Inverse Problems at the University of Washington
The University of Washington Mathematics Department has a very active group working on inverse problems and related topics. This group at present consists of Ken Bube, Edward Curtis, Jim Morrow, John Sylvester and Gunther Uhlmann. Activities of this large and diverse group include several ongoing research programs, a weekly seminar on inverse problems, and a summer Research Experiences for Undergraduates (REU) program on discrete inverse problems directed by E. Curtis and J. Morrow.  In the past six years, 11 students have written PhD theses on different aspects of inverse problems, and there are currently four graduate students working in the field.

Here are brief introductions to the research interests of the faculty members in this group.

SEISMIC INVERSE PROBLEMS AND SEISMIC TRAVELTIME TOMOGRAPHY

Ken Bube has been working for a number of years on inverse problems in reflection seismology and seismic traveltime tomography. In reflection seismology, seismic waves which travel into the subsurface of the Earth are generated at or near the surface. Some of the energy of these waves is reflected from structures in the subsurface back to the surface, where they are measured. The goal is to determine the structure of the subsurface. Bube and some of his Ph.D. students have worked on several problems in reflection seismology, analyzing and proving convergence for numerical methods in idealized situations in stratified media, and studying the effect of different aspects of the physics of models (attenuation, viscoelasticity, etc.) on our ability to recover the material coefficients (density, wave speed, etc.). Some of this work is focused on the effective discretization of PDEs near discontinuities in material coefficients using immersed interface methods; having a good solver for the forward problem is important in solving the inverse problem numerically.

In seismic traveltime tomography, traveltimes from many source locations to many receiver locations are derived from seismic data. In surface reflection tomography, reflection traveltimes are measured, and the slowness field (reciprocal of wave speed) and reflector depths are to be determined. In crosswell tomography, transmission traveltimes (sometimes augmented with reflection traveltimes) are used to determine the slowness field between the wells (and sometimes also reflector positions). Bube and colleagues in industry have been working on both numerical issues (e.g., effective discretization, regularization, and numerical algorithms) and theoretical issues (e.g., uniqueness results for reflector depths, characterization of the slowness null space in linearized tomography, and characterizing the nonuniqueness in anisotropic traveltime tomography) in seismic traveltime tomography.

DISCRETE INVERSE PROBLEMS

Ed Curtis and Jim Morrow have been working in the area of discrete inverse problems for the last ten years. Inverse electrical network problems are typical discrete inverse problems. For example, the shape (up to Y-Delta equivalence) of a planar electrical network can be recovered from information gathered at the boundary of the network (the "response matrix"). The conductivities of the individual resistors can be computed from the response matrix. In addition one can characterize which matrices can occur as response matrices. This area of research is elementary enough that talented undergraduates can make genuine contributions. Curtis and Morrow are directors of a Research Experiences for Undergraduates program in which undergraduates work on these problems. Some of these students have continued their work and written senior theses or Ph.D. theses on discrete inverse problem.

LAYER STRIPPING FOR THE HELMHOLTZ EQUATION

John Sylvester has been working on one-dimensional inverse problems motivated by layer-stripping algorithms.

The fundamental task of science is to probe the world around us. The most powerful method for accomplishing this goal is to direct energy, in the form of waves, at an object and to observe the waves after they have interacted with that object. For example, a conventional photograph is produced by directing light waves from the flash bulb to the object under study and recording the image formed by the reflected waves on film.

While the results of such an experiment can be readily understood by the human scientist (we can just look at the picture), results of analogous experiments using waves which penetrate more deeply into the medium under study (e.g. microwaves, X-rays, and some sound waves) are less directly meaningful.

A wave which penetrates deeply into a medium gives us the opportunity to see below the surface. However, the picture we obtain is a stack of images of the top surface and the various layers below it, all superimposed on the same "photograph". In addition, there are even more superimposed images, formed from internal reflections between layers (multiple reflections).

John Sylvester continues to study mathematical methods for turning a "microwave photograph", made up of a stack of superimposed images, into a stack of individual photographs, each containing the image of a single layer of the medium. From these individual photographs, we see the true structure of the underlying medium.

Together with Dale Winebrenner at the Applied Physics Lab, he has developed a mathematically rigorous and highly stable inverse scattering algorithm for a lossless stratified medium (1-D Helmholtz equation), which has successfully treated experimental remote sensing data with high noise levels. The method is a layer-stripping method, based on a nonlinear Riesz transform, rather than a trace formula. It features a nonlinear Plancherel equality and some nonlinear Paley-Weiner theorems. Together with the Riesz transform, these allow us to develop a rather complete Fourier analysis, even in the presence of strong multiple reflections. They continue to investigate the extensions of these methods to more complicated (e.g. lossy and non-stratified) media. For a more detailed overview see http://www.math.washington.edu/~sylvest/.

INVERSE BOUNDARY AND SCATTERING PROBLEMS

Gunther Uhlmann's work on inverse problems involves a variety of inverse boundary value problems and inverse scattering problems.

Inverse boundary value problems are a class of problems in which the unknown coefficients of a partial differential equation represent internal parameters of a medium, and the known information consists of boundary measurements of the solutions. A prototypical example to which Uhlmann has devoted a lot of attention is electrical impedance tomography (EIT). In this non-invasive inverse method one attempts to determine the conductivity of a medium by making voltage and current measurements at the boundary. This problem arose in the early part of the century in geophysics exploration. More recently it has been proposed as a diagnostic tool in medicine. EIT also arises in non-destructive evaluation of materials. Of particular interest are the problems of crack and corrosion identification and the determination of conductivities of high contrast. Other physically interesting boundary value problems in which Uhlmann is working involve the determination of electromagnetic parameters by measuring the boundary components of the electric and magnetic field, and the determination of elastic parameters by making displacement and traction measurements at the boundary. Uhlmann and coauthors have developed mathematical methods that lead to analytic reconstruction methods for several of these problems.

Uhlmann has also worked on X-ray tomography which revolutionized the practice of many parts of medicine. The mathematics of X-ray tomography has traditionally been viewed as a special branch of integral geometry. In recent years, another viewpoint has developed in which it is seen as an inverse boundary value problem for a special (Boltzmann) transport equation. This context also includes single emission tomography and the newer technique of optical tomography, which is based on boundary measurements of near-infrared light transmitted through a body. A related inverse boundary value problem is diffuse tomography, which refers to low-energy imaging in which the paths of the radiant energy are not necessarily straight and are unknown.

The setting for inverse scattering is as follows. Far away from the target having unknown physical properties, a wave field is sent in. The scattered field is measured, and from this data one attempts to determine the properties of the scatterer. A basic example arises in quantum mechanics. The problem is to determine a potential in the Schrödinger equation from scattering information. In particular, Uhlmann has worked on the inverse scattering problem at a fixed energy. Another important problem he has considered in quantum inverse scattering is the inverse backscattering problem.