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

This 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.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We 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.

MULTI-UNIT ACCIDENT CONTRIBUTIONS TO U.S. NUCLEAR REGULATORY COMMISSION QUANTITATIVE HEALTH OBJECTIVES: A SAFETY GOAL POLICY ANALYSIS USING MODELS FROM STATE-OF-THE-ART REACTOR CONSEQUENCE ANALYSES

The U.S. Nuclear Regulatory Commission implemented a safety goal policy in response to the 1979 Three Mile Island accident. This policy addresses the question “How safe is safe enough?” by specifying quantitative health objectives (QHOs) for comparison with results from nuclear power plant (NPP) probabilistic risk analyses (PRAs) to determine whether proposed regulatory actions are justified based on potential safety benefit. Lessons learned from recent operating experience—including the 2011 Fukushima accident—indicate that accidents involving multiple units at a shared site can occur with non-negligible frequency. Yet risk contributions from such scenarios are excluded by policy from safety goal evaluations—even for the nearly 60% of U.S. NPP sites that include multiple units. This research develops and applies methods for estimating risk metrics for comparison with safety goal QHOs using models from state-of-the-art consequence analyses to evaluate the effect of including multi-unit accident risk contributions in safety goal evaluations.

CHARACTERIZATION OF RADIATION DAMAGE TO A NOVEL PHOTONIC CRYSTAL SENSOR

New methods of nuclear fuel and cladding characterization must be developed and implemented to enhance the safety and reliability of nuclear power plants. One class of such advanced methods is aimed at the characterization of fuel performance by performing minimally intrusive in-core, real time measurements on nuclear fuel on the nanometer scale. Nuclear power plants depend on instrumentation and control systems for monitoring, control and protection. Traditionally, methods for fuel characterization under irradiation are performed using a “cook and look” method. These methods are very expensive and labor-intensive since they require removal, inspection and return of irradiated samples for each measurement. Such fuel cladding inspection methods investigate oxide layer thickness, wear, dimensional changes, ovality, nuclear fuel growth and nuclear fuel defect identification. These methods are also not suitable for all commercial nuclear power applications as they are not always available to the operator when needed. Additionally, such techniques often provide limited data and may exacerbate the phenomena being investigated. This thesis investigates a novel, nanostructured sensor based on a photonic crystal design that is implemented in a nuclear reactor environment. The aim of this work is to produce an in-situ radiation-tolerant sensor capable of measuring the deformation of a nuclear material during nuclear reactor operations. The sensor was fabricated on the surface of nuclear reactor materials (specifically, steel and zirconium based alloys). Charged-particle and mixed-field irradiations were both performed on a newly-developed “pelletron” beamline at Idaho State University's Research and Innovation in Science and Engineering (RISE) complex and at the University of Maryland's 250 kW Training Reactor (MUTR). The sensors were irradiated to 6 different fluences (ranging from 1 to 100 dpa), followed by intensive characterization using focused ion beam (FIB), transmission electron microscopy (TEM) and scanning electron microscopy (SEM) to investigate the physical deformation and microstructural changes between different fluence levels, to provide high-resolution information regarding the material performance. Computer modeling (SRIM/TRIM) was employed to simulate damage to the sensor as well as to provide significant information concerning the penetration depth of the ions into the material.