Generalized Cross Validation for Ill-Posed Inverse Problems Open Access

Wu, Hanyong (2017)

Permanent URL: https://etd.library.emory.edu/concern/etds/c247ds894?locale=en
Published

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

Rights statement
  • Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.
School
Department
Degree
Submission
Language
  • English
Research field
Keyword
Committee Chair / Thesis Advisor
Committee Members
Last modified

Primary PDF

Supplemental Files