A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems

Abstract

Abstract. This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient-type methods. Some theoretical properties of the preconditioner are discussed, and numerical experiments on test matrices from the Harwell-Boeing collection and from Tim Davis's collection are presented. Our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners. Furthermore, the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners.

A comparative study of sparse approximate inverse preconditioners

“…A first class of methods, described in this subsection, does not require any information about the triangular factors of A: the factorized approximate inverse preconditioner is constructed directly from A. Methods in this class include the FSAI preconditioner introduced by Kolotilina and Yeremin [61], a related method due to Kaporin [59], incomplete (bi)conjugation schemes [15,18], and bordering strategies [71]. Another class of methods first compute an incomplete triangular factorization of A using standard techniques, and then obtain a factorized sparse approximate inverse by computing sparse approximations to the inverses of the incomplete triangular factors of A.…”

“…This approach, hereafter referred to as the AINV method, is described in detail in [15] and [18]. The AINV method does not require that the sparsity pattern be known in advance, and is applicable to matrices with general sparsity patterns.…”

“…It is easy to see that, in exact arithmetic, the above process can be completed without divisions by zero if and only if all the leading principal minors of A are nonzeros or, equivalently, if and only if A has an LU factorization [18]. In this case, if A = LDU is the decomposition of A as a product of a unit lower triangular matrix L, a diagonal matrix D, and a unit upper triangular matrix U , it is easily checked that Z = U −1 and W = L −T (D being exactly the same matrix in both factorizations).…”

“…Because of the initialization chosen (step (1) in Algorithm 4), the z-and w-vectors are initially very sparse; however, the updates in step (8) cause them to fill-in rapidly. See [18] for a graph-theoretical characterization of fill-in in Algorithm 4. Sparsity in the inverse factors is preserved by carrying out the biconjugation process incompletely, similar to ILU-type methods.…”

On residual smoothing in ILUM-type preconditioning

“…As an alternative to ilu, some authors propose the idea of constructing the matrix M −1 explicitly by minimising a norm of the defect I − M −1 A for all matrices M −1 with a given sparsity pattern, see [2,3,6]. If the Frobenius norm is used, the problem can be subdivided nicely into independent subproblems which can be treated in parallel.…”