Mathematics of Revealing Inaccessible Objects Using Linear and Nonlinear Waves

The project is concerned with the mathematical theory of inverse problems. Inverse problems arise in numerous medical and seismic imaging applications as well as in exploration geophysics and non-destructive evaluation, where one is interested in producing images of an inaccessible interior of a medium from measurements performed in the exterior. Typically, in order to reveal the internal structure, one measures the response of the medium when probed with different kinds of waves, ranging from electromagnetic waves to X-rays.

Recently, it has been observed that nonlinear seismic responses may give additional information concerning the interior structure of the Earth. Similarly, nonlinear ultrasound techniques in medical imaging may provide better images since the contrast in nonlinear media parameters is usually larger than that in the linear parameters. The project strives to develop significantly prominent mathematical methods where the nonlinear interaction of waves is used to bear on challenging inverse problems coming from applications.

These novel mathematical techniques may lead to significant advances, in particular in medical and seismic imaging. One of the specific focuses of the project is the Electrical Impedance Tomography problem for nonlinear anisotropic media, an imaging modality with applications in biomedical imaging and non-destructive testing of mechanical parts. An integral part of the project is concerned with the educational training of graduate students.

The project consists of four research topics. The first topic deals with partial data inverse problems for nonlinear elliptic partial differential equations (PDE), where nonlinear parameters of an unknown medium are to be determined from measurements performed along a small portion of the boundary. Despite the great significance of such inverse problems and their ubiquity in applications, they are among some of the most fundamental open questions in the field.

The goal here is to solve such problems for important nonlinear PDE, including the quasilinear anisotropic conductivity equation, building upon the recent advances by the investigator and collaborators, and to work towards the solution of these open problems in the linear setting. The second topic is devoted to the geometric version of the anisotropic Calderon problem, where one seeks to determine potential in the Schrodinger equation on a compact Riemannian manifold from boundary measurements.

By introducing a nonlinearity, the goal is to solve the anisotropic Calderon problem for the nonlinear Schrodinger equation in geometric settings for which the corresponding inverse problem in the linear case is still open. The third topic deals with inverse problems for the fundamental systems in physics and geometry, such as the anisotropic Maxwell and Yang-Mill’s systems.

The fourth topic is concerned with inverse problems for nonlinear hyperbolic PDE on manifolds with boundary, aiming to exploit the nonlinearity as a tool to solve some of them in cases when the linear counterpart is open.

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.