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IMA Tutorial
Inverse Problems and the Quantification of Uncertainty
April 19, 2002

Mathematics in Geosciences, September 2001 - June 2002

Organizers:

Philip B. Stark
Department of Statistics
University of California Berkeley
stark@stat.berkeley.edu

William W. Symes
Department of Computational and Applied Mathematics
Rice University
symes@caam.rice.edu

Schedule

April 19, 2002, 9:30-11:00am, Lind 409

Deterministic measures of uncertainty in inverse problems
Slides

William W. Symes

Abstract: Inverse problems are often formulated as functional equations or optimization problems. Thus the estimation of errors in the solutions amounts to the study of error propagation for these two classes of mathematical problems, at least from the deterministic perspective. A solution algorithm produces a solution estimate in which data error translates into both intrinsic (to the problem) and algorithmic (or numerical) estimation error. All real world data exhibits both data uncertainty (measurement error) and model incompatibility. Therefore any solution estimate which matches the data with some tolerance must be accepted as a "solution". The sets of solutions defined in this way are often very large, and the problem becomes one of determining what inference can be drawn about common features of the models which these solutions represent. Several factors have a large influence on the quality of the answer to this question, for example the choice of data misfit measure, the amount of misfit tolerated, and the type of additional data supplied to specify an acceptable model. These influences are fairly well understood for linear inverse problems, in which model and predicted data are linearly related. I will review the linear theory and present some examples of its application. Deterministic and statistical approaches to error estimation for linear inverse problems are parallel in several respects, and I will note some of the similarities and differences. Nonlinear inverse problems can exhibit a much wilder range of error propagation effects, many of which are at present only poorly understood. Some of the additional subtlety supplied by nonlinearity is nicely illustrated by examples from seismology.

April 19, 2002, 2:00-3:00pm, Lind 409

Statistical measures of uncertainty in inverse problems
Slides:    html    pdf    powerpoint

Philip B. Stark

Abstract: Inverse problems can be viewed as special cases of statistical estimation problems. From that perspective, one often can study inverse problems using standard statistical measures of uncertainty, such as bias, variance, mean squared error and other measures of risk, confidence sets, and so on. It is useful to distinguish between the intrinsic uncertainty of an inverse problem and the uncertainty of applying any particular technique for "solving" the inverse problem. The intrinsic uncertainty depends crucially on the prior constraints on the unknown (including prior probability distributions in the case of Bayesian analyses), on the forward operator, on the statistics of the observational errors, and on the nature of the properties of the unknown one wishes to estimate. I will try to convey some geometrical intuition for uncertainty, and the relation between the intrinsic uncertainty of linear inverse problems and the uncertainty of some common techniques applied to them.

#### LIST OF CONFIRMED PARTICIPANTS

Name Department Affiliation
Santiago Betelu Mathematics University of North Texas
Jamylle Carter   Institute for Mathematics & its Applications
Christine Cheng Institute for Mathematics & its Applications University of Minnesota
Dacian Daescu University of Minnesota Institute for Mathematics and its Applications
Gregory S. Duane University of Minnesota Institute for Mathematics and its Applications
Michael Efroimsky University of Minnesota Institute for Mathematics and its Applications
Selim Esedoglu   Institute for Mathematics & its Applications
Daniel Kern
Anna Mazzucato Mathematics Yale University
Aurelia Minut University of Minnesota Institute for Mathematics and its Applications
M. Yvonne Ou University of Minnesota Institute for Mathematics and its Applications
Jianliang Qian   Institute for Mathematics & its Applications
Professor Philip Stark Statistics University of California, Berkeley
Toshio Yoshikawa University of Minnesota Institute for Mathematics and its Applications

2001-2002 IMA Thematic Year on Mathematics in the Geosciences

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