Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

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dc.contributor.advisorTadmor, Eitanen_US
dc.contributor.authorZhong, Mingen_US
dc.contributor.departmentApplied Mathematics and Scientific Computationen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2016-06-22T05:56:50Z
dc.date.available2016-06-22T05:56:50Z
dc.date.issued2016en_US
dc.description.abstractWe present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.en_US
dc.identifierhttps://doi.org/10.13016/M2GZ0N
dc.identifier.urihttp://hdl.handle.net/1903/18280
dc.language.isoenen_US
dc.subject.pqcontrolledApplied mathematicsen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledCompressed Sensingen_US
dc.subject.pquncontrolledDe-convolutionen_US
dc.subject.pquncontrolledInverse Problemsen_US
dc.subject.pquncontrollediterative methoden_US
dc.subject.pquncontrolledLinear Regressionen_US
dc.subject.pquncontrolledmultiscale methoden_US
dc.titleHierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problemsen_US
dc.typeDissertationen_US

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