Numerical solution of an inverse random source problem for the time fractional diffusion equation via PhaseLift

TL;DR

This paper addresses the inverse problem of recovering a diffusion coefficient in a stochastic time fractional diffusion equation using PhaseLift, demonstrating uniqueness and effectiveness through numerical experiments.

Contribution

It introduces a novel application of PhaseLift with random masks to solve the inverse random source problem for stochastic time fractional diffusion equations.

Findings

Fourier modulus of the diffusion coefficient is uniquely determined by boundary data variance.

PhaseLift method successfully recovers the diffusion coefficient from Fourier modulus.

Numerical experiments confirm the effectiveness of the proposed approach.

Abstract

This paper is concerned with the inverse random source problem for a stochastic time fractional diffusion equation, where the source is assumed to be driven by a Gaussian random field. The direct problem is shown to be well-posed by examining the well-posedness and regularity of the solution for the equivalent stochastic two-point boundary value problem in the frequency domain. For the inverse problem, the Fourier modulus of the diffusion coefficient of the random source is proved to be uniquely determined by the variance of the Fourier transform of the boundary data. As a phase retrieval for the inverse problem, the PhaseLift method with random masks is applied to recover the diffusion coefficient from its Fourier modulus. Numerical experiments are reported to demonstrate the effectiveness of the proposed method.