An Inverse Potential Problem for Subdiffusion: Stability and Reconstruction

TL;DR

This paper investigates the inverse problem of recovering a potential in a subdiffusion model with fractional time derivatives, establishing stability results and proposing an effective numerical reconstruction algorithm.

Contribution

The work extends stability analysis of inverse potential problems from classical to fractional subdiffusion models and introduces a practical algorithm for coefficient recovery.

Findings

Proved local Lipschitz stability for small terminal time

Extended classical results to fractional subdiffusion case

Demonstrated efficiency and accuracy of the numerical algorithm

Abstract

In this work, we study the inverse problem of recovering a potential coefficient in the subdiffusion model, which involves a Djrbashian-Caputo derivative of order $\alpha\in(0,1)$ in time, from the terminal data. We prove that the inverse problem is locally Lipschitz for small terminal time, under certain conditions on the initial data. This result extends the result in Choulli and Yamamoto (1997) for the standard parabolic case to the fractional case. The analysis relies on refined properties of two-parameter Mittag-Leffler functions, e.g., complete monotonicity and asymptotics. Further, we develop an efficient and easy-to-implement algorithm for numerically recovering the coefficient based on (preconditioned) fixed point iteration and Anderson acceleration. The efficiency and accuracy of the algorithm is illustrated with several numerical examples.