Large Scale Inverse Problems: Low-Rank Approximations and Optimization Pubblico

Abstract

Inverse problems can be found in a variety of scientific applications, and the development of efficient and reliable methods remain an essential and challenging task. In this thesis, we introduce novel low-rank solvers for linear systems that arise from large scale inverse problems, which are usually ill-posed and require the use of regularization to obtain meaningful solutions. The new methods are developed around the concept of regularization: i) the low-rank, Kronecker product based forward model approximation method involves the approximation of a truncated singular value decomposition; and ii) the low-rank Krylov subspace methods are based on nuclear norm regularization. We explore the performance of these novel low-rank methods in various imaging applications such as image deblurring, inpainting and computer tomography. Besides applications where the forward model is known and fixed, we also consider an extended application, where the forward model is not exactly known and requires calibration. In this context, we are able to not only apply our new low-rank methods, but also propose a new hybrid machine learning and block coordinate descent algorithm that effectively improves solution accuracy.

Table of Contents

1 Introduction 1

1.1 Contributions of Work . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Overview of Regularization Methods 9

2.1 Tikhonov Regularization and TSVD . . . . . . . . . . . . . . . . . . . 10

2.2 Choosing the Regularization Parameter for Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Other Regularization Terms . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Iterative Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 GMRES and LSQR . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.2 Hybrid Krylov Methods . . . . . . . . . . . . . . . . . . . . . 22

2.4.3 Flexible Krylov Methods . . . . . . . . . . . . . . . . . . . . . 26

3 TSVD approximations 29

3.1 The Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Kronecker Product Summation . . . . . . . . . . . . . . . . . . . . . 32

3.3 A New TSVD Approximation Method via Kronecker Product Summations . . . . . . . . . . . . . . . . . . . 34

3.3.1 Approximation Quality . . . . . . . . . . . . . . . . . . . . . . 37

4 Low Rank Regularization: Derivation of New Methods 42

4.1 Nuclear Norm Regularization . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Derivation of New Methods . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Solution of problem (4.8) via Krylov methods . . . . . . . . . . . . . 49

4.3.1 Methods Based on the GKB Algorithm . . . . . . . . . . . . . 50

4.3.2 Methods Based on the Arnoldi Algorithm . . . . . . . . . . . 51

4.4 Solution through flexible Krylov subspaces . . . . . . . . . . . . . . . 54

4.5 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Low Rank Regularization: Numerical Experiments 62

5.1 Low Rank Projection Methods: classical and new approaches . . . . . 64

5.1.1 Low-rank flexible GMRES (LR-FGMRES) and low-rank flexible LSQR (LR-FLSQR) . . . . . . . . . . . . . . 66

5.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 A Note on Regularization Parameters . . . . . . . . . . . . . . 84

6 Extended Application: Model Calibration for Computed Tomography 88

6.1 Parallel Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Fan-Beam Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2.1 Hybrid ML-BCD Method . . . . . . . . . . . . . . . . . . . . 108

6.2.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 110

7 Conclusions and Future Work 121

Bibliography 124

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Applied Mathematics
Parola chiave	Numerical Analysis Inverse Problems Low-Rank Approximation Optimization Scientific Computing
Committee Chair / Thesis Advisor	James Nagy, Emory University
Committee Members	Lars Ruthotto, Emory University Yuanzhe Xi, Emory University

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