High-Performance Numerical Computation of Multidimensional Integral using Random Sampling

Abstract

This study examines the use of high-performance computing to carry out multidimensional integral calculation based on stochastic techniques, particularly in the context of Monte Carlo integration. Considering that traditional methods are facing extreme difficulty especially in high-dimension when encountered with "dimensionality curse", random sampling technique to estimate integral values is used. This technique is superior in many aspects, for example in terms of scalability and flexibility, even in complex and irregular domains. In particular, the work concentrates on the case of calculating the volume of a multi-dimensional sphere using random sampling or Monte Carlo techniques. It also introduces a framework that employs the Graphics Processing Unit (GPU) to carry out these computations more effectively. Using dimensionalities from 2 to 24, the work compares both accuracy and computation time of the method. The results show that the random sampling method attains high accuracy in the computation of π which is used as a benchmark. The computational model is implemented in CUDA C/C++, and it takes advantage of GPU parallelism to process large sample sizes as well as execute calculations efficiently. Here it is shown in general that Monte Carlo integration is a viable approach to high-dimensional problems when combined with very rapid GPU parallelism.