M. Benzi, D. B. Szyld and A. VanDuin, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., 20-5 (1999), pp. 1652-1670.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....on the ordering selected, a natural question concerns the sensitivity of the quality of the preconditioner to this. In particular in [14] it is shown that the numerical behaviour of IC is very much dependent on the ordering and a similar study and comparable conclusion with AINV is described in [8]. In Table 9, we display the number of iterations with SQMR, selecting the same density parameters as those used for the experiments reported in Table 9, but using di#erent orderings to permute the original pattern of MSym Frob . More precisely we consider the reverse Cuthil MacKee ordering [13] ....

M. Benzi, D.B. Szyld, and A. van Duin. Orderings for incomplete factorization preconditioning of nonsymmetric problems. SIAM J. Scientific Computing, 20:1652--1670, 1999.

.... 26] Intuitively, if one is computing an incomplete factorization, an ordering which tends to minimize the ll in in a complete factorization should tend to minimize the error E = P t AP (L D)D 1 (D U) For particular classes of matrices, specialized ordering schemes have been developed [34, 15, 37, 36]. For example, for matrices arising from convection dominated problems, ordering along the ow direction has been used with great success. However, in this general setting, we prefer to use just one strategy for all matrices. This reduces the complexity of the implementation, and avoids the ....

M. Benzi, D. B. Szyld, and A. van Duin, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., 20 (1999), pp. 1652-1670.

....Immediately following this was a paper by Duff and Meurant [56] which concluded, similarly, that ICCG does not in general benefit in any significant manner form reordering. These studies were limited to certain types of reorderings and certain types of preconditioners. It is now known [18] that certain reorderings, such as Reverse Cuthill McKee are beneficial in preconditioning methods, in particular with some form of dropping strategy. The beneficial impact of well chosen fill ins was already demonstrated in [56] for some orderings. What seems to be also clear is that the best ....

M. Benzi, D.B. Szyld, and A. van Duin. Orderings for incomplete factorization preconditioning of nonsymmetric problems. SIAM Journal on Scientific Computing, 20:1652--1670, 1999.

....prescribed sparsity pattern. See Saad (1996) for an overview. Incomplete factorizations are used so that the resulting factors are more economical to store, to compute, and to solve with. One of the reasons why incomplete factorizations can behave poorly is that pivots can be arbitrarily small (Benzi, Szyld and van Duin 1997, Chow and Saad 1997) Pivots may even be zero in which case the incomplete factorization fails. Small pivots allow the numerical values of the entries in the incomplete factors to become very large, which leads to unstable and therefore inaccurate factorizations. In such cases, the norm of the ....

Benzi, M., Szyld, D. B. and van Duin, A. (1997), Orderings for incomplete factorization preconditioning of nonsymmetric problems, Technical Report LA-UR-97-3525, Los Alamos National Laboratory, Los Alamos, NM.

....can be seen in the form of full or partial pivoting strategies in Gaussian elimination. In ILU factorization, column pivoting strategy has been implemented with Saad s ILUT, resulting in an ILUTP variant [35] However, ILUTP has not always been helpful in dealing with nonsymmetric matrices [3, 9]. As Chow and Saad pointed out [9] a poor pivoting sequence can occasionally trap a factorization into a zero pivot, even if the factorization would have succeeded without pivoting. In addition, existing pivoting strategies for incomplete factorization cannot guarantee that a nonzero pivot always ....

M. Benzi, D. B. Szyld, and A. van Duin. Orderings for incomplete factorization preconditioning of nonsymmetric problems. SIAM J. Sci. Comput. to appear.

....[3] 3. Block ordering strategies for the reduced grid. The question of ordering is of major importance, as a good ordering strategy can lead to fast convergence. An excellent overview of the literature that deals with ordering strategies is found in a recent report by Benzi, Szyld, and van Duin [1]. For 3D problems it seems useful to consider the ordering of blocks of unknowns, rather than pointwise ordering. Such a strategy could be particularly useful for the cyclically reduced problem, as the reduced grid is somewhat irregular. Instead of ordering the unknowns directly in the 3D ....

M. Benzi, D. B. Szyld, and A. van Duin, Orderings for incomplete factorization preconditioning of nonsymmetric problems. SIAM J. Sci. Comput., 20 (1999), pp. 1652--1670.

....algorithms. Reordering methods are originally devised for symmetric positive definite systems to reduce fill in in the factorization. With proper modification, they can also be used in nonsymmetric systems. But experiments show reordering methods are ine#ective to diagonally dominant matrices [17]. Since the construction of e#cient general purpose preconditioner is not possible, we use the combination of several criterions to choose S. 1. Band Criterion All nonzeros in the diagonal zone with width of BANDWD #n will be Figure 3.1: Preconditioner structures in the 1st iteration 20 ....

M. Benzi, D. B. Szyld, A. V. Duin, Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems, SIAM J. SCI. COMPUT. , Vol.20, No.5, 1652-1670, 1999.

....can be seen in the form of full or partial pivoting strategies in Gaussian elimination. In ILU factorization, column pivoting strategy has been implemented with Saad s ILUT, resulting in an ILUTP techniques [32] However, ILUTP has not always been helpful in dealing with nonsymmetric matrices [3, 9]. As Chow and Saad pointed [9] a poor pivoting sequence can occasionally trap a factorization into a zero pivot, even if the factorization would have succeeded without pivoting. In addition, existing pivoting strategies for incomplete factorization cannot guarantee that a nonzero pivot can always ....

M. Benzi, D. B. Szyld, and A. van Duin. Orderings for incomplete factorization preconditioning of nonsymmetric problems. SIAM J. Sci. Comput. to appear.

....reasonable speed ups being achieved by Jones and Plassmann (1994) The situation for unsymmetric systems is, however, much less clear. Although there have been many experiments on using incomplete factorizations and there have been studies of the effect of orderings on the number of iterations (Benzi, Szyld and van Duin 1997, Dutto 1993) there is very little theory governing the behavior for general systems and indeed the performance of ILU preconditioners is very unpredictable. Allowing high levels of fill in can help but again there is no guarantee, as we have argued in Section 1. 3 Some Other Forms of ....

Benzi, M., Szyld, D. B. and van Duin, A. C. N. (1997), Orderings for incomplete factorization preconditioning of nonsymmetric problems, Technical Report 9791, Temple University, Department of Mathematics, Philadelphia, PA.

....Although the convection coefficients are constant, the solution is related to the Reynolds number. Even for moderate Re, the solution is nearly identically zero in Omega except for boundary layers of thickness O(1=2Re) near x = 0 and y = 1. This test problem was also used by Benzi, Szyld and Duin [3] to test the ordering effect on the incomplete factorization preconditioning of nonsymmetric problems. It was used in our experiments to test how the discretization schemes resolve the boundary layers. Figure 2 shows the exact solutions of the three test problems. Problem 3 was plotted twice with ....

....implementation usually requires a preconditioned iterative methods with high order preconditioner and high level of fill ins. Faster convergence, with less fill ins in the preconditioners, may be possible by doing a reordering of the coefficient matrix before the ILU factorization is performed [3]. Nevertheless, this seems to be the only choice (amongst the three schemes considered in this paper) Classical iterative methods do not converge with the central difference scheme in this case and it does not seem to have a report on the implementation of multigrid method either. If computer s ....

M. Benzi, D. B. Szyld, and A. van Duin, Ordering for incomplete factorization preconditioning of nonsymmetric problems, Technical Report LA-UR-97-3525, Los Alamos National Laboratory, NM, 1997.

.... preconditioners, and also blocks for the treatment of certain Markov chain problems [10] see also [23] It turns out that this reordering is also useful for point incomplete factorizations, where it is often better than the natural ordering [6] and some of the reorderings considered in this paper [7]. However, for most cases treated in this paper, the performance of TPABLO is inferior to that of some of the reorderings designed for direct methods, and thus we do not report results with it here. TPABLO might prove useful in the context of block incomplete factorizations, but this topic is ....

....Special thanks go to Miroslav Tuma, who not only provided us with some of his software but also was very generous in sharing his insight on sparse matrix reorderings at various stages of this project. We are indebted to Gerard Meurant, whose questions and detailed reading helped us turn report [7] into this paper. Part of this research took place while the first author was with CERFACS and the second author was a visitor there. CERFACS s support and warm hospitality are greatly appreciated. ....

M. Benzi, D. B. Szyld, and A. van Duin, Orderings for Incomplete Factorization Preconditionings of Nonsymmetric Problems, Research Report 97-91, Department of Mathematics, Temple University, 1997; also available online from http://www.math.temple.edu/#szyld.

....use of nonsymmetric (one sided) and symmetric (fill reducing) permutations. The effect of symmetric reorderings on preconditioned iterative methods has been considered by many authors before, but mostly on problems which are either symmetric [28] or at least structurally symmetric or nearly so [7], 12] 15] On the other hand, the effect of nonsymmetric permutations has received little attention so far, and the same is true for the use of symmetric permutations on matrices that are highly nonsymmetric, structurally as well as numerically. Note that nonsymmetric permutations, unlike ....

....kind of instability. We stress that the instability here is not in the incomplete factorization process, but in the incomplete factors. Unstable (more accurately, ill conditioned) ILU factors occur frequently for matrices that are far from symmetric and lack diagonal dominance; see [30] 16] [7]. For certain PDE problems, symmetric permutations aimed at reducing the bandwidth of A or the fill in in the factors have been shown to have a stabilizing effect on the ILU factors, resulting in small kEkF and rapid convergence; see [7] For general sparse matrices, it is frequently the case that ....

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M. Benzi, D. B. Szyld, and A. van Duin, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., 20 (1999), pp. 1652--1670.

....viable. Notice that this situation is not specific to factorized approximate inverse preconditioners: exactly the same argument applies to standard ILU type preconditioners. In particular, this helps explain the generally poor performance of Minimum Degree for ILU(0) and ILUT preconditioning (see [2], 13] Hence, we face the following dilemma: graph theoretical considerations suggest the use of orderings that will cause a small amount of inverse fill, like Nested Dissection or Minimum Degree, whereas a look at the decay rates suggests that we use band reducing orderings (like RCM) which ....

....nz = 4992 nonzero coefficients. The right hand side is chosen so that the solution to the discrete system is the vector (1; 2; n) The parameter 0 controls the difficulty of the problem the smaller is , the harder it is to solve the discrete problem by iterative methods (see also [2]) For our experiments, we generated ten linear systems of increasing difficulty, corresponding to Gamma1 = 100; 200; 1000. The coefficient matrix A becomes increasingly nonsymmetric and far from diagonally dominant as gets smaller. Moreover, Green function arguments can be used to ....

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M. Benzi, D. B. Szyld, and A. van Duin, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., to appear.

....in the preconditioner, like FGMRES [67] Another issue that deserves to be mentioned is the sensitivity of these preconditioners to reorderings. It is well known that incomplete factorization preconditioners are very Sparse Approximate Inverse Preconditioners 11 sensitive to reorderings; see [16], 41] On the other hand, the SPAI and MR preconditioners are scarcely sensitive to reorderings. This is, at the same time, good and bad. The advantage is that A can be partitioned and reordered in whichever way is more convenient in practice, for instance to better suite the needs of a ....

M. Benzi, D. B. Szyld, and A. van Duin. Orderings for incomplete factorization preconditioning of nonsymmetric problems. Technical Report LA-UR-97-3525, Los Alamos National Laboratory, 1997.

....denoted by ILUT( p) where is the dropping threshold and p is the maximum number of nonzeros of fill allowed in a row above those present in the original matrix. The effect of reorderings on the performance of ILU type preconditioners has been studied by Duff and Meurant [7] Benzi et al. [2], and Saad [19] among others. Duff and Meurant have studied the impact of reorderings on incomplete factorization preconditioning for SPD problems. The sources of inaccuracy in incomplete factorizations and the effect of reorderings on accuracy and stability have been analyzed by Chow and Saad ....

....others. Duff and Meurant have studied the impact of reorderings on incomplete factorization preconditioning for SPD problems. The sources of inaccuracy in incomplete factorizations and the effect of reorderings on accuracy and stability have been analyzed by Chow and Saad [4] and Benzi et al. [2]. Let A = L U . The residual matrix R = A Gamma A measures the accuracy of the incom M. Benzi, W. Joubert, and G. Mateescu 10003 plete factorization. Let k Delta kF denote the Frobenius norm of a matrix. Chow and Saad [4] and Benzi et al. 2] have shown that for nonsymmetric ....

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M. BENZI, D. B. SZYLD, AND A. VAN DUIN, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., 20 (1999), to appear.

....this situation is not specific to factorized approximate inverse preconditioners: exactly the same argument applies to standard ILU type preconditioners. In particular, this suggests a possible explanation of the generally poor performance of Minimum Degree for ILU(0) and ILUT preconditioning (see [3], 14] Hence, we face the following dilemma: graph theoretical considerations suggest the use of orderings that will cause a small amount of inverse fill, like Nested Dissection or Minimum Degree, whereas a look at the decay rates suggests that we use band reducing orderings (like RCM) which ....

....nz = 4992 nonzero coefficients. The right hand side is chosen so that the solution to the discrete system is the vector (1; 2; n) The parameter 0 controls the difficulty of the problem the smaller is , the harder it is to solve the discrete problem by iterative methods (see also [3]) For our experiments, we generated ten linear systems of increasing difficulty, corresponding to Gamma1 = 100; 200; 1000. The coefficient matrix A becomes increasingly nonsymmetric and far from diagonally dominant as gets smaller. Moreover, Green function arguments can be used to ....

[Article contains additional citation context not shown here]

M. Benzi, D. B. Szyld and A. van Duin, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., to appear.

....2 we survey some of the contributions on the effect of permutations on the convergence of preconditioned Krylov subspace methods. In Sect. 3 we present our numerical experiments and comment on those results. Finally, in Sect. 4 we present our conclusions. This paper is an abbreviated version of [3] 2 OVERVIEW OF THE LITERATURE The influence of reorderings on the convergence of preconditioned iterative methods has been considered by a number of authors. Several of these papers are concerned with symmetric problems only [4, 8, 12, 14, 21, 22, 23, 27] Duff and Meurant [12] have performed a ....

....unstructured grids) was possibly the first to observe that MD and other direct solver reorderings can have a positive effect on the convergence of GMRES with ILU(0) preconditioning. This is mostly consistent with some of our own experiments reported here. For further review of the literature, see [3]. We also mention the Minimum Discarded Fill (MDF) algorithm (see [8] which takes into account the numerical values of the entries of A. This method can be very effective, but it is often too expensive to be practical, except for rather simple problems. In addition to reorderings which were ....

[Article contains additional citation context not shown here]

M. Benzi, D.B. Szyld, and A.C.N. van Duin. Orderings for Incomplete Factorization Preconditionings of Nonsymmetric Problems. Research Report 97-91, Department of Mathematics, Temple University, September 1997

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M. Benzi, D. B. Szyld and A. VanDuin, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., 20-5 (1999), pp. 1652-1670.

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M. Benzi, D. B. Szyld, A. van Duin, 1999. Orderings for incomplete factorization preconditioning of nonsymmetric problems. SIAM J. Sci. Comp., vol. 20, no. 5, pp. 16521670, 1999.

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