A computational method for the inverse transmission eigenvalue problem

In this work, we consider the inverse transmission eigenvalue problem to determine the refractive index from transmission eigenvalues. We adopt a weak formulation of the problem and provide a Galerkin scheme in to compute transmission eigenvalues. Using a proper operator representation of the problem, we show convergence of the method. Next, we define the inverse transmission problem and show that numerically the problem can be considered as a discrete inverse quadratic eigenvalue problem. First, we investigate the case of a spherically symmetric piecewise constant refractive index and confirm our results with analytic computations. Then, we show that a relative small number of eigenvalues are sufficient for simple cases of a few layers by just minimizing the total error between measured and computed eigenvalues to reconstruct the refractive index. Finally, we propose a computational method based on a Newton-type algorithm for reconstructions of a general piecewise constant refractive index for any domain from transmission eigenvalues. The algorithm can be performed without having knowledge of the exact position of the eigenvalues in the spectrum.