Multi-level Monte Carlo Finite Difference Methods for Fractional Conservation Laws with Random Data

TL;DR

This paper introduces a multi-level Monte Carlo finite difference approach to efficiently approximate the average solutions of fractional conservation laws with randomness in initial data and fluxes, supported by theoretical analysis and numerical validation.

Contribution

It develops a novel MLMC-FDM framework for fractional conservation laws with random data, including convergence analysis and error-work estimates.

Findings

MLMC-FDM achieves efficient approximation of ensemble averages.

Theoretical convergence rates are established and validated.

Numerical experiments confirm the method's effectiveness.

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 multi-level 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 it with the convergence rates for the deterministic case. Additionally, we formulate error vs. work estimates for the multi-level estimator. Finally, we present several numerical experiments to demonstrate the efficiency of these schemes and validate the theoretical estimates obtained in this work.