BRIDGING THE PERFORMANCE GAP IN EXASCALE ARCHITECTURES: MONTE CARLO APPROACHES TO HIGH-DIMENSIONAL PROBLEMS

Abstract

The paper uses Monte Carlo algorithms applications to solve large-scale and sparse linear systems that have a significant spectrum of application in the modern field of computation science. Monte Carlo techniques can be used to compute accurate approximations by building up the expected value as the trajectory of random walks and the techniques naturally lend themselves to parallel computation by splitting the trajectory among several processors. The necessity of scalable and memory and parallelism numerical approaches has become obvious as the high-performance and exascale computing architectures develop to a new level. The Neumann-Ulam stochastic methods may indeed offer a valid alternative to the traditional solvers, including direct and iterative ones. This research article uses Python-program-based applications to analyse the speed of computation on the systems of varying size (n = 100, 500 and 1000). The data indicates that the method would perform comparable in terms of accuracy with the more expensive standard direct solvers and that the absolute error can be reduced with the number of samples used. The algorithm has good scaling properties suggesting that it possibly can be efficient in running large-scale scientific and engineering tasks. The results support the general agenda of introducing probabilistic numerical approaches into the computational pipelines, especially in the context of cases where memory restrictions or heavy needs of parallelization are essential factors.