# Multi-frequency iterative methods for the inverse medium scattering problems in elasticity

## TL;DR

This paper develops multi-frequency iterative algorithms to accurately reconstruct elastic parameters of an inhomogeneous medium in elasticity using scattering data, with numerical validation showing effectiveness.

## Contribution

It introduces two Landweber iterative algorithms leveraging multi-frequency data and phaseless measurements for elastic parameter reconstruction.

## Key findings

- Algorithms successfully reconstruct elastic parameters with high accuracy.
- Plane pressure waves improve reconstruction quality.
- Numerical examples validate the proposed methods.

## Abstract

This paper concerns the reconstruction of multiple elastic parameters (Lam\'e parameters and density) of an inhomogeneous medium embedded in an infinite homogeneous isotropic background in $\mathbb{R}^2$. The direct scattering problem is reduced to an equivalent system on a bounded domain by introducing an exact transparent boundary condition and the wellposedness of the corresponding variational problem is established. The Fr\'{e}chet differentiability of the near-field scattering map is studied with respect to the elastic parameters. Based on the multi-frequency measurement data and its phaseless term, two Landweber iterative algorithms are developed for the reconstruction of the multiple elastic parameters. Numerical examples, indicating that plane pressure incident wave is a better choice, are presented to show the validity and accuracy of our methods.