Numerical methods for stochastic conservation laws with multiplicative rough drivers in the flux

Overview

Abstract

Stochastic conservation laws (SCL) with quasilinear multiplicative``rough'' path dependence in the flux arise in modeling of mean field games. An impressive collection of theoretical results has been developed for SCL in recent years by Gess, Lions, Perthame, and Souganidis. We present the first fully computable numerical methods for pathwise solutions of scalar SCL with, for instance, "rough" paths in the form of Wiener processes. Convergence rates are derived for the numerical methods and we show that for strictly convex flux functions, "rough" path oscillations lead to cancellations in the solution flow map; a property we take advantage of to develop more efficient numerical methods.

Brief Biography

Håkon Hoel is a visiting researcher in the Stochastic Numerics Group at KAUST. He holds an MSc degree in Computational Science from the University of Oslo and a PhD in Numerical Analysis from KTH Royal Institute of Technology. His main research interests are numerical analysis of stochastic differential equations, nonlinear filtering and multilevel Monte Carlo (MLMC) methods.