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SIAM Journal on Numerical Analysis








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.


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© 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


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