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.

Inverse problems, foundational in applied sciences, involve deducing system inputs from specific output observations. These problems find applications in diverse domains such as aerospace engineering, weather prediction, and oceanography. However, their solution often requires complex numerical simulations and substantial computational resources. Modern machine learning based approaches have emerged as an alternative and flexible methodology for solving these types of problems, however their generalization power often comes at the cost of working with large descriptive datasets, a requirement that many applications cannot afford. This thesis proposes and explores the novel Deep Multi-resolution Operator Network (DMON), inspired by the recently developed DeepONet architecture. The DMON model is designed to solve inverse problems related to chaotic non-linear systems with low-resolution data through intelligently utilizing high-resolution data from a similar system. Performance of the DMON model and the proposed selection mechanisms are evaluated on two chaotic systems, a double pendulum and turbulent flow around a cylinder, with improvements observed under idealized scenarios whereby high and low-resolution inputs are manually paired, along with minor improvements when this pairing is conducted through the proposed the latent space comparison selection mechanism. / Master of Science / In everyday life, we often encounter the challenge of determining the cause behind something we observe. For instance, meteorologists infer weather patterns based on limited atmospheric data, while doctors use X-rays and CT scans to reconstruct images representing the insides of our bodies. Solving these so called ``inverse problems'' can be difficult, particularly when the process is chaotic such as the weather, whereby small changes result in much larger ones over time. In this thesis, we propose a novel method using artificial intelligence and high-resolution simulation data to aid in solving these types of problems. Our proposed method is designed to work well even when we only have access to a small amount of information, or the information available isn't very detailed. Because of this there are potential applications of the proposed method across a wide range of fields, particularly those where acquiring detailed information is difficult, expensive, or impossible.

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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.

Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This dissertation develops a complete framework for fully discrete adjoint sensitivity analysis and inverse problem solutions, in the context of time dependent, adaptive mesh, and adaptive step models. The discrete framework addresses all the necessary ingredients of a state–of–the–art adaptive inverse solution algorithm: adaptive mesh and time step refinement, solution grid transfer operators, a priori and a posteriori error analysis and estimation, and discrete adjoints for sensitivity analysis of flux–limited numerical algorithms. / Ph. D.