GPU Integral Computations in Stochastic Geometry

Stochastic geometry has applications in areas such as robotics, tomographic reconstruction with uncertainties, spatial Poisson-Vonoroi tessellations, and imaging from medical data using tetrahedral meshes. We examine numerical approaches for the computation of multivariate integrals for a family of problems where uniformly distributed points are picked as polyhedron vertices for tessellations, for example, tetrahedron vertices in a cube, tetrahedron or on a spherical surface. The classical cube tetrahedron picking problem yields the expected volume of a random tetrahedron in a cube, and helps furthermore assessing unsolved extremal problems (cf., A. Zinani, 2003). We demonstrate feasible numerical approaches including adaptive integration through region partitioning, quasi-Monte Carlo (based on a randomized Korobov lattice), and Monte Carlo techniques, which are the basic methods of our parallel integration package ParInt. We then describe our implementation of the Monte Carlo approach on GPUs (Graphics Processing Units) in CUDA C, and demonstrate its parallel performance for various stochastic geometry integrals.