Parallel Preconditioning with Sparse Approximate Inverses

Abstract

A parallel preconditioner is presented for the solution of general sparse linear systems of equations. A sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its application only requires a matrix-vector product. The sparsity pattern of the approximate inverse is not imposed a priori but captured automatically. This keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum. Rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system, and the proximity of the approximate to the true inverse is estimated. An extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach.

A comparative study of sparse approximate inverse preconditioners

“…Such an approach was first proposed by Cosgrove et al [33]. Slightly different strategies were also considered by Grote and Huckle [54] and by Gould and Scott [52].…”

“…The most successful of these approaches is the one proposed by Grote and Huckle, hereafter referred to as the SPAI preconditioner [54]. The algorithm runs as follows.…”

“…Our implementation is as close as possible to the description of the algorithm given in [54]. Input data structures include the original matrix A stored by columns (that is, A is stored in CSC, or Harwell-Boeing format) and the matrix structure stored by rows.…”

Prospects for CFD on Petaflops Systems

“…Another example of a latency-tolerant preconditioner is the form of the Sparse Approximate Inverse (SPAI) recently developed in [14]. Here B is formed in explicit, forward-multiply form by performing a sparsity-constrained norm minimization of ||AB − I|| F .…”

“…(The next public release of PETSc will contain an interface to this package.) The sparsity profiles of an original matrix A and its SPAI, with a comparable number of differently positioned nonzeros are shown below (from [14]):…”