Stochastic Preconditioning for Diagonally Dominant Matrices

Abstract

This paper presents a new stochastic preconditioning approach for large sparse matrices. For the class of matrices that are rowwise and columnwise irreducibly diagonally dominant, we prove that an incomplete $\rm{LDL^T}$ factorization in a symmetric case or an incomplete LDU factorization in an asymmetric case can be obtained from random walks and used as a preconditioner. It is argued that our factor matrices have better quality, i.e., better accuracy-size trade-offs, than preconditioners produced by existing incomplete factorization methods. Therefore a resulting preconditioned Krylov-subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance. The advantage increases for larger and denser matrices. These claims are verified by numerical tests, and we provide techniques that can potentially extend the theory to non-diagonally-dominant matrices.