Multi-way Monte Carlo Method for Linear Systems.

We study a novel variation on the Ulam-von Neumann Monte Carlo method for solving a linear system. This is an old randomized procedure that results from using a random walk to stochastically evaluate terms in the Neumann series. In order to apply this procedure, the variance of the stochastic estimator needs to be bounded. The best known sufficient condition for bounding the variance is that the infinity norm of the matrix in the Neumann series is smaller than one, which greatly limits the usability of this method. We improve this condition by proposing a new stochastic estimator based on a different type of random walk. Our multiway walk and estimator is based on a time-inhomogeneous Markov process that iterates through a sequence of transition matrices built from the original linear system. For our new method, we prove that a necessary and sufficient condition for convergence is that the spectral radius of the elementwise absolute value of the matrix underlying the Neumann series is smaller than one. This is a strictly weaker condition than currently exists. In addition, our new method is often faster than the standard algorithm. Through experiments, we demonstrate the potential for our method to reduce the time needed to solve linear equations by incorporating it into an outer iterative method.