Sparse Approximate Inverse Preconditioners

Abstract

Consider a preconditioner M based on an incomplete LU (or Cholesky) factorization of a matrix A. M−1, which represents an approximation of A−1, is applied by performing forward and back substitution steps; this can present a computational bottleneck. An alternative strategy is to directly approximate A−1 by explicitly computing M−1. Preconditioners of this kind are called sparse approximate inverse preconditioners. They constitute an important class of algebraic preconditioners that are complementary to the approaches discussed in the previous chapter. They can be attractive because when used with an iterative solver, they can require fewer iterations than standard incomplete factorization preconditioners that contain a similar number of entries while offering significantly greater potential for parallel computations.