Epilogue

Abstract

Stochastic partial differential equations usually have solutions of low regularity due to the nature of infinite dimensional rough noises. The low regularity results in an enormous amount of computational time spent on Monte Carlo simulations. Despite of the simplicity of Monte Carlo methods, the slow convergence of Monte Carlo methods is the main bottleneck in computing numerical solutions to SPDEs. Although substantial improvements in Monte Carlo methods have been made in recent years, it is still desirable to have further accelerated sampling techniques. Depending on the specific problem, the integration in random space can be made effective using different methods such as quasi-Monte Carlo methods, Wiener chaos methods, and stochastic collocation methods.