# An inverse acoustic-elastic interaction problem with phased or phaseless far-field data

## TL;DR

This paper develops a mathematical and numerical framework for solving inverse acoustic-elastic scattering problems using phased or phaseless far-field data, including techniques to address translation invariance in phaseless cases.

## Contribution

It introduces a coupled boundary value problem approach, boundary integral equations, and a reference ball technique for the first time to solve inverse elastic obstacle problems with phaseless data.

## Key findings

- The method achieves high accuracy in locating and shaping elastic obstacles.
- The reference ball technique effectively breaks translation invariance in phaseless data.
- Numerical experiments confirm the robustness and effectiveness of the proposed algorithms.

## Abstract

Consider the scattering of a time-harmonic acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous acoustic medium. This paper concerns an inverse acoustic-elastic interaction problem, which is to determine the location and shape of the elastic obstacle by using either the phased or phaseless far-field data. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The jump relations are studied for the second derivatives of the single-layer potential in order to establish the corresponding boundary integral equations. The well-posedness is discussed for the solution of the coupled boundary integral equations. An efficient and high order Nystr\"{o}m-type discretization method is proposed for the integral system. A numerical method of nonlinear integral equations is developed for the inverse problem. For the case of phaseless data, we show that the modulus of the far-field pattern is invariant under a translation of the obstacle. To break the translation invariance, an elastic reference ball technique is introduced. We prove that the inverse problem with phaseless far-field pattern has a unique solution under certain conditions. In addition, a numerical method of the reference ball technique based nonlinear integral equations is also proposed for the phaseless inverse problem. Numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed methods.