Numerical Reconstruction and Analysis of Backward Semilinear Subdiffusion Problems

TL;DR

This paper develops a numerical scheme for the backward semilinear subdiffusion problem, establishing stability, regularization, and error estimates, and demonstrates its effectiveness through convergence analysis and numerical experiments.

Contribution

It introduces a fully discrete regularized scheme using finite element and convolution quadrature methods with proven error bounds and a convergent iterative solver.

Findings

The scheme achieves optimal error estimates for smooth and nonsmooth data.

The iterative algorithm converges linearly under certain conditions.

Numerical examples confirm the theoretical stability and accuracy.

Abstract

This paper aims to develop and analyze a numerical scheme for solving the backward problem of semilinear subdiffusion equations. We establish the existence, uniqueness, and conditional stability of the solution to the inverse problem by applying the smoothing and asymptotic properties of solution operators and constructing a fixed-point iteration. This derived conditional stability further inspires a numerical reconstruction scheme. To address the mildly ill-posed nature of the problem, we employ the quasi-boundary value method for regularization. A fully discrete scheme is proposed, utilizing the finite element method for spatial discretization and convolution quadrature for temporal discretization. A thorough error analysis of the resulting discrete system is provided for both smooth and nonsmooth data. This analysis relies on the smoothing properties of discrete solution operators, some nonstandard error estimates optimal with respect to data regularity in the direct problem, and the arguments used in stability analysis. The derived a priori error estimate offers guidance for selecting the regularization parameter and discretization parameters based on the noise level. Moreover, we propose an easy-to-implement iterative algorithm for solving the fully discrete scheme and prove its linear convergence. Numerical examples are provided to illustrate the theoretical estimates and demonstrate the necessity of the assumption required in the analysis.