# Elastic energy regularization for inverse obstacle scattering problems

## TL;DR

This paper introduces a novel regularization method using elastic energy for inverse obstacle scattering, enabling the reconstruction of complex shapes including non-star-shaped curves with proven stability and convergence.

## Contribution

It proposes a shape manifold approach with elastic energy regularization and M"obius energy penalization, providing theoretical guarantees and a numerical method for non-star-shaped obstacle reconstruction.

## Key findings

- Successfully reconstructs non-star-shaped obstacles
- Proves regularization and convergence properties
- Demonstrates numerical feasibility with examples

## Abstract

By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how self-intersections can be avoided by penalization with the M\"obius energy and prove the regularizing property of our approach as well as convergence rates under variational source conditions. In the second part of the paper the discrete setting is introduced, and we describe a numerical method for finding the minimizer of the Tikhonov functional on a shape-manifold. Numerical examples demonstrate the feasibility of reconstructing non-star-shaped obstacles.