Effective Preconditioning through Ordering Interleaved with Incomplete Factorization

Consider the solution of a sparse linear system Ax = b when the matrix A is symmetric and positive definite. A typical iterative solver is obtained by using the method of Conjugate Gradients (CG) [15] preconditioned with an incomplete Cholesky (IC) factor L̂ [4]. The latter is an approximation to the (complete) Cholesky factor L, where A = LL . Consequently, the process of computing L̂ relies to a large extent on data structures and graph-theoretic methods used to compute L in a sparse direct solver. During sparse Cholesky factorization, some of the elements in A that are zeroes will fill-in and become nonzeroes in the factor L [9, 12]. An incomplete factor L̂ is obtained by performing Cholesky factorization while selectively discarding some fraction of the fill-in nonzeroes as soon as they are created. The two main types of incomplete Cholesky factorization schemes involve either a symbolic level-of-fill approach [20], or a drop-threshold approach [21, 30]. For L̂ generated from either approach, retaining more fill typically leads to a better preconditioner, i.e., one that can significantly reduce the number of CG iterations and/or prevent failure [22]. When such preconditioners are required for the application, both level of fill and drop-threshold schemes benefit from a fill-reducing ordering scheme. The fill incurred during Cholesky factorization depends on the sparsity structure of A and not on the numerical values of the elements. This fact is exploited in a first ordering step in direct solvers to control fill-in; ordering algorithms compute a symmetric permutation of A using graph-theoretic methods to limit the number of nonzeroes in the corresponding L. Although there has been earlier work on computing orderings tailored for incomplete factorizations by D’Azevedo et al. [5, 6], typical incomplete factorization schemes tend to use orderings computed to reduce fill in the complete sparse Cholesky factor L. Currently, level-of-fill (symbolic) and drop-threshold (numeric) incomplete factorization schemes use fill-reducing ordering schemes as follows. The ordering step is a first symbolic step (as in a direct solver) to compute a fill-reducing permutation; this permutation is directed toward reducing fill in the complete sparse Cholesky factor L and not in its incomplete form L̂. In a subsequent step, strategies are incorporated to select elements to retain or discard to obtain L̂ as an approximation to the sparse Cholesky factor L of the permuted matrix A. Thus, in broad terms, for a given permutation π of A to reduce fill in L(π), both symbolic and numeric forms of incomplete factorization compute L̂(π) as an approximation to the complete Cholesky factor L(π) with an error matrix E(π) = L(π)− L̂(π). This paper focuses on the role of fill-reducing orderings in computing effective incomplete factor preconditioners. Consider computing a permutation τ using an ordering scheme that is specifically directed towards reducing fill in the incomplete factor L̂(τ) corresponding to a level-of-fill or dropthreshold scheme. Now there exists a complete sparse Cholesky factor L(τ) for this permutation of A