Solving linear equations by Monte Carlo simulation

A new Monte Carlo estimator for solving the matrix equation x = Hx+b is presented, and theoretical results comparing this estimator with the traditional terminal and collision estimators are given. We then make a detailed investigation of the average complexity of the Monte Carlo estimators when several popular random variable generation techniques are used, and we show that the average complexity can be as low as C parallel to z parallel to(1)n+ N, where n is the number of random walks generated, N is the size of the matrix H, z is the solution of an associated matrix equation, and C is a small constant. As a consequence of the complexity results, we observe how scaling of matrices, a well-known technique in deterministic methods, can increase the efficiency of the Monte Carlo method. One advantage of the Monte Carlo method is the efficiency at which it can be parallelized. The algorithms we discuss can provide fast and approximate solutions to systems of linear equations in massively parallel computing environments. Surprisingly, sequential ( or adaptive) Monte Carlo methods can even be competitive in single-processor computing environments. We present numerical results and compare our Monte Carlo algorithms with Krylov subspace methods for some test matrices.