Monotonicity-Based Regularization for Shape Reconstruction in Linear Elasticity

TL;DR

This paper introduces a novel regularization method based on monotonicity principles for shape reconstruction in elastic bodies, providing rigorous convergence guarantees and demonstrating robustness through numerical comparisons.

Contribution

It converts monotonicity methods into a regularization framework with proven convergence, improving reliability over traditional data-fitting approaches.

Findings

The method guarantees convergence for noisy data.

Numerical results show robustness against noise.

Compared favorably with standard Tikhonov regularization.

Abstract

We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from [5], but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.