An improved incomplete Cholesky factorization

Reviewer: Maurice W. Benson

Two new incomplete Cholesky factorization algorithms suitable for “black-boxed” implementation are motivated with a brief theoretical discussion, described in detail (including pseudocode), and demonstrated to be effective for the generation of preconditioners for a variety of sparse problems. The new approaches do not require any user-supplied parameters and retain the same number of nonzero elements in the factors as in the original matrix (in a row or column manner depending on the method). This ensures the same computational cost for the use of the new factors as with those from the standard incomplete Cholesky factorization. The methods retain the required number of nonzeros in the factors by keeping only larger-magnitude values, while not enforcing any sparsity pattern. A good review of related work is provided, and the new features of the methods presented are described clearly. Extensive experimental results illustrate the utility of the new strategies. Iteration counts for the preconditioned conjugate gradient algorithm demonstrate behavior superior to that of the standard incomplete Cholesky factorization. Efficiency and the effect of varied orderings of the original system are among the features examined. This report should interest specialists developing new algorithms and researchers interested in applications of new, more efficient preconditioners. The presentation is clear, and the references provide a good base for anyone needing more detail.