Math Seminars.
Y. Kurylev (Loughborough)
Boundary Control method for anisotropic inverse boundary value problems I & II
The bulk of these talks is devoted to the development of basic
ideas of the boundary control method in inverse problems. We consider the
example of the inverse problem of the reconstruction of a Riemannian
manifold with a boundary, i.e. its differentiable and metric structures,
from the so-called Gel'fand boundary spectral data. These are the
eigenvalues and restrictions to the boundary (or part of the boundary) of
the eigenfunctions of the Laplace operator with Neumann boundary
conditions.
We prove the uniqueness (up to an isometry) and describe a reconstruction
algorithm. The method is based on elements of the PDE-control, spectral
theory, properties of the hyperbolic PDE's, as well as a special
representation of the Riemannian manifold. If time permits, the rest of
talks will be devoted to more advanced aspects of the boundary control
method including invers problems for the Maxwell and Dirac systems,
equivalence of inverse problems of different types, stability in inverse
problems, etc.