Quasi-Monte Carlo algorithms for diffusion equations in high dimensions

Abstract

Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is N−1/2 accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve N−1 convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.