Quasi Monte Carlo Methods For Hyperbolic Conservation Laws With Uncertain Initial Data

Abstract

We consider solving stochastic hyperbolic conservation laws with a quasi Monte Carlo method based on Sobol sequences. As far as we are aware, no one has done any research on this specific combination before. We extend an already available tool for uncertainty quantification, ALSVID-UQ, with the ability to perform quasi Monte Carlo using Sobol sequences. We perform numerical experiments with uncertain initial values for Burgers equation and most of the results converge with the rate $O(1/M)$ where $M$ is the amount of samples, compared to $O(1/\sqrt{M})$ for conventional Monte Carlo methods. The convergence is measured in variance and mean. The problems are easily parallelizable, and simulations are done on a supercomputer with up to 1024 CPU cores used simultaneously.