Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems
For researchers solving such discretized PDEs, it explains a known performance issue and predicts when it will happen.
The paper analyzes why Cholesky factorization performs poorly for linear systems from singularly perturbed reaction-diffusion equations, providing bounds on parameter ranges where poor performance occurs.
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.