Analytic and Geometric Inverse Problems and Related Topics

University Of California-Irvine, Irvine CA

Investigators

Abstract

The goal of the project is to develop new mathematical techniques to attack central and challenging questions in inverse problems. Being at the core of technological advances and modern scientific investigations, inverse problems are concerned with determining properties of a medium from indirect, incomplete, or noisy observations. Applications of inverse problems range over a broad spectrum of geophysical, sonar, and medical imaging techniques, used for finding location of oil and mineral deposits in the interior of the Earth, early detection of pulmonary edema, as well as monitoring of lung function. A major topic for this project is the fundamental Calderon problem, which forms the basis of Electrical Impedance Tomography, an imaging modality with applications in biomedical imaging and nondestructive testing of mechanical parts. The Calderon problem asks whether the conductivity of a medium can be determined by performing voltage and current measurements on the surface of the medium. The specific focus of the project is on recovering the conductivity of a rough medium by performing measurements along possibly small portions of the boundary surface, a situation ubiquitous in practice, as well as on reconstructing the conductivity of an anisotropic medium, such as the muscle tissue in the human body, say. The investigator will also attack significant inverse problems arising in fluid dynamics, elasticity, and anisotropic linear and non-linear electromagnetism. The novel mathematical approaches to be developed in the project may lead to important advances in medical and seismic imaging. The project will focus on the following four significant research topics. The first one is inverse boundary problems with partial data in the low regularity setting. In this class of problems one seeks to determine low regularity coefficients of partial differential equations (PDE), representing characteristics of a medium, from boundary measurements performed along small portions of the boundary. Motivated by recovery of material parameters in anisotropic media, the second topic is concerned with geometric inverse boundary problems, where one wishes to recover also topological and differential characteristics of the underlying Riemannian manifold. The third topic proposes a broad and systematic attack on the phenomenon of increasing stability for partial data inverse boundary problems for elliptic PDE in the high frequency regime, driven by the need to design reconstruction algorithms with high resolution. The investigator aims to develop the innovative approach of bringing the powerful techniques of semiclassical analysis to bear on this important class of problems. Inspired by the study of the stability of matter and non-hermitian quantum mechanics, the fourth topic proposes to investigate the distribution of complex eigenvalues for non-self-adjoint operators of Schrodinger type, in the spirit of the classical Keller and Lieb-Thirring bounds. This research is to be carried out in the general setting of non-compact manifolds, whose structure at infinity generalizes the Euclidean one, relying upon the powerful techniques of geometric microlocal analysis. While seemingly quite independent of each other, the different parts of the project are in fact intimately connected, there being a synergy amongst them, and tools developed in one part, such as Carleman and resolvent estimates, say, can lead to progress in the others. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.