Incomplete LU Preconditioner Based on Max-Plus Approximation of LU Factorization

Abstract

Abstract. We present a new method for the a priori approximation of the orders of magnitude of the entries in the LU factors of a complex or real matrix A. This approximation is used to determine the positions of the largest entries in the LU factors of A, and these positions are used as the sparsity pattern for an incomplete LU factorization preconditioner. Our method uses max-plus algebra and is based solely on the moduli of the entries of A. We also present techniques for predicting which permutation matrices will be chosen by Gaussian elimination with partial pivoting. We exploit the strong connection between the field of Puiseux series and the max-plus semiring to prove properties of the max-plus LU factors. Experiments with a set of test matrices from the University of Florida Sparse Matrix Collection show that our max-plus LU preconditioners outperform traditional level of fill methods and have similar performance to those preconditioners computed with more expensive threshold-based methods.Key words. max-plus algebra, LU factorization, Hungarian scaling, linear systems of equations, sparse matrices, incomplete LU factorization, preconditioning AMS subject classifications. 65F08, 65F30, 15A23, 15A80 DOI. 10.1137/16M10945791. Introduction. Max-plus algebra is the analogue of linear algebra developed for the binary operations max and plus over the real numbers together with −∞, the latter playing the role of additive identity. Max-plus algebraic techniques have already been used in numerical linear algebra to, for example, approximate the orders of magnitude of the roots of scalar polynomials [19], approximate the moduli of the eigenvalues of matrix polynomials [1,10,14], and approximate singular values [9]. These approximations have been used as starting points for iterative schemes and in the design of preprocessing steps to improve the numerical stability of standard algorithms [3,6,7,14,20]. Our aim is to show how max-plus algebra can be used to approximate the sizes of the entries in the LU factors of a complex or real matrix A and how these approximations can subsequently be used in the construction of an incomplete LU (ILU) factorization preconditioner for A.In order to be able to apply max-plus techniques to the matrix A ∈ C n×n we must first transform it into a max-plus matrix. We do this using the valuation map

A Max-Plus Approach to Incomplete Cholesky Factorization Preconditioners

“…, n}. In [20], Hook and Tisseur make repeated use of the heuristic that V det(A) ≈ perm V(A) . This approximation can be intuitively justified by expressing the determinant as a sum of terms coming from each permutation and the permanent as the maximum of those terms.…”

“…If both the numerator and denominator in either (2.3) or (2.4) are zero, then the convention 0/0 = 0 is used. Using this fact and heuristic (2.2), Hook and Tisseur [20] define the max-plus LU factors of A ∈ R n×n max as the lower triangular max-plus matrix L and upper triangular max-plus matrix U with entries…”

“…where L is the max-plus lower Cholesky factor of H. [20] present several algorithms for computing max-plus LU factors for nonsymmetric matrices. Their algorithms all work on a directed bipartite graph, with a pair of vertices for each row in the matrix.…”

“…For general nonsymmetric systems, incomplete LU (ILU) factorization preconditioners are frequently used as they work well for a wide range of problems, and, again, many variants have been proposed (see [31] for an introduction). The basic idea is to compute a factorization A ≈ LU (or A ≈ LL T in the positive-definite case) with L and U sparse triangular matrices with the fill-in (that is, the entries in L and U that lie outside the sparsity pattern of A) restricted to some sparsity pattern S. Recently, Hook and Tisseur [20] have shown how max-plus algebra can be used to approximate the order of magnitude of the moduli of the entries in the LU factors of A and have used this to construct the sparsity pattern of ILU preconditioners. Max-plus algebra is the analogue of linear algebra developed for the binary operations max and plus over the real numbers together with −∞, the latter playing the role of additive identity; an introduction and a large number of references to the literature may be found in [16,Chap.…”

“…In the past few years, max-plus algebra has been used to examine a number of numerical linear algebra problems [2,12,15,28]. While the numerical experiments reported on in [20] were limited to modest-sized sparse problems (of order up to 10…”