Inverse Problems - the Real World's reaction to our inquiries

http://www.inverse-problems.com/
http://web.ift.uib.no/~antonych/invG.html
http://www.ipgp.jussieu.fr/~tarantola/
http://sepwww.stanford.edu/sep/berryman/main.html
http://www.me.ua.edu/inverse/2icipe.html
Apps to Image Estimation: http://sepwww.stanford.edu/sep/prof/gee/toc_html/

What is an Inverse Problem?

Inverse Theory is concerned with the problem of making inferences about physical systems from data (usually remotely sensed). Since nearly all data are subject to some uncertainty, these inferences are usually statistical. Further, since one can only record finitely many (noisy) data and since physical systems are usually modeled by continuum equations (at least geophysical ones are) no geophysical inverse problems are really uniquely solvable: if there is a single model that fits the data there will be an infinity of them. (A model is a parameterization of the system, usually a function.) Our goal then is to characterize the set of models that fit the data and satisfy our prejudices as well as other information.

To make these inferences quantitative one must answer three fundamental questions. How accurately are the data known? I.e., what does it mean to ``fit'' the data. How accurately can we model the response of the system? In other words, have we included all the physics in the model that contribute significantly to the data. Finally, what is known about the system independent of the data? This is called a priori information and is essential since for any sufficiently fine parameterization of a system there will be unreasonable models that fit the data too. Prior information is the means by which we reject or down-weight unreasonable models.

This course focuses on the theoretical and practical aspects of inverse problems. We will use a broad range of examples to illustrate the basic ideas of how one makes inferences about physical systems from real data. Specific applications may be selected from the students' areas of interest. The mathematical tools that we will use will be primarily those of probability and statistics and linear algebra.
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