Course Announcement

Course:  Math  496/896 (Seminar in Mathematics)

Topic: Inverse Theory

Time:  3:30 - 4:45, TR,  Spring Semester 2006 (or a mutually agreed alternate time).

Place: 12 Avery Hall

Instructor: Thomas Shores

Text:  Parameter Estimation and Inverse Problems by Aster, Borchers and Thurber, Elsivier 2005. 

Preq:  Math 314 (Linear Algebra) and 221 (Differential Equations), or permission.

Description: This topics course is intended to be an introduction to the field of inverse theory.  The typical order of business in a math course is: “Here is a problem, what is the solution?”  This is termed the “forward problem” in inverse theory.  Contrast this with the order in inverse theory, where the question is: “Here is a solution, what is the problem?”  It's rather like a  mathematical version of the popular television game Jeopardy.  A typical example is tomography, where we try to infer the physical properties of a solid by passing a wave  of some sort through the solid and examining the output.  Here the forward problem is to determine how waves behave when they pass through a material, and the inverse problem is to determine the material properties, given the behavior of the waves.  An example from engineering is to determine the thermal characteristics of a medium, given a set of heat measurements in an experiment with controlled heat sources.  Mathematical biology abounds with examples.  For instance, in ecology we might be given population measurements and want to to determine the growth parameters of the population model we use. Or in pharmacokinetics, we might introduce a drug into a body in a controlled fashion, take measurements of concentrations and hope determine exchange rates of the drug between organs in the body from the measurements.

All of these inverse problems share some characteristics: they are typically ill-posed (may have no solution, more than one or solutions that do not vary continuously with the parameters), they are more difficult to solve than direct problems and will almost certainly have error in the experimental measurement data. These difficulties require a body of special techniques, and that is what this course is all about.  Some of these techniques involve statistical or functional analytic ideas, but we will not assume any prerequisites in these topics.

There will be a midterm, a final and homework assignments.  Some Matlab programming will be required, but all of it can be done in the Mathematics Computer Lab and no prior experience with Matlab is required.

If anyone has questions about the course, contact Thomas Shores (tshores@math.unl.edu).