A Max-Plus Approach to Incomplete Cholesky Factorization Preconditioners

Abstract

We present a new method for constructing incomplete Cholesky factorization preconditioners for use in solving large sparse symmetric positive-definite linear systems. This method uses max-plus algebra to predict the positions of the largest entries in the Cholesky factor and then uses these positions as the sparsity pattern for the preconditioner. Our method builds on the max-plus\nincomplete LU factorization preconditioner\nrecently proposed in [J. Hook and F. Tisseur, Incomplete LU preconditioner based on max-plus approximation of LU factorization, MIMS Eprint 2016.47, Manchester, 2016] but applied to symmetric positive-definite matrices, which comprise an important special case for the method and its application. A attractive feature of our approach is that the sparsity pattern of each column of the preconditioner can be computed in parallel. Numerical comparisons are made with other incomplete Cholesky factorization preconditioners using problems from a range of practical applications. We demonstrate that the new preconditioner can outperform traditional level-based preconditioners and offer a parallel alternative to a serial limited-memory based approach.