Numerical Approaches for Large-Scale Ill-Posed Inverse Problems Open Access

Abstract

Abstract Numerical Approaches for Large-Scale Ill-Posed Inverse Problems By Julianne Chung

Ill-posed inverse problems arise in a variety of scientifc applications. Regularization methods exist for computing stable solution approximations, but many of these methods are inadequate or insufficient for solving large-scale problems. This work addresses these limitations by developing advanced numerical methods to solve ill-posed inverse problems and by implementing high-performance parallel code for large-scale applications. Three mathematical models that frequently arise in imaging applications are considered: linear least squares, nonlinear least squares, and nonlinear Poisson maximum likelihood. Hybrid methods are developed for regularization of linear least squares problems, variable projection algorithms are used for nonlinear least squares problems, and reconstruction algorithms are investigated for nonlinear Poisson-based models. Furthermore, an efficient parallel implementation based on the Message Passing Interface (MPI) library is described for use on state-of-the-art computer architectures. Numerical experiments illustrate the effectiveness and efficiency of the proposed methods on problems from image reconstruction, super-resolution imaging, cryo-electron microscopy reconstruction, and digital tomosynthesis.

Table of Contents

1 Introduction...1

1.1 Mathematical Models...2

1.1.1 Linear Least Squares...2 1.1.2 Separable Nonlinear Least Squares...3 1.1.3 Nonlinear Poisson Maximum Likelihood...3

1.2 Ill-Posed Problems...4 1.3 Outline of Work...6 1.4 Contributions...7

2 Linear Least Squares Problems...10

2.1 Test Problems...10 2.2 Regularization...14

2.2.1 Tikhonov Regularization and GCV...16 2.2.2 Iterative Regularization: LSQR...18

2.3 Hybrid Methods...21

2.3.1 Tikhonov and GCV in Hybrid Methods...26 2.3.2 Diculty in using GCV in Hybrid Methods...27

2.4 Weighted GCV Method...29

2.4.1 W-GCV for Tikhonov Regularization...31 2.4.2 Interpretations of the W-GCV Method...32 2.4.3 W-GCV for the Bidiagonal System...33 2.4.4 Choosing ω...34 2.4.5 Stopping Criteria for LBD...37

2.5 Numerical Results...39

2.5.1 Results on Various Test Problems...39 2.5.2 Eect of Noise on ω...43

2.6 Remarks and Future Directions...45

3 Separable Nonlinear Least Squares Problems...48

3.1 Motivating Examples...49

3.1.1 Super-Resolution Imaging...49 3.1.2 Blind Deconvolution...54

3.2 Solution through Optimization...56

3.2.1 General Gauss-Newton Approach...57 3.2.2 Variable Projection Method...58 3.2.3 Jacobian Construction...61

3.3 Numerical Results...63 3.4 Summary and Future Work...75

4 A Nonlinear Poisson-based Inverse Problem...76

4.1 Background Information...77 4.2 Polyenergetic Tomosynthesis Model...81

4.2.1 Polyenergetic Model Development...81 4.2.2 Poisson-based Likelihood Function...83

4.3 Iterative Reconstruction Algorithms...84

4.3.1 Gradient Descent Algorithm...87 4.3.2 Newton Approach...88

4.4 Numerical Results...90 4.5 Signicance and Future Directions...95

5 Large-Scale Implementation...100

5.1 Motivating Application: Cryo-EM...101

5.1.1 Mathematical Framework...106 5.1.2 Iterative Reconstruction Methods...107

5.2 Large-Scale Implementation...108

5.2.1 Compact Volume Representation...109 5.2.2 Parallelization using 1D Data Distribution...110 5.2.3 Parallelization using 2D Data Distribution...113

5.3 Numerical Results...118

5.3.1 Quality of Iterative Reconstruction Algorithms...118 5.3.2 Single Processor Performance...123 5.3.3 Parallel Performance...126

5.4 Research Impact...131

6 Concluding Remarks...133 Appendix...135

A.1 Weighted-GCV...135 A.2 Choosing ω in W-GCV...140 A.3 Convexity for Tomosynthesis...143

Bibliography...147

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	iterative methods imaging applications inverse problems large-scale regularization ill-posed
Committee Chair / Thesis Advisor	Nagy, James, Emory University
Committee Members	Haber, Eldad, Emory University Benzi, Michele, Emory University

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