Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values

TL;DR

This paper develops an efficient numerical framework for estimating expected values of solutions to time-fractional diffusion equations with stochastic parameters, combining high-order quasi-Monte Carlo, finite element discretization, and error analysis.

Contribution

It introduces a novel combination of numerical methods for efficiently approximating expected values in stochastic time-fractional diffusion models, with theoretical error estimates and numerical validation.

Findings

High-order quasi-Monte Carlo effectively estimates high-dimensional integrals.

Finite element and time-stepping schemes achieve second-order accuracy.

Numerical experiments confirm theoretical error bounds.

Abstract

In this work, we explore a time-fractional diffusion equation of order $\alpha \in (0,1)$ with a stochastic diffusivity parameter. We focus on efficient estimation of the expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of our model problem. To estimate the expected value computationally, the infinite expansions of the random parameter need to be truncated. Then we approximate the high-dimensional integral over the random field using a high-order quasi-Monte Carlo method. This follows by approximating the deterministic solution over the space-time domain via a second-order accurate time-stepping scheme in combination with a spatial discretization by Galerkin finite elements. Under reasonable regularity assumptions on the given data, we show some regularity properties of the continuous solution and investigate the errors from estimating the expected value. We report on numerical experiments that complement the theoretical results.