This proposal concerns inverse problems for anisotropic media, in particular, questions on stability, global uniqueness, and boundary determination. In the first project (joint with A. Sa Barreto) the PI will study the continuous dependence of a smooth Riemannian metric g (defined on a bounded, n-dimensional domain) on the Dirichlet-to-Neumann map associated with the wave equation for the Laplace-Beltrami operator, up to the pullback by a diffeo-morphism that fixes the boundary. The second project is concerned with global uniqueness for a certain class of anisotropic media. Techniques developed here will be applied to an inverse problem for elastodynamics with initial stress. The aim of the third project is to describe conditions under which the coefficients of the operator for elastodynamics with initial stress are determined to infinite order at the boundary by the Dirichlet-to-Neumann map.
Arising in diverse settings, inverse problems stand out as compelling examples of the potential for application of mathematical techniques to practical problems. Inverse problems involve describing internal properties, such as the conductivity of an airplane wing, the elasticity of subterranean ore deposits, or the density of human tissue, given only measurements made externally. For objects studied via an inverse problem in elasto-dynamics, for example, it is known in some cases that the density and elastic properties of the objects are uniquely determined by measurements made at the surface. That is, if two objects with the same size, shape, and orientation respond in the same way to certain tests made only at the surface, then it is guaranteed that they do, in fact, have the same density and elastic properties throughout. In the second and third projects mentioned above the PI will consider elastic objects with initial stress. Events in the past may have built up an initial stress in the object, for example, through welding or seismic fault activity. Some error is inherent in any real measurements, though, and so the aim of the first project mentioned above is to show that only certain ambiguities must be taken into account in order to reconstruct properties of the object arbitrarily accurately from approximate surface measurements.
