Quasi-Monte Carlo methods for transport equations

Monte Carlo simulation has provided an efficient technique for estimating the solution of a great many problems that have resisted solution by analytical means. In the last two decades or so, many scientific workers have also studied various ways to improve the efficiency of Monte Carlo simulation. The convergence of Monte Carlo methods can often be improved by replacing pseudo-random numbers with more uniformly distributed numbers known as quasi-random numbers. This replacement also leads to deterministic, rather than statistical, error bounds. The primary goal of this dissertation is to study the convergence characteristics of quasi-Monte Carlo methods for solving the integral transport equation.;Based on the above theoretical ideas, we have developed several computer programs to display the efficiency of our results. Our results will prove to be useful in the implementing of Monte Carlo simulation methods based on the transport equation in the areas of nuclear reactor design, operations research, and economic models as well as to oil well logging problems.;In this dissertation, we first develop deterministic error bounds for estimating a functional of the solution of the integral transport equation via random walks generated by several low-discrepancy sequences such as quasi-pseudo mixed and scrambled quasi sequences. Use of these sequences in the construction of the random walks is intended to take advantage of optimal uniformity properties of quasi-random numbers and the statistical (independence) properties of pseudo-random numbers. The quasi-Monte Carlo techniques developed in this dissertation improve upon the convergence rate of Monte Carlo method. Model problem computations verify these improved convergence properties. We also study sequential sampling methods that are designed to build information drawn from early batches of random walk histories into the random walk process used to generate later histories in order to accelerate convergence. In previously published work, such methods have been applied within the pseudo-random, rather than the quasi-random, only to matrix problems. In this dissertation, the sequential methods have been improved by modifying the transition function on each iteration, utilizing information from the previous iteration. These techniques are formulated in an abstract space. Then by specifying the abstract space, we are able to generalize this application to integral transport equations. Finally, we study discretizations of the integral transport equation by using a low-discrepancy sequence as nodal points, instead of the traditional grid-partition method. Thus, we are able to combine the advantages of the quasi-Monte Carlo methods and the sequential method to speed up the convergence rate.