Bayesian inversion for an inverse spectral problem of transmission eigenvalues

The transmission eigenvalue problem plays an important role in the inverse scattering theory of inhomogeneous media. In particular, transmission eigenvalues can be reconstructed from scattering data and used to obtain qualitative information about the material properties of the scattering medium. In this paper, we consider the inverse spectral problem to determine the material properties given a few transmission eigenvalues. The lack of theoretical results motivates us to propose a Bayesian approach. The inverse problem is first formulated as a statistical inference problem using the Bayes’ theorem. Then, the MCMC algorithm is used to compute the posterior density. Due to the non-uniqueness nature of the problem, we adopt the local conditional means (LCM) to characterize the posterior density function. Numerical examples show that the proposed method can provide useful information about the unknown material properties.