Author ORCID Identifier
https://orcid.org/0000-0002-1789-2629
Date of Award
Spring 5-2026
Document Type
Thesis (Ph.D.)
Department or Program
Mathematics
First Advisor
Anne Gelb
Abstract
Inverse problems arise throughout science and engineering, where indirect, incomplete, and noisy observations are used to recover unknown parameters of interest. In these applications, the corresponding forward or measurement models are often ill-conditioned or underdetermined, so direct inversion is unstable and regularization is required. This thesis develops computational methods for linear inverse problems in which the unknown is assumed to be approximately sparse in a transformed domain defined by a linear, possibly rank-deficient operator, such as a finite-difference matrix, with particular emphasis on large-scale problems.
The thesis makes three main contributions. First, it generalizes hierarchical Bayesian maximum a posteriori estimation by extending the iterative alternating sequential algorithm to accommodate rank-deficient sparsifying transformations and to estimate an unknown noise variance. Second, it provides a self-contained treatment of the generalized singular value decomposition as a tool for theory and computation for Tikhonov regularization and priorconditioning (prior-preconditioning), together with accompanying Python software. Third, it develops priorconditioned projection methods for sparsity-promoting inverse problems, including restarted and recycled variants for memory-constrained large-scale settings. It provides rigorous spectral analysis that explains their improved convergence behavior, efficient numerical procedures for the weighted pseudoinverse matrix-vector and matrix-matrix products required by these methods, and detailed numerical comparisons with existing methods in the literature.
Numerical demonstrations throughout the thesis, including signal and image reconstruction and computed tomography examples, illustrate the behavior and computational advantages of these methods and techniques. Together, these results make deterministic and Bayesian sparsity-promoting regularization more practical for large-scale inverse problems, particularly those with rank-deficient sparsifying transformations.
Recommended Citation
Lindbloom, Jonathan, "Advances in Computational Methods for Sparsity-Promoting Linear Inverse Problems" (2026). Dartmouth College Ph.D Dissertations. 520.
https://digitalcommons.dartmouth.edu/dissertations/520
