New “Walk on Equations” Monte Carlo Algorithm for Linear Systems

Abstract

Abstract A novel version of Monte Carlo algorithm for solving systems of linear algebraic equations is presented and studied. The algorithm is similar to the “Walk on Equations” Monte Carlo method recently developed by Ivan Dimov, Sylvain Maire and Jean Michel Sellier. It is done a comparison with the Gauss-Seidel method for matrices up to size of 212. The algorithm could be drastically improved by choosing appropriate values for the relaxation parameters, which in turn leads to dramatic reduction in time and lower relative errors for a given number of iterations. What is more, a sequential Monte Carlo method of John Halton based on an iterative use of the control variate method has been applied. Some of the most important numerical applications are the large system, coming from a finite element approximation of problems, describing a beam structure in constructive mechanics, and the block-diagonal matrices, which come from discretization of models in a regime-switching economy.