Inverse problems, and the algorithms that can solve them, are of great importance in mathematics, engineering, and science. Mathematical inverse problems are at the core of the technology in tomography and CAT scans, MRI, spectroscopy, ultrasound, radar, and many other technologies.
One famous inverse problem is the problem of phase retrieval which arises in the experimental uses of diffraction to determine intrinsic structure. The diffracted wave form contains complete information about the object that causes the diffraction. But most methods of measuring this wave form yields only the intensity of the signal and not the phase information. Also, in many experimental situations one is measuring the intensity of the Fraunhofer diffraction, not the intensity of the Fresnel diffraction. But the Fraunhofer diffraction is approximately a Fourier transform of the original distribution that caused the diffraction. So the phase retrieval problem becomes the mathematical problem of determining a distribution from the modulus of its Fourier transform.
This problem is a very interesting one, flexible not only in what context one considers it, but also flexible in how much technical mathematical background one needs to study it and make significant advances. Also, depending on the focus one takes, the phase retrieval problem becomes almost exclusively an algebraic or analytic problem. For example, discrete distribution yield phase retrieval problems in which one is trying to determine an element D in the group ring K[Zn] given one knows only D*D*. Very interesting factorization questions arise if K is the field of rational numbers, but one wants only solutions D with integer, or whole number, coefficients. The role of unique factorization and the use of the prime factors to create solutions to this phase retrieval problem are incompletely solved when one puts coefficient constraints on the solutions. As another example, but one where real and complex analysis play a central role, one can try to reconstruct an analytic function f knowing only the value of g(z) = f(x)f(x). One finds then that what one needs is to know the location of the complex zeros of f and go through a process called zero-flipping to find possible solutions f given knowledge only of g. But given only some data at irregularly spaced points on the real line (a typical situation in an experimental context), one would have to extrapolate and use a numerical algorithm to find these zeros. The questions of which algorithms work effectively and which ones do not are very interesting questions in computational analysis.
There is a great deal of physics, optics, and chemistry that comes into practical, effective algorithms that deal with inverse problems in general, and phase retrieval in particular. Behind these algorithms are algebraic and analytic mathematical techniques and theorems that are accessible to undergraduate students for study and research. This type of mathematics also has many interesting aspects that are incompletely understood by engineers, scientists, and mathematicians. This makes inverse problems an excellent theme for a summer research experience.
References
1. The Structure of Homometric Sets (joint with P. Seymour), SIAM
Journal of Algebraic and Discrete Methods 3 (1982) 343-350.
2. Phase Retrieval, Communications
in Mathematical Physics 5 (1984) 317-343.
3. Determining a Distribution from the Modulus
of its Fourier Transform, Complex Variables
and Applications 10 (1988) 319-326.
4. Homometric Elements in Semisimple Rings
(joint with D. Shapiro), Communications in
Algebra 17(12) (1989) 3043-3051.