Recovery of Multiple Parameters in Subdiffusion from One Lateral Boundary Measurement

TL;DR

This paper develops a numerical method to simultaneously recover the fractional order and diffusion coefficient support in a subdiffusion model from a single boundary measurement, even in an unknown medium.

Contribution

It proves uniqueness of parameter recovery from limited boundary data and introduces a robust algorithm combining asymptotic expansion, analytic continuation, and level set methods.

Findings

Successful numerical recovery of fractional order and diffusion support.

Validation of the method through extensive numerical experiments.

Discussion on extending recovery to general coefficients with specialized boundary excitation.

Abstract

This work is concerned with numerically recovering multiple parameters simultaneously in the subdiffusion model from one single lateral measurement on a part of the boundary, while in an incompletely known medium. We prove that the boundary measurement corresponding to a fairly general boundary excitation uniquely determines the order of the fractional derivative and the polygonal support of the diffusion coefficient, without knowing either the initial condition or the source. The uniqueness analysis further inspires the development of a robust numerical algorithm for recovering the fractional order and diffusion coefficient. The proposed algorithm combines small-time asymptotic expansion, analytic continuation of the solution and the level set method. We present extensive numerical experiments to illustrate the feasibility of the simultaneous recovery. In addition, we discuss the uniqueness of recovering general diffusion and potential coefficients from one single partial boundary measurement, when the boundary excitation is more specialized.