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48 (1986), pp. 479--498.

.... CGNR and CGNE, CGS, BiCGstab, GMRES, TFQMR, templates version of CGS, templates version of GMRES, Jacobi, Gauss Seidel, SOR, Orthomin Preconditioners Identity, ILU(s) MILU(s; r) ILUT(s; t) SSOR( TRID(s) ILU0, ECIMGS; where s is the number of levels of fill, r is the relaxation parameter [1], t is the drop tolerance. Data structures Compressed Sparse Row Manual 26 pages Example codes Driver program that read a Harwell Boeing matrix and solves a problem with it. 15 ftp: ftp.cs.indiana.edu pub bramley splib.tar.gz 3.15 Templates The templates codes are meant as example ....

Owe Axelsson and Gunhild Lindskog. On the eigenvalue distribution of a class of preconditioning matrices. Numer. Math., 48:479--498, 1986.

....equation Re1 1D Richard s equation Table 0.1: Identifiers for the test problems. Identifier Solution method Reference(s) BG Gauss elim. on banded matrix [16] J Jacobi iterations [26] 27] GS Gauss Seidel iterations [26] 27] CG Conjugate gradient method [14] 20] MPCG CG MILU prec. [4], 17] RPCG CG RILU prec. 0:9) 4] 17] FPCG CG fast Fourier trans. prec. 25] NMG Nested multigrid cycles [18] Table 0.2: Identifiers for the solution methods for linear systems. CPU MEASUREMENTS OF SOME NUMERICAL PDE SIMULATIONS 3 1 Introduction The primary purpose of this work ....

....0.1: Identifiers for the test problems. Identifier Solution method Reference(s) BG Gauss elim. on banded matrix [16] J Jacobi iterations [26] 27] GS Gauss Seidel iterations [26] 27] CG Conjugate gradient method [14] 20] MPCG CG MILU prec. 4] 17] RPCG CG RILU prec. 0:9) [4], 17] FPCG CG fast Fourier trans. prec. 25] NMG Nested multigrid cycles [18] Table 0.2: Identifiers for the solution methods for linear systems. CPU MEASUREMENTS OF SOME NUMERICAL PDE SIMULATIONS 3 1 Introduction The primary purpose of this work is to collect information on CPU time ....

[Article contains additional citation context not shown here]

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48:479--498, 1986.

....(that is, one often takes c = 0) this may lead to ineffective preconditioning or even break down of the preconditioner, see Eijkhout (1992) In our context, the rowsum requirement in (2.3) amounts to an additional correction to the diagonal entries d i , compared to those computed in (2. 2) Axelsson and Lindskog (1986) describe a relaxed form of this modified incomplete decomposition that, for the five diagonal A, leads to the following relations for the d i : d i = a i;1 Gamma a i Gamma1;2 (a i Gamma1;2 ffa i Gamma1;3 ) d i Gamma1 Gamma a i Gamman x ;3 (a i Gamman x ;3 ffa i Gamman x ;2 ) d i Gamman x ....

....elliptic problems, in practice MILU does not always perform better than ILU. This may have to do with the higher sensitivity of MILU to round off errors (van der Vorst 1990a) This provides motivation for an interpolated version between ILU and MILU, see for example Ashcraft and Grimes (1988) and Axelsson and Lindskog (1986). The idea is that in the MILU algorithm, the update of 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 ....

Axelsson, O. and Lindskog, G. (1986), `On the eigenvalue distribution of a class of preconditioning methods', Numerische Mathematik 48, 479--498.

....requirements. iterations CPU time memory method vectors full GMRES 31 0.61 31 GMRESR(5) 7 0.35 19 GMRESR(8) 4 0. 37 16 Table 8: Iterative methods applied to the momentum equations We have solved the pressure equations with a combination of GMRESR(m) with a (M)ILU preconditioner (see [10] 16] [1], and [8] In Table 9 we show results, using an average of an ILU and a MILU preconditioner with ff = 0:975 (for a motivation of this, see [19] p.8) For large problems (32 Theta 128) GMRESR(m) is much better than full GMRES. Since GMRES(m) converges very slowly for these examples we have not ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48 (1986), pp. 479--498.

....in this case. Hence, the eigenvalues of M Gamma1 h A ffi;h satisfy 2 fffig [ 0 ; 1] 3.9) where we recall that 0 is a positive constant independent of ffi and h. Positive de nite problems with eigenvalue distributions in the form (3. 9) have been thoroughly studied by Axelsson and Lindskog [3, 4]. They showed that an error bound of the form kp h Gamma p (k) kA ffi kp h Gamma p (0) kA ffi will be reached in at most k = ln(2= ln(1=ffi) ln(oe Gamma1 ) 1 (3.10) CG iterations. Here, 0 is the error level and oe is given by oe = 1 Gamma p 0 1 p 0 : Thus we ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48:479498, 1986.

....of problems. Among all these preconditioners the Incomplete LU factorizations [69, 21] are the most popular ones, and attempts have been made to improve them, for instance by including more fill [70] or by modifying the diagonal of the ILU factorization in order to force rowsum constraints [58, 6, 5, 73, 95, 34], or by changing the ordering of the matrix [96, 97] A collection of experiments with respect to the effects of ordering is contained in [30] More recently, it was discovered that a multigrid inspired ordering can be very effective for discretized diffusion convection equations, leading in some ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48:479--498, 1986.

....the test problems included in the report. Identifier Solution method Reference(s) BG Gauss elim. on banded matrix [16] SG Gauss elim. on sparse matrix [13] 15] J Jacobi iterations [25] 26] GS Gauss Seidel iterations [25] 26] CG Conjugate gradient method [14] 19] MPCG CG MILU prec. [3], 17] RPCG CG RILU prec. 0:9) 3] 17] FPCG CG fast Fourier trans. prec. 24] NMG Nested multigrid cycles [18] Table 0.2: Identifiers for the solution methods for linear systems. 1 Introduction The primary purpose of this report is to collect some information on the CPU time ....

....Identifier Solution method Reference(s) BG Gauss elim. on banded matrix [16] SG Gauss elim. on sparse matrix [13] 15] J Jacobi iterations [25] 26] GS Gauss Seidel iterations [25] 26] CG Conjugate gradient method [14] 19] MPCG CG MILU prec. 3] 17] RPCG CG RILU prec. 0:9) [3], 17] FPCG CG fast Fourier trans. prec. 24] NMG Nested multigrid cycles [18] Table 0.2: Identifiers for the solution methods for linear systems. 1 Introduction The primary purpose of this report is to collect some information on the CPU time consumptions in a series of typical numerical ....

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O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48:479--498, 1986.

....not work for the typical application shown in Sect. 4. Therefore, in this paper we use a new and efficient implementation of a Block Incomplete LU (BILU) decomposition as a preconditioner for the restarted version of GMRES [9] It will be shown that modification of the blocks on the main diagonal [7, 3] can improve the convergence behaviour. Section 2 describes the time discretisation that results in the linear systems, in Sect. 3 the new preconditioner is introduced, and in Sect. 4 we demonstrate the performance of the method on a two dimensional test problem. In the last section we draw some ....

....are known to be powerful for reducing the number of iteration steps. Unfortunately, the performance of such methods is sometimes spoiled by the largest eigenvalues of the preconditioned matrix. Therefore, we use a block form of the so called relaxed modified incomplete decomposition [3]. The fillin blocks that are not allowed in the factors L and U are first multiplied by a constant factor ff, before they are added to the main diagonal blocks. This constant ff is chosen in the interval [0,1] For ff = 0 we obtain the standard BILUdecomposition without modification of the main ....

Axelsson, O., G. Lindskog, G.: On the eigenvalue distribution of a class of preconditioning methods. Numer. Math. 48 (1986) 479--498

....ICCG(1) or ICCG(1,2) method. The incomplete factorization can be modified in various way: adding the deleted entries to the diagonal entries in the same row, or modifying the pivot entry by adding a positive (small) number to it. For further study, see Gustafsson [18, 17] Axelsson and Lindskog [4] and Axelsson [7] The ICCG method needs the solution of lower and upper triangular matrices, which is not readily vectorizable or parallelizable. Johnson and Paul [30] and van der Vorst [54] have proposed some variants for vector machines. These variants have ample data parallelism, however these ....

.... a[i] j 1] and a[i 1] j] In this case, the number of the inner points is O(n) and the number of the boundary for (i = 1; i N 1; i ) for (j = 1; j M 1; j ) b[i] j] dd[i] j] 0] a[i 1] j] dd[i] j] 1] a[i] j 1] dd[i] j] 2] a[i] j] dd[i] j] 3] a[i] j 1] dd[i] j][4] a[i 1] j] Figure 9.1: Fragment of pentadiagonal matrix vector multiplication compute inner points send data receive data compute edge points Node ID 0 network latency compute inner points send data receive data compute edge points Node ID 1 Figure 9.2: Nodal code for overlapping ....

Axelsson, O. and G. Lindskog, "On the eigenvalue distribution of a class of preconditioning methods," Numer. Math., vol. 48, pp. 479--498, 1986.

....a noticeable effect on (B ( Gamma1 A ( Therefore, we included only two basic smoothing strategies in our numerical experiments, each of them with just one pre and one post smoothing iteration. These are damped Jacobi smoothing with = 2 on the one hand and Relaxed ILU( Gamma1) [9] on the other hand. Both satisfy max (R (k) A (k) 1 when A (k) is an M matrix [24, 30] Damped Jacobi is fully parallelizable but performs poorly in presence of anisotropy, whereas RILU based smoothers are robust for anisotropic problems, at least in 2D [36] For both choices, ....

....9n flops) the total number of operations per iteration is thus about 53n when c o = c s = 1 , 47n when c o = 1 ; c s = 0 and 40n when c o = c s = 0 . By comparison, it is about 29n flops with an ILU(0) like preconditioner and 33n flops with a block ILU like preconditioner (e.g. [9] ) For 3D problems and 7 point matrices one has, with a standard coarsening, nz(A (k) 7n k , nz(A (k) 11 ) 37 8 n k and n k 8 k Gamma n . Therefore, c( 2 7 (37c o 56 14(c s 1) 1 Gamma c s ) n = 21 10c o 4c s Gamma 5 2c s Gamma 4c o 7 n ; ....

O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numer. Math., 48 (1986), pp. 479--498.

....to reduce the condition number of the preconditioned operator W Gamma1 l A l , which is important for CG like methods. In [Gus78] MILU was applied to matrices for which the off diagonal elements of the rest matrix of the MILU decomposition are positive so that MILU equals ILU Gamma1 . In [AL86], MILU was generalized to RILU, characterized by r ii = P j 6=i r ij for some 2 [ Gamma1; 0) Numerische Mathematik Electronic Edition page numbers may differ from the printed version page 296 of Numer. Math. 68: 295 309 (1994) Modified ILU as a smoother 297 In [OS89] and [Kha89] ....

Axelsson, O., Lindskog, G. (1986): Eigenvalue distribution of a class of preconditioning methods. Numer. Math. 48, 479--498

....MILU does not always perform better than ILU. Part of the reason is the favourable eigenvalue distribution for ILU and part of it has to do with the higher sensitivity of MILU to round off errors [94] This provides motivation for an interpolated version between ILU and MILU, first proposed in [10, 1]. Namely, in the MILU algorithm, the 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 ....

O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning matrices. Numer. Math., 48:479--498, 1986.

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O. Axelsson and G. Lindskog, On the eigenvalue distribution of a class of preconditioning methods, Numerische Mathematik, 48 (1986), pp. 479--498.

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O. Axelsson and G. Lindskog. On the eigenvalue distribution of a class of preconditioning methods. Numer. Math., 48:479--498, 1986.

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