Generalized Cross Validation for Ill-Posed Inverse Problems Pubblico

Abstract

In this thesis, we will introduce two popular regularization tools for ill-posed inverse problem, truncated SVD and Tikhonov regularization. After that we will implement them with GCV ltering parameter choosing method and do some numerical experiment to see their performance in test and real-world problems.

Table of Contents

1 Introduction 5

1.1 Background ............................ 6

1.2 Filtering.............................. 9

2 Regularization Methods 11

2.1 Truncated Singular Value Decomposition . . . . . . . . . . . . 11

2.2 Tikhonov Regularization ..................... 13

3 Choosing Regularization Parameters 16

3.1 Generalized cross validation for TSVD . . . . . . . . . . . . . 18

3.2 Generalized cross validation for Tikhonov. . . . . . . . . . . . 19

3.3 Optimal Regularization Parameter ............... 21

3.4 Optimal Regularization for TSVD................ 22

3.5 Optimal Regularization for Tikhonov . . . . . . . . . . . . . . 23

4 Numerical Experiments 24

4.1 Test problem experiments .................... 25

4.1.1 With error in only b ................... 25

4.1.2 With error in only A ................... 31

4.1.3 With errors in both of A and b ............. 37

4.2 Application to Image Deblurrings ................ 42

4.2.1 AtmosphericBlur10.................... 46

4.2.2 AtmosphericBlur30.................... 48

4.2.3 AtmosphericBlur50.................... 52

5 Conclusions 55

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Parola chiave	Inverse Problems Regularization Methods GCV Function Image Restoration
Committee Chair / Thesis Advisor	Nagy, James, Emory University
Committee Members	Curran, Christopher, Emory University Hertzberg, Vicki S, Emory University

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