Inverse Boundary Value Problems

University Of Washington, Seattle WA

Investigators

Abstract

The ability to determine the internal properties of a medium by making measurements at the boundary of the medium provides important insight in a wide range of scientific applications. The question is whether one can one "see" what is inside the medium by making measurements on the outside. This project involves establishing a deeper mathematical understanding of the inverse imaging technique called electrical impedance tomography (EIT), which arises both in medical imaging and geophysics. EIT attempts to determine the electrical properties of an object by making voltage and current measurements from electrodes located at the boundary of the object. This project will also investigate the question of determining the inner structure of the Earth by measuring the travel times of earthquakes measured at different seismic stations located throughout the Earth. Graduate students will be trained and contribute to these projects. This project will address the mathematical theory of several fundamental inverse problems arising in many areas of science and technology including medical imaging, geophysics, astrophysics and nondestructive testing, to name a few. Three topics of research will be addressed. The first one is Electrical Impedance Tomography (EIT), also called Calderon’s problem. The second topic is travel time tomography in anisotropic media. The third topic is inverse problems for non-linear hyperbolic equations. EIT is an inverse method used to determine the conductivity of a medium by making voltage and current measurements at the boundary. Specific projects will address mathematical challenges in developing and understanding the frameworks that address the case of partial data, anisotropic conductors, the recovery of discontinuities of a medium from boundary information, quasilinear model equations, and high frequencies for anisotropic media. An understanding of travel time tomography involves the determination of a Riemannian metric (anisotropic sound speed) in the interior of a domain from the lengths of geodesics joining points of the boundary (travel times) and from other kinematic information. This project will address the two dimensional scenario, the range characterization and boundary rigidity for simple manifolds, and a novel metric from the area of minimal surfaces bounded by closed curves on the boundary. The investigator will also develop a framework for using the interaction of waves to create new waves that will give information about the object being probed. Specific topics include the study of an inverse problem for the non-linear Klein Gordon equation and inverse problems arising in fluid dynamics. 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.