Multilevel Monte Carlo Finite Difference Methods for Fractional Conservation Laws with Random Data

Abstract

We establish a notion of random entropy solution for degenerate fractional conservation laws incorporating randomness in the initial data, convective flux, and diffusive flux. In order to quantify the solution uncertainty, we design a multilevel Monte Carlo finite difference method (MLMC-FDM) to approximate the ensemble average of the random entropy solutions. Furthermore, we analyze the convergence rates for MLMC-FDM and compare them with the convergence rates for the deterministic case. Additionally, we formulate error vs. work estimates for the multilevel estimator. Finally, we present several numerical experiments to demonstrate the efficiency of these schemes and validate the theoretical estimates obtained in this work.