GPU accelerated Monte Carlo sampling for SPDEs

Monte Carlo sampling methods is a broad class of computational algorithms, that rely on repeated random sampling to obtain numerical results. The idea of such algorithms is to introduce randomness to solve problems, even in the deterministic case. Such algorithms are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches due to limitations, such as cost of performing experiment or inability to take direct measures. The problems typically require solving a stochastic partial differential equations (SDPEs), where an uncertainty is incorporated in to the model. For example: as an input parameter, or initial boundary condition. Extensive efforts have been devoted to the development of accurate numerical algorithms, so that simulation predictions are reliable in since that the numerical errors are well understood and under control for practical problems. Multilevel Monte Carlo (MLMC) is a novel idea. Instead of sampling from the true solution, a sampling is done at different levels. Such approach is beneficial in terms of convergence rate. However for practical simulations a large number of problems has to be solved, with huge number of unknowns. Such computational restrictions naturally leads to challenging parallel algorithms. To overcome some of the limitations, here we consider a parallel implementation of MLMC algorithm for a model SPDE, that uses GPU acceleration for the permeability generation.