Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

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dc.contributor.advisorCzaja, Wojciechen_US
dc.contributor.authorHafftka, Arielen_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.accessioned2016-06-22T05:52:03Z
dc.date.available2016-06-22T05:52:03Z
dc.date.issued2016en_US
dc.description.abstractThis thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.en_US
dc.identifierhttps://doi.org/10.13016/M2WB7T
dc.identifier.urihttp://hdl.handle.net/1903/18246
dc.language.isoenen_US
dc.subject.pqcontrolledApplied mathematicsen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pqcontrolledMedical imagingen_US
dc.subject.pquncontrolledCompressed Sensingen_US
dc.subject.pquncontrolledMultidimensional Inverse Problemsen_US
dc.subject.pquncontrolledNuclear Magnetic Resonanceen_US
dc.subject.pquncontrolledRegularizationen_US
dc.subject.pquncontrolledRelaxometryen_US
dc.subject.pquncontrolledTensor Completionen_US
dc.titleTensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometryen_US
dc.typeDissertationen_US

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