Stochastic Preconditioning for Iterative Linear Equation Solvers

Abstract

This paper presents a new stochastic preconditioning approach. For symmetric diagonally-dominant M-matrices, we prove that an incomplete LDL factorization can be obtained from random walks, and used as a preconditioner for an iterative solver, e.g., conjugate gradient. It is argued that our factor matrices have better quality, i.e., better accuracy-size tradeoffs, than preconditioners produced by existing incomplete factorization methods. Therefore the resulting preconditioned conjugate gradient (PCG) method requires less computation than traditional PCG methods to solve a set of linear equations with the same error tolerance, and the advantage increases for larger and denser sets of linear equations. These claims are verified by numerical tests, and we provide techniques that can potentially extend the theory to more general types of matrices.