# An inverse medium problem using Stekloff eigenvalues and a Bayesian approach

## TL;DR

This paper introduces a novel approach combining integral equations, Bayesian estimation, and spectral indicators to reconstruct Stekloff eigenvalues and the index of refraction of inhomogeneous media from boundary data, validated by numerical experiments.

## Contribution

It presents a new integral equation for the reciprocity gap method and integrates Bayesian and spectral indicator techniques for inverse spectral problems.

## Key findings

- Effective reconstruction of Stekloff eigenvalues demonstrated
- Bayesian method accurately estimates the index of refraction
- Numerical experiments confirm the approach's robustness

## Abstract

This paper studies the reconstruction of Stekloff eigenvalues and the index of refraction of an inhomogeneous medium from Cauchy data. The inverse spectrum problem to reconstruct Stekloff eigenvalues is investigated using a new integral equation for the reciprocity gap method. Given reconstructed eigenvalues, a Bayesian approach is proposed to estimate the index of refraction. Moreover, since it is impossible to know the multiplicities of the reconstructed eigenvalues and since the eigenvalues can be complex, we employ the recently developed spectral indicator method to compute Stekloff eigenvalues. Numerical experiments validate the effectiveness of the proposed methods.