L1 scheme for solving an inverse problem subject to a fractional diffusion equation

TL;DR

This paper analyzes the convergence of the L1 scheme for a fractional diffusion inverse problem, establishing error estimates and demonstrating robustness as the fractional order approaches one through theoretical and numerical validation.

Contribution

It provides a rigorous convergence analysis of the L1 scheme for fractional diffusion inverse problems, including error estimates and robustness as the fractional order nears one.

Findings

L1 scheme converges for sectorial operators with spectral angle less than π/2

Error estimates are explicitly related to the fractional order α

Numerical results confirm theoretical convergence and robustness

Abstract

This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle $< \pi/2 $, that is the resolvent set of this operator contains $ \{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< \theta\}$ for some $ \pi/2 < \theta < \pi $. The relationship between the time fractional order $\alpha \in (0, 1)$ and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as $ \alpha $ approaches $ 1 $. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results.