An inverse medium problem using Stekloff eigenvalues and a Bayesian approach††Dedicated to the memory of Armin Lechleiter, a dear friend of the third author.

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 of Stekloff eigenvalues is investigated by the reciprocity gap method. Then a Bayesian approach is proposed to estimate the index of refraction using a few reconstructed eigenvalues. Since the multiplicities of the eigenvalues, which can be complex, are unknown, we employ the recently developed spectral indicator method to compute Stekloff eigenvalues. Numerical experiments validate the effectiveness of the proposed method.