Inverse Problems and Their Applications: Deterministic and Statistical Methods for Variable Local Regularization

Michigan State University, East Lansing MI

Investigators

Abstract

Abstract: 0405978 P Lamm, Michigan State University Inverse Problems and their Applications: Deterministic and Statistical Methods for Variable Local Regularization In this project the principal investigator plans to develop theoretically-sound methods for the practical selection of variable regularization parameters as part of local regularization methods for inverse problems. The PI first proposes to undertake a deterministic approach to the problem of variable parameter selection via the coordination of local discrepancy principles and under the assumption that some local information about data error-level is available. Parts of this work will entail wavelet-based approximations used in conjunction with the local regularization ideas. In addition, the PI plans to study a statistical parameter estimation idea currently being tested by researchers working on the problem of detecting ozone levels in the atmosphere, and to give these ideas the theoretical basis they are currently lacking. There is hope that powerful new methods for the selection of variable local regularization parameters, methods not requiring information about local noise-levels in the data, will emerge from this work. As part of the project the PI plans to apply this new class of methods to the ozone detection problem. Inverse problems occur widely in many applications, including problems of biomedical imaging (CT scans and X-rays), image reconstruction (from satellites or other sources), the detection of ozone levels in the atmosphere, and geophysical exploration. While classical methods exist for for solving such problems, classical methods are often very inefficient and lead to overly expensive solution techniques. A second disadvantage of classical solution methods can be seen in imaging applications where reconstructed images may have blurred edges and inadequately detailed features. The PI has been working to address these difficulties with the development of new solution methods based on the ideas of local regularization. The use of these newer methods can lead to a significant decrease in cost for the solution of a wide class of practical inverse problems, with improved resolution of detailed features of solutions.