Please cite this article in press as: A. Srinivasan, Monte Carlo linear solvers with non-diagonal splitting, Math. Comput. Simul. (2009), doi:10.1016/j.matcom.2009.03.010 ARTICLE IN PRESS +Model MATCOM-3202; No. of Pages 11 Available online at www.sciencedirect.com Mathematics and Computers in Simulation xxx (2009) xxx–xxx Monte Carlo linear solvers with non-diagonal splitting A. Srinivasan ∗ Department of Computer Science, Florida State University, Tallahassee, FL 32312, United States Received 11 July 2007; received in revised form 25 November 2008; accepted 21 March 2009 Abstract Monte Carlo (MC) linear solvers can be considered stochastic realizations of deterministic stationary iterative processes. That is, they estimate the result of a stationary iterative technique for solving linear systems. There are typically two sources of errors: (i) those from the underlying deterministic iterative process and (ii) those from the MC process that performs the estimation. Much progress has been made in reducing the stochastic errors of the MC process. However, MC linear solvers suffer from the drawback that, due to efficiency considerations, they are usually stochastic realizations of the Jacobi method (a diagonal splitting), which has poor convergence properties. This has limited the application of MC linear solvers. The main goal of this paper is to show that efficient MC implementations of non-diagonal splittings too are feasible, by constructing efficient implementations for one such splitting. As a secondary objective, we also derive conditions under which this scheme can perform better than MC Jacobi, and demonstrate this experimentally. The significance of this work lies in proposing an approach that can lead to efficient MC implementations of a wider variety of deterministic iterative processes. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Monte Carlo; Linear solver 1. Introduction The use of Monte Carlo (MC) in linear algebra dates back to the work of von Neumann and Ulam (described by Forsythe and Leibler [6] in 1950). However, with the development of modern deterministic numerical techniques, MC started losing its appeal in numerical linear algebra. There has been a recent revival of interest in MC linear algebra, partly because of advances in MC techniques, but more importantly, due to the increasing importance of applications where the use of MC techniques is attractive [14]. For example, the use of MC is promising in applications where approximate solutions are sufficient, such as in preconditioning, graph partitioning, information retrieval, and feature extraction. Furthermore, parallel MC is very latency tolerant, and so should be effective in a Grid-like environment. MC can also yield specific components of the solution. In addition, the convergence rate is independent of the size of the matrix. A major problem with current MC linear solver techniques, which we summarize in Section 2, is that they are fundamentally based on the Jacobi method (a diagonal splitting). We proposed a different iterative process and evaluated it empirically with dense matrices in [11,12], and presented a sparse implementation in [13]. Here, we present a more detailed discussion, including theoretical analysis, a better sparse implementation, and more exhaustive empirical ∗ Corresponding author. Tel.: +1 850 644 0559; fax: +1 850 644 0058. E-mail address: asriniva@cs.fsu.edu. 0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.03.010