The Studies And Applications Of Quasi Monte Carlo Methods

This dissertation covers the applications of quasi-Monte Carlo methods in global optimization and unbounded weighted integration problems. The disser-tation consists of three chapter.Chapter 1 Introduce the development and main ideal of Monte Carlo and quasi Monte CarloChapter 2 Adaptive quasi-Monte Carlo methods for global optimization prob-lems based on the ideal of multistart.Quasi Monte Carlo method is widely applied in nondifferentiable optimization. We borrow the ideals of adaptive quasi random search and multistart algorithms to solve global optimization problems; Search direction and search step size are adjusted according to previous search result in the adaptive quasi random search method. Using the ideal of adaptive quasi random search for the major iterations and com-plete iteration of multistart algorithms, it's well to balance the global and local problems. In order to speed up the method, the worse points will be replaced by the new samples according to results of iteration. Comparing with the other methods, our method has better result.Chapter 3 Quasi-Monte Carlo methods for unbounded integration problems based on the ideal of rejection sampling.In this article, we investigate quasi-Monte Carlo methods for weighted integration problems, in which the inte-grand is allowed to be unbounded at the lower boundary of the integration domain. Hlawka Muck proposed a method to generate the H-discrepancy sequences and proved convergence theorems for the distributions with in-dependent marginals, which we will extend to an arbitrary distribution. We proved that it's sufficient for rejection sampling method in quasi-Monte Carlo to generate H-discrepancy sequences. Considering the accuracy may be lost due to the discontinuity of the characteristic functions, smoothing technique will be used to modify the standard rejection sampling methods. To effectively use rejection sampling method, B-spline rejection sampling methods is described. In the end, the rejection methods will be replaced by B-spline rejection methods without changing the value of integrals to simulate the unbounded weighted integration.