Solving Inverse Problems with Spectral Data

We consider a two dimensional membrane. The goal is to find properties of the membrane or properties of a force on the membrane. The data is natural frequencies or mode shape measurements. As a result, the functional relationship between the data and the solution of our inverse problem is both indirect and nonlinear. In this paper we describe three distinct approaches to this problem. In the first approach the data is mode shape level sets and frequencies. Here formulas for approximate solutions are given based on perturbation results. In the second approach the data is frequencies and boundary mode shape measurements; uniqueness results are obtained using the boundary control method. In the third approach the data is frequencies for four boundary value problems. Local existence, uniqueness results are established together with numerical results for approximate solutions.