The adaptivity refines approximate solutions of ill-posed problems due to the relaxation property

TL;DR

This paper proves that adaptive finite element methods improve solutions of ill-posed problems through local mesh refinement, especially in coefficient inverse problems, supported by numerical results on simulated and experimental data.

Contribution

The paper provides a rigorous proof of adaptivity's image improvement property for ill-posed problems and demonstrates its effectiveness in a two-stage coefficient inverse problem approach.

Findings

Adaptive refinement enhances solution accuracy for ill-posed problems.

The two-stage procedure effectively approximates coefficients with limited initial information.

Numerical results confirm the method's applicability to real and simulated data.

Abstract

Adaptive Finite Element Method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property is the central part of this paper. In terms of Coefficient Inverse Problems with single measurement data, the authors consider the adaptivity as the second stage of a two-stage numerical procedure. The first stage delivers a good approximation of the exact coefficient without an advanced knowledge of a small neighborhood of that coefficient. This is a necessary element for the adaptivity to start iterations from. Numerical results for the two-stage procedure are presented for both computationally simulated and experimental data.