Date of Award
Doctor of Philosophy
Dr. Elise de Doncker
Dr. John Kapenga
Dr. Joseph McKean
parallel algorithms, lattice rules, Monte Carlo, multivariate integration, Bayesian inference, Feynman loop diagrams
While adaptive integration by region partitioning is generally effective in low dimensions, quasi-Monte Carlo methods can be used for integral approximations in moderate to high dimensions. Important application areas include high-energy physics, statistics, computational finance and stochastic geometry with applications in robotics, tessellations and imaging from medical data using tetrahedral meshes.
Lattice rule integration is a class of quasi-Monte Carlo methods, implemented by an equal-weight cubature formula and suited for fairly smooth functions. Successful methods to construct these rules are the component-by-component (CBC) algorithm by Sloan and Restsov (2001) and the fast algorithm for CBC by Nuyens and Cools (2006). As the ability to invoke a large number of function evaluations is an important factor in high-dimensional integration, we investigate the acceleration of the CBC construction for large rank-1 lattice rules using the CUDA (cuFFT) Fast Fourier Transform procedure.
A major part of this study is the development of high-performance lattice rule algorithms for approximating moderate- to high-dimensional integrals on GPUs. Lattice rules are incorporated with a periodizing transformation. We show that rank-1 lattice rules on GPUs (possibly with an integral transformation to alleviate the effects of boundary singularities) yield better accuracy and efficiency for various classes of integrals compared to classic Monte Carlo and adaptive methods. The computational power of GPU accelerators also leads to significant improvements in efficiency and accuracy for integration based on embedded (composite) lattices.
These methods have been motivated as possible contributions to high-performance computing software such as the ParInt multivariate integration package developed at WMU. We further show an application in Bayesian analysis, leading to a class of problems where the integrand has a dominant peak in the integration domain. We demonstrate a black-box approach provided by the adaptive strategies in the ParInt package.
Almulihi, Ahmed Hassan H., "High-Performance Quasi-Monte Carlo Integration and Applications" (2019). Dissertations. 3452.