Document Type

Article

Publication Date

2013

DOI

10.1137/130904867

Publication Title

SIAM Journal on Numerical Analysis

Volume

51

Issue

4

Pages

2107-2122

Abstract

The convergence of Markov chain--based Monte Carlo linear solvers using the Ulam--von Neumann algorithm for a linear system of the form x = Hx + b is investigated in this paper. We analyze the convergence of the Monte Carlo solver based on the original Ulam--von Neumann algorithm under the conditions that ||H|| < 1 as well as ρ(H) < 1, where ρ(H) is the spectral radius of H. We find that although the Monte Carlo solver is based on sampling the Neumann series, the convergence of Neumann series is not a sufficient condition for the convergence of the Monte Carlo solver. Actually, properties of H are not the only factors determining the convergence of the Monte Carlo solver; the underlying transition probability matrix plays an important role. An improper selection of the transition matrix may result in divergence even though the condition ||H|| <1 holds. However, if the condition ||H|| < 1 is satisfied, we show that there always exist certain transition matrices that guarantee convergence of the Monte Carlo solver. On the other hand, if ρ(H) <1 but ||H|| ≥ 1, the Monte Carlo linear solver may or may not converge. In particular, if the row sum ∑ n/j= 1|Hij > 1 for every row in H or, more generally, ρ(H+) >1, where H+ is the nonnegative matrix where H+ij = |Hij|, we show that transition matrices leading to convergence of the Monte Carlo solver do not exist. Finally, given H and a transition matrix P, denoting the matrix H* via H*ij = H2ij/Pij, we find that ρ(H*) < 1 is a necessary and sufficient condition for convergence of the Markov chain--based Monte Carlo linear solvers using the Ulam--von Neumann algorithm.

Comments

"SIAM does not currently publish Open Access journals, but we do allow authors to post their article - which we provide - on their personal website, their institution’s website, and/or an online repository."

"The author may post the final published version of the work on the author's personal web site and on the web server of the author's institution, provided that proper notice of the publisher's copyright is included and that no separate or additional fees are collected for access to or distribution of the work."

© 2013, Society for Industrial and Applied Mathematics

Original Publication Citation

Ji, H., Mascagni, M., & Li, Y. (2013). Convergence analysis of Markov chain Monte Carlo linear solvers using Ulam--von Neumann Algorithm. SIAM Journal on Numerical Analysis, 51(4), 2107-2122. doi: 10.1137/130904867

ORCID

0000-0003-3058-4580 (Michael Mascagni), 0000-0003-0178-1876 (Yaohang Li)

Download

Share

COinS