#### Title

Convergence Analysis of Markov Chain Monte Carlo Linear Solvers Using Ulam--von Neumann Algorithm

#### 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* = H*x* + 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|H_{ij } > 1_{ } for every row in H or, more generally, ρ(H^{+}) >1, where H^{+} is the nonnegative matrix where H+_{ij} = |H_{ij}|, 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 = H2_{ij}/P_{ij}, 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.

#### 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

#### Repository 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)

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