Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems

Abstract

Abstract. Numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear systems is shown. The preconditioners used in this study are different variants of incomplete factorizations. It is shown that certain reorderings for direct methods, such as reverse Cuthill-McKee, can be very beneficial. The benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity.Key words. linear systems, nonsymmetric matrices, reorderings, permutation of sparse matrices, preconditioned iterative methods, incomplete factorizations, Krylov subspace methods AMS subject classifications. Primary, 65F10, 65N22, 65F50; Secondary, 15A06 PII. S10648275973268451. Introduction. In this paper, we study experimentally how different reorderings affect the convergence of Krylov subspace methods for nonsymmetric systems of linear equations when incomplete LU factorizations are used as preconditioners. In other words, given a sparse linear system of equations Av = b, where v and b are ndimensional vectors, we consider symmetric permutations of the matrix A, i.e., of the form P T AP , and then solve the equivalent system P T AP w = P T b, with v = P w, by way of some preconditioned iterative method. Our focus is on linear systems arising from the discretization of second order partial differential equations, which often are structurally symmetric (or very nearly so) and have a zero-free diagonal. For these matrices, it is usually possible to carry out an incomplete factorization without pivoting for stability (that is, choosing the pivots from the main diagonal). Such properties are preserved under symmetric permutations of A, but not necessarily under nonsymmetric ones. Hence, we restrict our attention to symmetric permutations only. We stress the fact that very different conclusions may hold for matrices which are structurally far from being symmetric, although we have little experience with such problems. If A is structurally symmetric, the reorderings are based on the (undirected) graph associated with the structure of A; otherwise, the structure of A + A T is used. We consider several iterative methods for nonsymmetric systems, including GMRES [43], Bi-CGSTAB [48], and transpose-free QMR (TFQMR) [28]; for a description of these, as well as a description of incomplete factorizations, see, e.g., [3], [42].In this paper, we mainly concentrate on orderings originally devised for matrix factorizations, i.e., those used to reduce fill-in in the factors; see, e.g., [18] or [29]. We want to call attention to the fact that a permutation of the variables (and equations)

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