An inverse random source problem for the Helium production-diffusion equation driven by a fractional Brownian motion

TL;DR

This paper investigates the inverse problem of reconstructing a helium source in a diffusion equation driven by fractional Brownian motion, establishing well-posedness, uniqueness, and stability, along with numerical validation.

Contribution

It introduces a novel approach to determine the statistical properties of the source from final-time data in a fractional Brownian motion-driven diffusion model.

Findings

Well-posedness of the direct problem with unique mild solution

Uniqueness and instability results for the inverse problem

Numerical methods successfully reconstruct the source properties

Abstract

In this paper, we consider the prediction of the helium concentrations as function of a spatially variable source term perturbed by fractional Brownian motion. For the direct problem, we show that it is well-posed and has a unique mild solution under some conditions. For the inverse problem, the uniqueness and the instability are given. In the meanwhile, we determine the statistical properties of the source from the expectation and covariance of the final-time data u(r,T). Finally, numerical implements are given to verify the effectiveness of the proposed reconstruction.