Monte-Carlo and Quasi-Monte Carlo Methods 1998 pp 252-272 | Cite as

# Error Analysis of Sequential Monte Carlo Methods for Transport Problems

Conference paper

## Abstract

In 1962, Halton introduced a sequential correlated sampling algorithm for the efficient solution of certain matrix problems. We have extended Halton’s method to the solution of certain simple transport problems and the resulting algorithm is capable of producing geometric convergence for these problems. In our algorithm, random walks are processed in groups, called stages, and the result of each stage is a small correction that is added to the solution at the previous stage. It is then of interest to determine conditions that guarantee strict error reduction at each stage forjrarious transport problems. Specifically, if in a certain probabilistic sense, where e is an error term that tends to zero as both the number of terms representing the global solution and the number of random walks per stage tend to infinity. We will indicate how to find such a λ, which is defined in terms of the number of random walks per stage and the coefficients of the transport problem in a rather natural way.

*Φ*(x) is the true transport solution and \( \tilde \Phi ^{n - 1} (x) \) and \( \tilde \Phi ^n (x) \) are the estimated solutions from the (n-l)^{st}and n^{th}stages, respectively, we demonstrate the existence of a number λ, 0 < λ < 1, which is independent of the stage number n, such that$$\left\| {{{\tilde \Phi}^n}(x)-\Phi (x)} \right\| \leqslant \lambda \left\| {{{\tilde \Phi}^{n-1}}(x)-\tilde \Phi (x)} \right\| + \in$$

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