Recent studies have examined the use of Monte Carlo methods for solving systems of linear equations. In analogy with the synthetic acceleration methods commonly used in the particle transport community, this approach has been termed Monte Carlo synthetic acceleration (MCSA). The existing MCSA algorithm is a preconditioned Richardson iteration in which the preconditioner consists of performing random walks along the Neumann series expansion of the inverse of a matrix.
In this study we consider approximating the matrix inverse with random walks along polynomial expansions other than the Neumann series, in particular polynomial expansions based on Chebyshev polynomials. While the use of Chebyshev polynomials leads to a dramatic reduction in the number of MCSA iterations required for convergence, the number of Monte Carlo histories that must be executed per iteration to guarantee stability is significantly larger than for the equivalent Neumann series.