Inexact Generalized Golub-Kahan Methods for Large-Scale Inverse Problems Open Access

Abstract

Solving large-scale Bayesian inverse problems presents significant challenges, particularly when key components, such as the forward operator or prior covariance matrix, are not known exactly. In image processing tasks, for instance, unknown defects in the forward process may result in varying degrees of inexactness in the forward model, complicating the reconstruction process. Moreover, challenges also arise from modeling the prior. Gaussian priors are commonly used, but the prior covariance matrix can be difficult to work with, as its square root or inverse could be computationally infeasible when the covariance kernel is defined on irregular grids, or it might be accessible only through matrix-vector products. Furthermore, the optimal parameter values that define the covariance matrix are often not known a priori. This thesis introduces an efficient approach to handle these challenges by developing an inexact generalized Golub-Kahan decomposition that can incorporate varying degrees and sources of inexactness to solve large-scale generalized Tikhonov regularized problems. Further, a hybrid iterative projection scheme is developed to automatically select Tikhonov regularization parameters. Numerical experiments on simulated tomography reconstructions demonstrate the stability and effectiveness of this novel hybrid approach.

Table of Contents

1 Introduction 1

1.1 Motivating examples of inverse problems . . . . . . . . . . . . . . . . 2

1.2 Contributions and outline ........................ 3

2 Inverse Problems 5

2.1 Background................................ 5

2.1.1 Bayes’ formula .......................... 6

2.1.2 Multivariate distribution..................... 7

2.2 Problem Setup .............................. 8

2.2.1 Construction of prior and likelihood function . . . . . . . . . . 9

2.2.2 Choice of covariance kernel.................... 10

2.2.3 Computing the MAP estimate.................. 12

2.2.4 Sources of inexactness ...................... 14

3 Iterative methods 16

3.1 Golub-Kahan Bidiagonalization ..................... 17

3.2 Generalized Golub-Kahan Bidiagonalization . . . . . . . . . . . . . . 20

3.3 Inexact Golub-Kahan Decomposition .................. 22

3.4 Hybrid projection methods........................ 24

3.4.1 Theoretical insights........................ 26

3.4.2 A general framework ....................... 30

4 Iterative methods based on inexact generalized Golub-Kahan decomposition 33 

4.1 Inexact forward model .......................... 33

4.1.1 Inexact generalized Golub-Kahan decomposition . . . . . . . . 34

4.1.2 Solving the LS problem...................... 36

4.2 Inexactness in the covariance kernel................... 38

4.2.1 Inexact generalized Golub-Kahan decomposition . . . . . . . . 38

4.2.2 Solving the LS problem...................... 40

4.3 Hybrid methods for inexact iterative methods . . . . . . . . . . . . . 41

5 Numerical Experiments 44

5.1 Experiments with inexactness in forward model . . . . . . . . . . . . 44

5.1.1 Comparison of iterative methods without regularization . . . . 46

5.1.2 Comparison of hybrid methods with optimal regularization . . 48

5.1.3 Comparison of regularization parameters . . . . . . . . . . . . 49

5.1.4 Inexactness in projection angles................. 51

5.2 Inexactness in the covariance kernel................... 53

5.2.1 Comparison of iterative method without regularization . . . . 54

5.2.2 Comparisonofhybridmethods ................. 56

6 Conclusion 59

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.A.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Krylov subspace methods Tikhonov regularization Inverse problems
Committee Chair / Thesis Advisor	Julianne Chung, Emory University
Committee Members	Matthias Chung, Emory University Alex Fedorov, Emory University

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