Math 581F/582G/583F
Inverse Problems

Gunther Uhlmann

Autumn 2005/Winter 2006/Spring 2006, Monday/Wednesday, 2:30--3:50 (10:00-11:15 in Spring)
We will consider the mathematical developments in the study of several inverse problems. The fall will be devoted to the mathematical study of the X-ray transform and Radon transform. In this case, the inverse problem consists of determining a function from its integral along lines or planes. The Radon transform and the X-ray transform are the fundamental transforms studied in integral geometry. We will develop the theory of these transforms from scratch. X-ray tomography played a fundamental role in the development of CT scans and therefore revolutionized the practice of many parts of medicine. During the past few years, there have been notable developments in the mathematics of X-ray tomography, including an inversion formula for the attenuated X-ray transform arising in SPECT (single positron emission tomography) and an inversion formula for the X-ray transform with sources on a helix (helical tomography) or more general curves.

In the second quarter of the course, we will consider several inverse boundary value problems associated to partial differential equations (PDEs). In general terms, the physical situation at hand is modeled by a PDE. The inverse boundary value problem is to determine the internal parameters of a medium (the coefficients of the PDE) given some information at the boundary of the medium. A prototypical example is Electrical Impedance Tomography which consists in the determination of the electrical conductivity of a body by making current and voltage measurements at the boundary. In this case, we study an inverse boundary problem associated to a second order elliptic equation. Another example is Optical Tomography which is based on boundary measurements of near infrared light transmitted through a body. In mathematical terms, this consists in studying an inverse boundary value problem for a special (Boltzmann) transport equation. No previous knowledge of PDE will be assumed; we will develop the required background in the class.

The third quarter will be urn as a graduate seminar with the students giving lectures from a selected list of articles on inverse problems, including inverse scattering, inverse spectral problems, boundary rigidity, and reflection seismology.

Prerequisites. The only prerequisite for the course is the material usually covered in Real Analysis (Math 524/5/6) or Linear Analysis (Math 554/5/6). No previous knowledge of PDEs will be assumed; we will develop the required background in class.

References. For the first quarter: Mathematics of Medical Imaging, by C. Epstein; The Mathematics of Computerized Tomography by F. Natterer, The Radon Transform and Some of Its Applications by S. Deans, and The Radon Transform by S. Helgason. For the second quarter, I will supply notes with further references. In the third quarter, we will study research articles.