In this talk we address the issue of stability for the first order perturbation of the biharmonic operator from partial data, in a bounded domain of dimension three or higher. Specifically, we shall consider two partial data settings: (1) Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement. (2) Assuming that the perturbations are known in a neighborhood of the boundary, measurements are performed only on arbitrarily small open subsets of the boundary. In both settings we obtain log type stability estimates. Part of this talk is based on a joint work with Salem Selim.
In this talk I will survey the classical Boundary Control method, originally developed by Belishev and Kurylev, which can be used to reduce an inverse problem for a hyperbolic equation, on a complete Riemannian manifold, to a purely geometric problem involving the so-called travel time data. For each point in the manifold the travel time data contains the distance function from this point to any point in a fixed a priori known closed observation set. If the Riemannian manifold is closed then the observation set is a closure of an open and bounded set, and in the case of a manifold with boundary the observation set is an open subset of the boundary. We will survey many known uniqueness and stability results related to the travel time data.
I will discuss inverse problems for metric spaces and their relation to more familiar types of inverse problems. Some non-metric problems have been recently solved with metric tools, and I will explain the benefits of this approach.
