Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems

TL;DR

This paper analyzes why Cholesky factorization performs poorly on linear systems from finite difference discretizations of singularly perturbed reaction-diffusion equations, providing bounds on parameters affecting performance.

Contribution

It offers an analysis of the entry distribution in Cholesky factors for these systems, explaining performance issues and establishing bounds related to perturbation and discretization parameters.

Findings

Cholesky factors can have entries with large magnitude causing computational issues

Performance degradation occurs within specific parameter ranges

Bounds on parameters where poor performance is expected are provided

Abstract

We consider the solution of large linear systems of equations that arise when two-dimensional singularly perturbed reaction-diffusion equations are discretized. Standard methods for these problems, such as central finite differences, lead to system matrices that are positive definite. The direct solvers of choice for such systems are based on Cholesky factorisation. However, as observed by MacLachlan and Madden (SIAM J. Sci. Comput. 35-5 (2013), pp. A2225-A2254), these solvers may exhibit poor performance for singularly perturbed problems. We provide an analysis of the distribution of entries in the factors based on their magnitude that explains this phenomenon, and give bounds on the ranges of the perturbation and discretization parameters where poor performance is to be expected.