An inverse potential problem for subdiffusion: stability and reconstruction

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 is an element of (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 [6] 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.

Automated summary

The study focuses on the inverse problem of recovering a potential coefficient in the subdiffusion model from terminal data, extending the result from the standard parabolic case to the fractional case. The analysis relies on refined properties of two-parameter Mittag-Leffler functions and an efficient algorithm for numerically recovering the coefficient is developed and illustrated with numerical examples.