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Numerical Methods for the Solution of Linear Ill-posed Problems

Alqahtani, Abdulaziz Mohammed M
http://orcid.org/0000-0003-0820-3948
http://rave.ohiolink.edu/etdc/view?acc_num=kent1669128334570823

Abstract Details

2022, PHD, Kent State University, 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 solution of an ill-posed operator equation is obtained by first determining an approximation of the operators of generally fairly small dimension by carrying out a few steps of a continuous version of the Golub–Kahan bidiagonalization (GKB) process to the noisy operator. Then Tikhonov regularization is applied to the low-dimensional problem so obtained and the regularization parameter is determined by solving a low-dimensional nonlinear equation. The effect of replacing the original operator by the low-dimensional operator obtained by the GKB process on the accuracy of the solution is analyzed, as is the effect of errors in the operator and data.
Lothar Reichel (Advisor)
Jing Li (Committee Member)
Barry Dunietz (Committee Member)
Qiang Guan (Committee Member)
Jun Li (Committee Member)
111 p.
Applied Mathematics
Golub–Kahan block bidiagonalization; large-scale discrete ill-posed problem; symmetric Lanczos block tridiagonalization; Tikhonov regularization; Ill-posed problem; Inverse problem; Chebfun; Truncated SVE.

Recommended Citations

Citations

  • Alqahtani, A. M. M. (2022). Numerical Methods for the Solution of Linear Ill-posed Problems [Doctoral dissertation, Kent State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=kent1669128334570823

    APA Style (7th edition)

  • Alqahtani, Abdulaziz. Numerical Methods for the Solution of Linear Ill-posed Problems . 2022. Kent State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=kent1669128334570823.

    MLA Style (8th edition)

  • Alqahtani, Abdulaziz. "Numerical Methods for the Solution of Linear Ill-posed Problems ." Doctoral dissertation, Kent State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=kent1669128334570823

    Chicago Manual of Style (17th edition)

kent1669128334570823
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This open access ETD is published by Kent State University and OhioLINK.