Numerical Analysis of Simultaneous Reconstruction of Initial Condition and Potential in Subdiffusion

This paper investigates the simultaneous identification of a spatially dependent potential and the initial condition in a subdiffusion model based on two terminal observations. The existence, uniqueness, and conditional stability of the inverse problem are established under weak regularity assumptions through a constructive fixed-point iteration approach. The theoretical analysis further inspires the development of an easy-to-implement iterative algorithm. A fully discrete scheme is then proposed, combining the finite element method for spatial discretization, convolution quadrature for temporal discretization, and the quasi-boundary value method to handle the ill-posedness of recovering the initial condition. Inspired by the conditional stability estimate, we demonstrate the linear convergence of the iterative algorithm and provide a detailed error analysis for the reconstructed initial condition and potential. The derived \textsl{a priori} error estimate offers a practical guide for selecting regularization parameters and discretization mesh sizes based on the noise level. Numerical experiments are provided to illustrate and support our theoretical findings.