I. Gustafsson, A class of first order factorization methods, BIT, 18 (1978), pp. 142--156.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....given where the usage of a fixed but deflated preconditioner is still very efficient for the twentieth eigenvalue (when ordered algebraically) notwithstanding the increased costs per iteration involved with deflation against the previously determined 19 eigenvectors. It is well known that MILU [5] often leads to a significant reduction in iteration steps when used as a preconditioner for Krylov subspace iteration methods for the solution of discretized PDE s with a relatively smooth solution. For an incomplete decomposition of the 5 point discretized Poisson operator, over a rectangular ....

I. Gustafsson, A Class of First Order Factorization Methods, BIT, 18 (1978), pp. 142--156.

.... Gamma x 1 ( Delta ) x 2 ) 0 subject to the same boundary conditions. This equation was discretized with finite differences on a 25 Theta 25 grid, see Fig. 4. The grid lines are distributed as the roots of the Chebychev polynomial of degree 25. As preconditioner we used the Modified ILU(2) [13] decomposition of the biharmonic operator Delta . Starting from the solution for Re = 0, we computed several solutions, using the the arc length continuation method (cf. the previous example, and [12] with step sizes Deltas = 100 for 0 Re 1400, Deltas = 200 for 1400 Re 2600, and ....

I. Gustafsson, A class of first order factorizations methods, BIT, 18 (1978), pp. 142--156.

....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 ....

I. Gustafsson. A class of first order factorization methods. BIT, 18:142-- 156, 1978.

....developments along this line were made by Beauwens [19] Axelsson [2] considered these methods as generalized relaxation methods. Meijerink and Van der Vorst [75] considered these methods as incomplete factorizations and they proved the existence of ILU preconditioners for M matrices. Gustafsson [63] proposed a modified version of the ILU preconditioner with improved spectral properties. Finally, the paper of Kershaw [69] provided convincing numerical experiments to show the effectiveness of these methods. Besides these historical references, there are several more easily accessible ....

....observed by Dupont, Kendall and Rachford [53] for elliptic PDEs and they proposed a simple modification which dramatically improves the performance as h tends to zero. We shall next describe the generalization of this modified ILU (MILU) preconditioner to a general matrix A due to Gustafsson [63]. The basic idea is extremely simple: in the condition (3) for ILU, the condition m i;i = a i;i is removed and a new row sum condition is added. That is, 3) is replaced by: m i;j = a i;j 8i and m i;j = a i;j if i 6= j and (i; j) 2 S: 5) Again, for certain classes of matrices, the ....

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I. Gustafsson. A class of first-order factorization methods. BIT, 18:142--156, 1978.

....[19] The rate of convergence and the efficiency of Krylov subspace methods is often improved by preconditioning. The identification of an effective preconditioner may be a problem. For instance, for interior eigenvalues the construction of an effective incomplete LU factorization [10] [8] for A Gamma e I may require much fill in , which makes the construction expensive. As we will argue in x2.7, it may be a good strategy to compute a preconditioner K for A Gamma I for a fixed value of only, and to use K j (I Gamma e q e q ) 13) as the preconditioner for various e q ....

I. Gustafsson, A class of first order factorizations methods, BIT, 18 (1978), pp. 142--156.

....Several popular preconditioners require the solution of two sparse triangular systems, usually by substitution. Such preconditioners include Symmetric Successive Over Relaxation (SSOR) 120] Incomplete Cholesky preconditioners [84] and Modified Incomplete Cholesky preconditioners [56]. We focus on preconditioning linear systems of equations Ax = b, where A is derived from a 5 point stencil in two dimensions using a preconditioner of the form , where L is a sparse lower triangular matrix, whose sparsity pattern is identical to the sparsity pattern in the lower ....

Ivar Gustafsson. A class of first order factorization methods. BIT, 18:142--156, 1978.

....that are sufficiently small to satisfy ja ij j cja ii a jj j ; 4.49) 23 are neglected, where a ij is (i; j) element at the (r 1)th elimination stage and 0 c 1. A common choice of the sparsity set is to let S = S 0 = f(i; j) j a ij 6= 0g: 4. 50) According to Gustafsson [18], the sparsity set of order q is defined in the following way. Let R q be the sparsity set that is defined by the position of the fill in entries of the incomplete factorization based on the sparsity set S q , and let S q 1 = S q R q (q = 0; 1; Discretizing two dimensional rectangular ....

....factorization is called the 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 ....

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Gustafsson, I., "A class of first order factorization methods," BIT, vol. 18, pp. 142--156, 1978.

....has a physical justification [4, Section 3.4. 2] More importantly, it has been shown that when A represents a regular 5 point discretization of Laplace s equation with Dirichlet boundary conditions, a no fill modified incomplete factorization reduced the condition number from O(n) to O( # n) see [4, 6, 16] and their reference) An unmodified factorization reduces the condition number by a constant factor, but not asymptotically [4, 16] 1.5. REORDERING MATRICES FOR SPARSITY 9 1.4. Matrices And Graphs We normally think of matrices as two dimensional arrays of numbers or as representations of linear ....

.... of Laplace s equation with Dirichlet boundary conditions, a no fill modified incomplete factorization reduced the condition number from O(n) to O( # n) see [4, 6, 16] and their reference) An unmodified factorization reduces the condition number by a constant factor, but not asymptotically [4, 16]. 1.5. REORDERING MATRICES FOR SPARSITY 9 1.4. Matrices And Graphs We normally think of matrices as two dimensional arrays of numbers or as representations of linear transformations. We can also represent matrices as graphs. Definition 1.4.1. The underlying graph GA = V A , EA ) of an n by n ....

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I. Gustafsson. A class of first-order factorization methods. BIT, 18:142--156, 1978.

....that it performs. It can perform complete, no fill incomplete (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 ....

I. Gustafsson, A class of first-order factorization methods, BIT, 18 (1978), pp. 142--156.

....for this project. The code can perform complete, no fill incomplete (sometimes known as ICC(0) or ICCG(0) 12] and droptolerance 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 [8]. The code implements a sparse left looking algorithm. The code is e#cient in the sense that its running time is proportional to the number of floating point operations that it performs. However, the code is not supernodal and is not blocked in any other way, so its performance is poorer than that ....

I. Gustafsson. A class of first-order factorization methods. BIT, 18:142--156, 1978.

....is not considered in this paper. If the subdomain problems are solved (inaccurately) using GMRES, this method reduces to GMRESR [46] for the single domain case. In case A ii = L i U i is the (relaxed) incomplete LU factorization of A ii , we obtain a blocked version of the subdomain RILU(ff) [27] postconditioner (with parameter ff) here called RIBLU(ff) Relaxed Incomplete Block LU) The parameter ff may be varied to improve convergence. The RIBLU(ff) preconditioner is investigated for parallel implementation in for example [21, 32, 18, 19] The present paper also investigates the ....

....Poisseuille problem 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 ....

I. Gustafsson. A class of first order factorization methods. BIT, 18:142--156, 1978.

.... 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 ....

I. Gustafsson. A class of first order factorization methods. BIT, 18:142--156, 1978.

....another family of preconditioners. This paper explains the technique, extends it further, and uses it to analyze two classes of known preconditioners for model problems. Specifically, we use the extended technique to analyze certain modified incomplete Cholesky (MICC) preconditioners (see [8]) and multilevel diagonal scaling (MDS) preconditioners (see [10] for example) The principle goal of this paper is to bring these techniques to the attention of a wider community of researchers. By doing so, we hope to encourage further work in this promising area. The primary original content ....

....problem, a Laplace equation with Neumann boundary conditions on a regular grid. The following result is, to the best of our knowledge, new. It is known, however, that this same asymptotic condition number bound holds for modified incomplete factorizations of a perturbed matrix A c n with c 0 [8]. We are aware of no previous proof for the case when c = 0. Consider the regular grid depicted by the solid lines in Figure 6.1. If we perform an elimination of the vertices in the natural order and discard all fill, then the discarded values will correspond to the dashed diagonals in the ....

I. Gustafsson, A class of first-order factorization methods, BIT, 18 (1978), pp. 142--156.

....for this project. The code can perform complete, no fill incomplete (sometimes known as ICC(0) or ICCG(0) 12] 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 [8]. The code implements a sparse left looking algorithm. The code is e#cient in the sense that its running time is proportional to the number of floating point operations that it performs. However, the code is not supernodal and is not blocked in any other way, so it s performance is poorer than ....

I. Gustafsson. A class of first-order factorization methods. BIT, 18:142--156, 1978.

....another family of preconditioners. This paper explains the technique, extends it further, and uses it to analyze two classes of known preconditioners for model problems. Specifically, we use the extended technique to analyze certain modified incomplete Cholesky (MICC) preconditioners (see [8]) and multilevel diagonal scaling (MDS) preconditioners (see [10] for example) The principle goal of this paper is to bring these techniques to the attention of a wider community of researchers. By doing so, we hope to encourage further work in this promising area. The primary original content ....

....problem, a Laplace equation with Neumann boundary conditions on a regular grid. The following result is, to the best of our knowledge, new. It is known, however, that this same asymptotic condition number bound holds for modified incomplete factorizations of a perturbed matrix A c=n with c 0 [8]. We are aware of no previous proof for the case when c = 0. Consider the regular grid depicted by the solid lines in Figure 6.1. If we perform an elimination of the vertices in the natural order and discard all fill, then the discarded values will correspond to the dashed diagonals in the ....

I. Gustafsson, A class of first-order factorization methods, BIT, 18 (1978), pp. 142--156.

....triangular, DA is a diagonal matrix and UA is strictly upper triangular. In this case we take M = D LA )D Gamma1 (D UA ) with D such that A ii = M ii , i.e. A and M have identical main diagonals, D ii = A ii Gamma X j i A ij D Gamma1 jj A ji : 1) A variation on ILUD, called MILUD [5] uses a slightly different criterion to determine the elements of D. We still use M = D LA )D Gamma1 (D UA ) but now D is such that P j A ij = P j M ij , i.e. A and M have identical 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 ....

I.A. Gustafsson. A class of first order factorization methods. BIT, 18:142--156, 1978.

....fill. Many variations are possible. For example, we could define S so that L is a banded matrix with predetermined bandwidth. These strategies have predictable memory requirements but are independent of the entries of A because the dropped elements depend only on the structure of A. Gustafsson [20] introduced the ILU(p) factorization, where p is the level of fill. In the ILU(0) factorization the sparsity pattern S is set to the sparsity pattern of the original matrix, but additional fill is allowed for p 0. The original definition of level of fill did not take into account the entries ....

....sparsity patterns of L and U T are di#erent. In particular, the product LU produced by ILUT is unlikely to be symmetric for a symmetric matrix A. There are several variations of the approaches that we have presented. In particular, in the modified incomplete Cholesky factorization of Gustafsson [20], dropped elements are added to the diagonal entries of the column. With this modification Re = 0, where e is the vector of all ones. For additional information on modified incomplete Cholesky factorizations, see Gustafsson [21, 22] Hackbusch [23] and Saad [37] Other variations arise from the ....

I. Gustafsson, A class of first order factorization methods, BIT, 18 (1978), pp. 142--156.

....to the whole area of iterative methods. The Dupont Kendall Rachford splitting can be viewed as an incomplete LU factorization with zero fillin, in which the elimination errors are compensated by corrections to the diagonal of the decomposition. In 1977 this procedure was generalized by Gustafsson [87] in 1978, as a modified form of the incomplete LU factorizations: MILU. Several developments marked the years that followed. Two distinct ways of developing incomplete factorization preconditioners with improved accuracy were developed. The first approach is based on a symbolic factorization ....

.... 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 ....

I. Gustafsson. A class of first order factorization methods. BIT, 18:142--156, 1978.

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I. Gustafsson, A class of first order factorization methods, BIT, 18 (1978), pp. 142--156.

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I. Gustafsson, A Class of First Order Factorization Methods, BIT 18 (1978), pp. 142-156.

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I. Gustafsson, A Class of First Order Factorization Methods, BIT 18 (1978), pp. 142-156.

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I. Gustafsson, A Class of First Order Factorization Methods, BIT 18, pp. 142-156, 1978.

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I. Gustafsson, A Class of First Order Factorization Methods, BIT 18, (1978) 142--156.

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I. Gustafsson, A class of first order factorization methods, BIT, 12 (1978), pp. 142--156.

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I. Gustafsson. A class of first order factorization methods. BIT, 18:142--156, 1978.

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