Identification of potential in diffusion equations from terminal observation: analysis and discrete approximation

TL;DR

This paper develops a theoretical and numerical framework for recovering spatially dependent potentials in diffusion equations from final time data, including stability analysis, a discrete reconstruction scheme, and convergence guarantees.

Contribution

It introduces a monotone operator approach for potential identification, proves uniqueness and stability, and develops a convergent discrete reconstruction algorithm with error estimates.

Findings

The fixed point method guarantees unique potential recovery.

The discrete scheme converges linearly under certain conditions.

Numerical experiments validate the theoretical error estimates.

Abstract

The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using Galerkin finite element method in space and finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem, and present a thorough error analysis for the reconstructed potential. Our derived \textsl{a priori} error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis.