PHD, Kent State University, 2022, College of Arts and Sciences / Department of Mathematical Sciences

Linear ill-posed problems arise in various fields of science and engineering. Their solutions, if they exist, may not depend continuously on the observed data. To obtain stable approximate solutions, it is required to apply a regularization method. The main objective of this dissertation is to investigate regularization approaches and develop some numerical methods for solving problems of this kind. This work begins with an overview of linear ill-posed problems in continuous and discrete formulations. We review the most common regularization methods relying on some factorizations of the system matrix. Several iterative regularization strategies based on Krylov subspace methods are discussed, which are well-suited for solving large-scale problems. We then analyze the behavior of the symmetric block Lanczos method and the block Golub–Kahan bidiagonalization method when they are applied to the solution of linear discrete ill-posed problems. The analysis suggests that it generally is not necessary to compute the more expensive singular value decomposition when solving problems of this kind. The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. The numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from finite-dimensional linear algebra. We explore the feasibility of applying the Chebfun package to solve ill-posed problems with a regularize-first approach numerically. This allows a user to work with functions instead of vectors and with integral operators instead of matrices. The solution process is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods. The difficult process of explicitly choosing a suitable discretization is not required. The solution of linear ill-posed operator equations with the presence of errors in the operator and the data is discussed. An approximate solut (open full item for complete abstract)

Committee: Lothar Reichel (Advisor); Jing Li (Committee Member); Barry Dunietz (Committee Member); Qiang Guan (Committee Member); Jun Li (Committee Member) Subjects: Applied Mathematics

PHD, Kent State University, 2023, College of Arts and Sciences / Department of Mathematical Sciences

Ill-posed inverse problems arise in many fields of science and engineering in linear and nonlinear problems. Their solution, if it exists, is extremely sensitive to perturbations in the data (small perturbations in the data cause large oscillations in the obtained solution). The main challenge of working with these problems comes from the ill-conditioning, nonlinearity, and the possible large dimension of the problems. A well known approach such as regularization aims to reduce the sensitivity of the problem by replacing the given problem with a nearby one, whose solution is less affected by perturbations in the available data. In this dissertation we mainly consider nonlinear discrete ill-posed problems that arise from the Electrical Impedance Tomography (EIT) problem. EIT is a cheap, non-invasive, radiation-free imaging technique which is used to recover the internal conductivity of a body using measurements from electrodes from its surface. The typical technique is to place electrodes in the body and measure the conductivity inside the object. Low frequency current is applied on the electrodes below a threshold, making the technique harmless for the body. Mathematically, the reconstruction of the internal conductivity is a severely ill-posed inverse problem and yields a poor-quality solution. Moreover, the desired solution has sharp changes in the electrical properties that are typically challenging to be reconstructed by traditional smoothing regularization methods. To remedy this difficulty, one solves a regularized problem that is better conditioned than the original problem by imposing constraints on the regularization term. In this work we propose a method to solve the general ℓp nonlinear EIT problem through an iteratively reweighted majorization-minimization strategy combined with the Gauss-Newton approach. Simulated numerical examples from a complete electrode model illustrate the effectiveness of our approach.

Committee: Jing Li (Advisor); Lothar Reichel (Committee Member); Xiang Lian (Committee Member); Qiang Guan (Committee Member); Jun Li (Committee Member) Subjects: Applied Mathematics

PHD, Kent State University, 2022, College of Arts and Sciences / Department of Mathematical Sciences

This work is concerned with the development of accurate and efficient iterative methods for the solution of linear discrete ill-posed problems when the matrix involved is nonsymmetric. These problems often arise in science and engineering through the discretization of Fredholm integral equations of the first kind. The matrices that define these problems are typically severely ill-conditioned and may be rank-deficient. Because of this, the solution of linear discrete ill-posed problems may not exist or are very sensitive to perturbations caused by errors in the available data. These difficulties can be reduced, for example, by applying iterative regularization techniques. Krylov subspace projection strategies have been used in tandem with iterative methods to form efficient and accurate solution methods. Specifically, the Arnoldi iteration is a well known iterative process that constructs an orthonormal basis of a Krylov subspace. The opening focus is on the development of a novel "approximate Tikhonov regularization" method based on constructing a low-rank approximation to the matrix in the linear discrete ill-posed problem by carrying out a few steps of the Arnoldi process. The subsequent chapter focuses on the description of three iterative methods that modify the generalized minimum residual (GMRES), block GMRES, and global GMRES methods for the solution of appropriate linear systems of equations. The primary contribution to this field of this work is through the introduction of two block variants for when there are multiple right-hand sides in the linear system. In the final chapter the limitations of applying block GMRES methods to linear discrete ill-posed problems are discussed. While block algorithms can be executed efficiently on many computers, the work herein shows that available block algorithms may yield computed approximate solutions of unnecessarily poor quality. A termed "local'' block GMRES method that can overcome the problems associated with b (open full item for complete abstract)

Committee: Alessandro Buccini (Advisor); Lothar Reichel (Advisor) Subjects: Applied Mathematics

PHD, Kent State University, 2021, College of Arts and Sciences / Department of Mathematical Sciences

This work is concerned with structure preserving and other techniques for the solution of linear discrete ill-posed problems with transform-based tensor-tensor products, e.g., the t-product and the invertible linear transform product. Specifically, we focus on two categories of solution methods, those that involve flattening, i.e., reduce the tensor equation to an equivalent equation involving a matrix and a vector, and those that preserve the tensor structure by avoiding flattening. Various techniques based on Krylov subspace-type methods for solving third order tensor ill-posed problems are discussed. The data is a laterally oriented matrix or a general third order tensor. Regularization of tensor ill-posed problem by Tikhonov's approach and truncated iterations are considered. Golub-Kahan bidiagonalization-type, Arnoldi-type, and Lanczos-type processes are applied to reduce large-scale Tikhonov minimization problems to small-sized problems. A few steps of the t-product bidiagonalization process can be employed to inexpensively compute approximations of the singular tubes of the largest Frobenius norm and the associated left and right singular matrices. A less prohibitive computation of approximations of eigentubes of the largest Frobenius norm and the corresponding eigenmatrix by a few steps of the t-product Lanczos process is considered. The interlacing of the Frobenius norm of the singular tubes is shown and applied. The discrepancy principle is used to determine the regularization parameter and the number of iterations by a chosen method. Several truncated iteration techniques, e.g., SVD-like, and those based on the above processes are considered. Solution methods for the weighted tensor Tikhonov minimization problem with weighted global and non-global bidiagonalization processes are discussed. The weights on the fidelity and regularization parts of this problem are suitably defined symmetric positive definite (SPD) tensors. The computation of the inverse of (open full item for complete abstract)

Committee: Lothar (Advisor) Reichel (Committee Chair); Xiaoyu Zheng (Committee Member); Barry Dunietz (Committee Member); Sergij Shyanovskii (Committee Member); Jing Li (Committee Member) Subjects: Applied Mathematics

PHD, Kent State University, 2020, College of Arts and Sciences / Department of Mathematical Sciences

Ill-posed inverse problems arise in many fields of science and engineering. Their solution, if it exists, is very sensitive to perturbations in the data. In this thesis we consider linear discrete ill-posed problems. The challenge of working with these problems comes from the ill-conditioning and the possible large dimension of the problems. Regularization methods try to reduce the sensitivity by replacing the given problem with a nearby one, whose solution is less affected by perturbations. For small to medium size problems, we describe how the generalized singular value decomposition can be combined with iterated Tikhonov regularization and we illustrate that the method so obtained determines approximate solutions of higher quality than the more commonly used approach of pairing generalized singular value decomposition with (standard) Tikhonov regularization. The regularization parameter is determined with the aid of the discrepancy principle. In the remainder of the thesis we focus on large scale problems. They are solved by projecting them into a Krylov or generalized Krylov subspace of fairly small dimension. Bregman-type iterative methods have attracted considerable attention in recent years due to their ease of implementation and the high quality of the computed solutions they deliver. However, these iterative methods may require a large number of iterations and this reduces their attractiveness. We develop a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. The projection reduces both the number of iterations and the computational effort required for each iteration. A variant of this solution method, in which nonnegativity of each computed iterate is imposed, also is described. Recently, the use of a $p$-norm to measure the fidelity term and a $q$-norm to meas (open full item for complete abstract)

Committee: Lothar Reichel (Advisor); Alessandro Buccini (Advisor); Jing Li (Committee Member); Sergij Shiyanovskii (Committee Member); Arvind Bansal (Committee Member); Xiaoyu Zheng (Committee Member) Subjects: Applied Mathematics

PHD, Kent State University, 2018, College of Arts and Sciences / Department of Mathematical Sciences

The symmetric Lanczos method is commonly applied to reduce large-scale symmetric linear discrete ill-posed problems to small ones with a symmetric tridiagonal matrix. We investigate how quickly the nonnegative subdiagonal entries of this matrix decay to zero. Their fast decay to zero suggests that there is little benefit in expressing the solution of the discrete ill-posed problems in terms of the eigenvectors of the matrix compared with using a basis of Lanczos vectors, which are cheaper to compute. We will also show that a truncated singular value decomposition, made up of a few of the largest singular values and associated left and right singular vectors, of the matrix of a large-scale linear discrete ill-posed problems can be computed quite inexpensively by an implicitly restarted Golub-Kahan bidiagonalization method. Extensions to hybrid methods for the solution of linear discrete ill-posed problems with several right-hand side vectors will be made. Applications include multi-channel image restoration when the image degradation model is described by a linear system of equations with multiple right-hand sides that are contaminated by errors.

Committee: Lothar Reichel PhD (Advisor); Li Jing PhD (Committee Member); Li Jun PhD (Committee Member); Ruttan Arden PhD (Committee Member); Bansal Arvind PhD (Committee Member) Subjects: Applied Mathematics; Mathematics

PHD, Kent State University, 2014, College of Arts and Sciences / Department of Mathematical Sciences

The Tikhonov regularization method is a popular method to solve linear discrete illposed problems. The regularized problems can be solved with the aid of the generalized singular value decomposition (GSVD) when the problem is of small to medium size. This decomposition is not practical to use when the problem is of large size since the computation of the GSVD then is too expensive. The idea is to construct a solution subspace of small size with the aid of a generalized Krylov subspace method and find a solution in the solution subspace as an approximation to the solution in the full space. We refer to this as a reduction method. Several reduction methods for solving large Tikhonov regularization problems have been developed and are discussed in the lliterature. In this work we will add three novel reduction methods to this family. Our methods can give approximate solutions of higher accuracy than the GSVD and, therefore are attractive alternatives to the GSVD also when the matrices are small enough for the latter to be computed. In the context of ε-pseudospectrum computations, we propose a new rational Arnoldi method that is well suited for the situation when the rational functions involved have few distinct poles that are applied in a cyclic fashion.

Committee: Lothar Reichel Dr. (Advisor); Xiaoyu Zheng Dr. (Committee Member); Jing Li Dr. (Committee Member); Arden Ruttan Dr. (Committee Member); Paul A. Farrell Dr. (Committee Member) Subjects: Mathematics

PhD, University of Cincinnati, 2010, Engineering : Nuclear and Radiological Engineering

This work presents a method to empirically determine the photon spectrum of a megavoltage bremsstrahlung beam. The method makes use of the fact that scatter cross sections vary in a known fashion with incident photon energy. The distribution of scatter produced by a scattering object placed in a good geometry photon beam was measured. The scatter distribution was simulated for a series of monoenergetic good geometry photon beams. A system of linear equations was generated to combine the polyenergetic measurements with the monoenergetic simulations. Regularization techniques were applied to solve the system for the incident photon spectrum. Monte Carlo simulations were used to predict the ratio of primary to scattered photons for narrow mono-energetic photon beams at 9 different locations, with 10 degree increments and 15 cm from a scattering material. Measurements were performed in the same geometry using the photon beams produced by linear accelerators. A linear matrix system, A×F=T, was developed to describe the scattering interactions and their relationship to the primary spectrum. A is the monoenergetic scatter kernel determined from the Monte Carlo simulations, F is the incident photon spectrum, and T represents the scatter distribution characterized by empirical measurement. Direct matrix inversion methods produce results that are not physically consistent due to errors inherent in the system. Tikhonov regularization methods were applied to address the effects of these errors and solve the system for physically consistent bremsstrahlung spectra. The results of this research provide a method to empirically characterize the primary radiation energy spectrum produced by a linear accelerator. Key words: photon energy, spectrum, scatter analysis, regularization, Tikhonov

Committee: Henry Spitz PhD (Committee Chair); Adrian Miron PhD (Committee Member); Sam Glover PhD (Committee Member); Michael Lamba PhD (Committee Member); Howard Elson PhD (Committee Member) Subjects: Biomedical Research