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

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Abstract: 0405978 P Lamm, Michigan State UniversityInverse Problems and their Applications: Deterministic and Statistical Methods for Variable Local Regularization In this project the principal investigator plans to developtheoretically-sound methods for the practical selection of variableregularization parameters as part of local regularization methods forinverse problems. The PI first proposes to undertake a deterministicapproach to the problem of variable parameter selection via thecoordination of local discrepancy principles and under the assumptionthat some local information about data error-level is available.Parts of this work will entail wavelet-based approximations used inconjunction with the local regularization ideas. In addition, the PIplans to study a statistical parameter estimation idea currently beingtested by researchers working on the problem of detecting ozone levelsin the atmosphere, and to give these ideas the theoretical basis theyare currently lacking. There is hope that powerful new methods forthe selection of variable local regularization parameters, methods notrequiring information about local noise-levels in the data, willemerge from this work. As part of the project the PI plans to applythis new class of methods to the ozone detection problem.Inverse problems occur widely in many applications, including problemsof biomedical imaging (CT scans and X-rays), image reconstruction(from satellites or other sources), the detection of ozone levels inthe atmosphere, and geophysical exploration. While classical methodsexist for for solving such problems, classical methods are often veryinefficient and lead to overly expensive solution techniques. Asecond disadvantage of classical solution methods can be seen inimaging applications where reconstructed images may have blurred edgesand inadequately detailed features. The PI has been working toaddress these difficulties with the development of new solutionmethods based on the ideas of local regularization. The use of thesenewer methods can lead to a significant decrease in cost for thesolution of a wide class of practical inverse problems, with improvedresolution of detailed features of solutions.