Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / In this dissertation, the author presents two independent researches on inverse problems: (1) creating materials in which heat propagates a long a line and (2) 3D inverse scattering problem with non-over-determined data. The theories of these methods were developed by Professor Alexander Ramm and are presented in Chapters 1 and 3. The algorithms and numerical results are taken from the papers of Professor Alexander Ramm and the author and are presented in Chapters 2 and 4.

Abstract The dissertation considers an inverse acoustic obstacle scattering problem in which the incident field is generated by a point source and the measurements are made in the near field region. Three methods to solve the problem of reconstructing the support of an unknown sound-soft or sound-hard scatterer from the near field measurements are presented. Methods are modifications of Kirsch factorization and modified Kirsch factorization methods. Numerical examples are given to show the practicality of one of the methods.

There are two commonly discrete approximations for the inverse conductivity problem. Finite element models are heavily used in electrical impedance tomography research as they are easily adapted to bodies of irregular shapes. The other approximation is to use electrical resistor networks for which several uniqueness results and reconstruction algorithms are known for the inverse problem. In this thesis the link between finite element models and resistor networks is established. For the planar case we show how resistor networks associated with a triangular mesh have an isotropic embedding and we give conditions for the uniqueness of the embedding. Moreover, a layered finite element model parameterized by thevalues of conductivity on the interior nodes is constructed. Construction of the finite element mesh leads to a study of the triangulation survey problem. A constructive algorithm is given to determine the position of the nodes in the triangulation with a knowledge of one edge and the angles of the finite element mesh. Also we show that we need to satisfy the sine rule as aconsistency condition for every closed basic cycle that enclosing interior nodes and this is a complete set of independent constraints.

EIT is a method of imaging that exists for a century, initially in geophysics and in recent years in medical imaging. Even though the practical applications of EIT go back to the early 20th century the systematic study of the inverse conductivity problem started in the late 1970s, hence many aspects of the problem remain unexplored. In the study of the inverse conductivity problem usually Finite Element Models are used since they can be easily adapted for bodies of irregular shapes. In this work though we use an equivalent approximation, the electrical resistor network, for which many uniqueness results as well as reconstruction algorithms exist. Furthermore resistor networks are important for EIT since they are used to provide convenient stable test loads or phantoms for EIT systems. In this thesis we study the transfer resistance matrix of a resistor network that is derived from n-port theory and review necessary and sufficient conditions for a matrix to be the transfer resistance of a planar network. The so called “paramountcy” condition may be useful for validation purposes since it provides the means to locate problematic electrodes. In the study of resistor networks in relation to inverse problems it is of a great importance to know which resistor networks correspond to some Finite Element Model. To give a partial answer to this we use the dual graph of a resistor network and we represent the voltage by the logarithm of the circle radius. This representation in combination with Duffin’s non-linear resistor network theory provides the means to show that a non-linear resistor network can be embedded uniquely in a Euclidean space under certain conditions. This is where the novelty of this work lies.

The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem). A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem). In this thesis, the effects of adding lumped masses to an Euler-Bernoulli beam on its frequencies and their corresponding mode shapes are investigated for simply-supported as well as fixed-free boundary conditions. This investigation paves the way for proposing a method to impose two frequencies on a system consisting of a beam and a lumped mass by determining the magnitude of the mass as well as its position along the beam.

In this paper we outline a novel method for training a generative adversarial network based denoising model from an exclusively corrupted and unpaired dataset of images. Our model can learn without clean data or corrupted image pairs, and instead only requires that the noise distribution is able to be expressed analytically and that the noise at each pixel is independent. We utilize maximum a posteriori estimation as the underlying solution framework, optimizing over the analytically expressed noise generating distribution as the likelihood and employ the GAN as the prior. We then evaluate our method on several popular datasets of varying size and levels of corruption. Further we directly compare the numerical results of our experiments to that of the current state of the art unsupervised denoising model. While our proposed approach's experiments do not achieve a new state of the art, it provides an alternative method to unsupervised denoising and shows strong promise as an area for future research and untapped potential.

Acoustic sensing provides a possibility of measuring propulsion flow fields non-intrusively, and is of great interest because it may be applicable to cases that are difficult to measure with traditional methods. In this work, some of the successes and limitations of this technique are considered. In the first main result, the acoustic time of flight is shown to be usable along with a calibration curve in order to accurately estimate the thrust of two turbofan engines (1.0-1.5%). In the second, it is shown that acoustic tomography methods that only use the first ray paths to arrive cannot distinguish some relevant propulsion flow fields (i.e., different flow fields can have the same times of flight). In the third result we demonstrate, via the first validated acoustic tomography experiment on a turbofan engine, that a reasonable estimate of the flow can be produced despite this challenge. This is also the first successful use of acoustic tomography to reconstruct a compressible, multi-stream flow. / Doctor of Philosophy / Sound may be used to measure air flows, and has been used for this purpose in studies of the atmosphere for decades. In this work, the extension of the method to measure air flows in aircraft engines is considered. This is challenging for two main reasons. The first challenge is that aircraft engines are very loud, which makes it harder to accurately measure the sounds that are needed to determine the speeds and temperatures. In this work, we show that the thrust (the force made by an engine) may be accurately measured using sound despite this difficulty. The second challenge is that the temperatures and velocities involved are very large compared to those in the atmosphere. We show that these large variations in temperature and velocity can make it impossible to distinguish between two different air flows in certain circumstances. We also show that despite this limitation, sound can be used to produce a reasonable, though imperfect, estimate of the flow. In particular, the technique was successfully used to measure the varying temperatures and velocities in a jet engine, which has not been done successfully before.

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Inverse problems arise in a wide variety of applications including biomedicine, environmental sciences, astronomy, and more. Computing reliable solutions to these problems requires the inclusion of prior knowledge in a process that is often referred to as regularization. Most regularization techniques require suitable choices of regularization parameters. In this work, we will describe new approaches that use deep neural networks (DNN) to estimate these regularization parameters. We will train multiple networks to approximate mappings from observation data to individual regularization parameters in a supervised learning approach. Once the networks are trained, we can efficiently compute regularization parameters for newly-obtained data by forward propagation through the DNNs. The network-obtained regularization parameters can be computed more efficiently and may even lead to more accurate solutions compared to existing regularization parameter selection methods. Numerical results for tomography demonstrate the potential benefits of using DNNs to learn regularization parameters. / Master of Science / Inverse problems arise in a wide variety of applications including biomedicine, environmental sciences, astronomy, and more. With these types of problems, the goal is to reconstruct an approximation of the original input when we can only observe the output. However, the output often includes some sort of noise or error, which means that computing reliable solutions to these problems is difficult. In order to combat this problem, we can include prior knowledge about the solution in a process that is often referred to as regularization. Most regularization techniques require suitable choices of regularization parameters. In this work, we will describe new approaches that use deep neural networks (DNN) to obtain these parameters. We will train multiple networks to approximate mappings from observation data to individual regularization parameters in a supervised learning approach. Once the networks are trained, we can efficiently compute regularization parameters for newly-obtained data by forward propagation through the DNNs. The network-obtained regularization parameters can be computed more efficiently and may even lead to more accurate solutions compared to existing regularization parameter selection methods. Numerical results for tomography demonstrate the potential of using DNNs to learn regularization parameters.

Probing methods for trace estimation of large, sparse matrices has been studied for several decades. In recent years, there has been some work to extend these techniques to instead estimate the diagonal entries of these systems directly. We extend some analysis of trace estimators to their corresponding diagonal estimators, propose a new class of deterministic diagonal estimators which are well-suited to parallel architectures along with heuristic arguments for the design choices in their construction, and conclude with numerical results on diagonal estimation and ordering problems, demonstrating the strengths of our newly-developed methods alongside existing methods. / Master of Science / In the past several decades, as computational resources increase, a recurring problem is that of estimating certain properties very large linear systems (matrices containing real or complex entries). One particularly important quantity is the trace of a matrix, defined as the sum of the entries along its diagonal. In this thesis, we explore a problem that has only recently been studied, in estimating the diagonal entries of a particular matrix explicitly. For these methods to be computationally more efficient than existing methods, and with favorable convergence properties, we require the matrix in question to have a majority of its entries be zero (the matrix is sparse), with the largest-magnitude entries clustered near and on its diagonal, and very large in size. In fact, this thesis focuses on a class of methods called probing methods, which are of particular efficiency when the matrix is not known explicitly, but rather can only be accessed through matrix vector multiplications with arbitrary vectors. Our contribution is new analysis of these diagonal probing methods which extends the heavily-studied trace estimation problem, new applications for which probing methods are a natural choice for diagonal estimation, and a new class of deterministic probing methods which have favorable properties for large parallel computing architectures which are becoming ever-more-necessary as problem sizes continue to increase beyond the scope of single processor architectures.