Quasi-monte Carlo Methods for Integral Equations

In this paper, we establish a deterministic error bound for estimating a functional of the solution of the integral transport equation via random walks that improves an earlier result of Chelson generalizing the Koksma-Hlawka inequality for nite dimensional quadrature. We solve such problems by simulation, using sequences that combine pseudorandom and quasirandom elements in the construction of the random walks in order to take advantage of the superior uniformity properties of quasirandom numbers and the statistical (independence) properties of pseudorandom numbers. We discuss implementation issues that arise when these hybrid sequences are used in practice. The quasi-Monte Carlo techniques described in this paper have the potential to improve upon the convergence rates of both (conventional) Monte Carlo and quasi-Monte Carlo simulations in many problems. Recent model problem computations connrm these improved convergence properties.