A new proof of geometric convergence for general transport problems based on sequential correlated sampling methods

In [J. Halton, Sequential Monte Carlo, Proc. Comb. Phil. Soc. 58 (1962), J. Halton, Sequential Monte Carlo Techniques for the Solution of Linear Systems, J. Sci. Comp. 9 (1994) 213–257] Halton introduced a strategy to be used in Monte Carlo algorithms for the efficient solution of certain matrix problems. We showed in [R. Kong, J. Spanier, Sequential correlated sampling methods for some transport problems, in: Harold Niederreiter, Jerome Spanier (Eds.), Monte-Carlo and Quasi Monte-Carlo Methods 1998: Proceedings of a Conference at the Claremont Graduate University, Springer-Verlag, New York, 2000, R. Kong, J. Spanier, Error analysis of sequential Monte Carlo methods for transport problems, in: Harold Niederreiter, Jerome Spanier (Eds.), Monte-Carlo and Quasi Monte-Carlo Methods 1998: Proceedings of a Conference at the Claremont Graduate University, Springer-Verlag, New York, 2000] how Halton’s method based on correlated sampling can be extended to continuous transport problems and established their geometric convergence for a family of transport problems in slab geometry. In our algorithm, random walks are processed in batches, called stages, each stage producing a small correction that is added to the approximate solution developed from the previous stages. In this paper, we demonstrate that strict error reduction from stage to stage can be assured under rather general conditions and we illustrate this rapid convergence numerically for a simple family of two dimensional transport problems.