Numerical Reconstruction and Analysis of Backward Semilinear Subdiffusion Problems

TL;DR

This work tackles the inverse problem of reconstructing the initial data $u_0$ from terminal observations in a semilinear time-fractional subdiffusion model with a Djrbashian--Caputo derivative. It develops a rigorous framework for the backward problem, proves existence, uniqueness, and a stability result via a fixed-point argument, and regularizes the mildly ill-posed problem using a quasi-boundary value method. A fully discrete numerical scheme is then built with finite elements in space and convolution quadrature in time, and a priori error estimates are derived for both smooth and nonsmooth data, guiding the choice of discretization and regularization parameters. An easy-to-implement iterative solver with linear convergence is proposed, and numerical experiments confirm the theoretical rates, illustrate the impact of data regularity and the necessity of a terminal-time restriction, and demonstrate the method’s practical viability for nonlinear subdiffusion. The results provide a comprehensive framework for stable numerical reconstruction in nonlinear subdiffusion and highlight important directions for extending the theory to larger times and more general data scenarios.

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.

Figures (5)

• Figure 1: Convergence histories of Algorithm \ref{['alg']} with different $T$, $\alpha$ and $L$.
• Figure 2: Convergence histories of Algorithm \ref{['alg']} with different $T$, and $L$.
• Figure 3: The numerical reconstruction $U_{h,\gamma}^{0,\delta}$ for $T=1$ with different $\alpha$ and $\delta$.
• Figure 4: The numerical reconstruction $U_{h,\gamma}^{0,\delta}$ for large $T=10$ with different $\alpha$ and $\delta$.
• Figure 5: The numerical reconstruction $U_{h,\gamma}^{0,\delta}$ for $T=0.1$ with $\alpha=1$ and different $\delta$.

Theorems & Definitions (50)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.1
• Lemma 2.4
• proof
• Theorem 2.1