SVD Approximations of Large Scale Inverse Problems 公开

Abstract

This thesis studies efficient methods for computing approximations of the singular value decomposition (SVD) for large matrices that arise in ill-posed inverse problems, with a focus on image deblurring. These methods are: the Lanczos method, the randomized method, and the Kronecker product approximation method. After introducing the SVD and describing the approximation methods, we show test results involving the accuracy and speed comparisons of these methods, and provide some deblurring examples.

Table of Contents

1 Introduction. 1

2 Singular Value Decomposition. 7

2.1 Definition and Properties. 7

2.2 Computational Difficulties. 9

2.3 Regularization Methods. 13

2.3.1 Truncated SVD. 13

2.3.2 Tikhonov Regularization. 14

3 SVD Approximations. 19

3.1 Lanczos Method. 19

3.2 Randomized Method. 20

3.3 Kronecker Product Approximation Method. 22

4 Numerical Experiments. 25

4.1 Accuracy Test. 25

4.2 Speed Test. 30

4.3 Deblurring Examples. 33

5 Conclusion and Discussion 40

About this Honors Thesis

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School	Emory College
Department	Mathematics and Computer Science
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Mathematics
关键词	Lanczos inverse problems regularization SVD Kronecker product Randomized methods
Committee Chair / Thesis Advisor	Nagy, James, Emory University
Committee Members	Veneziani, Alessandro, Emory University Brody, Jed, Emory University

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