2025/05/06 by Xu Wu, Yang Jiang, Wu, Xu +3 · 2 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2505.03625
openalex publication_date 2025/05/06 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
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.