Inverse Problems and Scattering Poles

East Carolina University, Greenville NC

Investigators

Abstract

ABSTRACT DMS-0070823 Under this grant, Stefanov plans to work in two areas: (1) Inverse Problems and (2) Resonances (scattering poles). In the area of Inverse Problems, he plans to study some inverse problems for partial differential equations and related inverse problems in geometry. Some of the problems of interest are inverse problems in anisotropic media. Mathematically, they lead to problems of recovering a Riemannian metric from boundary data or scattering data. One example is the inverse kinematic problem, where one has to recover a Riemannian metric in a bounded domain from the lengths of geodesics connecting each two boundary points. This problem has natural applications in geophysics and is closely connected to the inverse problem of recovering a metric from the associated hyperbolic Dirichlet-to-Neumann (DN) map on the boundary. The PI plans to study this problem for larger classes of metrics than already known and to prove also stability estimates. The related linearized problem - recovery of tensors from integrals along geodesics will also be studied. The PI will also work on the (elliptic) anisotropic inverse boundary value problem. The possibility of developing an abstract approach to obtaining conditional stability estimates will be also studied and the PI plans to find sufficient conditions that would guarantee that one can derive a stability estimate (and therefore uniqueness) directly from the injectivity of the linearized problem. The PI will also conduct research in the area of resonances (scattering poles). The PI plans to study some physical systems where one can construct non-real quasimodes and to show that those quasimodes are close to resonances not converging to the real axis. Two such systems are the transmission problem and scattering by a strictly convex obstacle. The motivation for this project comes from its possible applications. Inverse Problems have numerous applications to geology, medicine, non-destructive evaluation of materials, etc. The anisotropic inverse problems are related to recovering parameters of media with properties (conductivity, wave speed, etc.) depending not only on the position but also on the direction. Conditional stability estimates show that small changes of the parameters of the system lead to small changes in the measured data under additional assumptions usually requiring sufficient smoothness of the coefficients. This demonstrates that the corresponding inverse problem is conditionally well-posed. Resonances arise naturally as resonance frequencies for systems in unbounded domains. They correspond to peaks in the measured data. For some systems, we plan to describe the asymptotic distribution of the resonances in some neighborhood of the real axis, which provides computational and qualitative tools for understanding those systems.