Low Precision Preconditioning for Iterated Tikhonov Regularization Open Access

Abstract

Mixed precision arithmetic has gained significant interest in recent years, given its ability to reduce memory cost and accelerate computation while maintaining accuracy. Many mixed precision algorithms have been designed for solving large-scale, well-conditioned linear systems that arise in various scientific applications. Iterative refinement is a common scheme in the design of such algorithms. In this thesis, we aim to extend mixed precision to ill-conditioned problems using variations of iterated Tikhonov as regularization. Several numerical experiments are conducted on applications from signal and image processing, and the results are compared with those obtained from standard methods, such as CGLS and Hybrid LSQR. Analysis of the results show that the method is able to produce solutions of comparable quality to the standard methods, but at a significantly lower computational cost. 

Table of Contents

Contents

1 Introduction 1

2 Background 3

2.1 Inverse Problems and Regularization .................. 3

2.1.1 TruncatedSVD.......................... 5

2.1.2 Tikhonov Regularization..................... 7

2.1.3 Iterated Tikhonov Regularization ................ 9

2.2 Low Precision Arithmetic ........................ 11

3 Modified Iterated Tikhonov 13

3.1 Replacing the Original Matrix with a Close Approximation . . . . . . 13

3.1.1 Algorithm Overview ....................... 13

3.1.2 Circulant Approximation and Fast Fourier Transform . . . . . 15

3.2 Arnoldi-based Preconditioner ...................... 16

3.2.1 Arnoldi Process.......................... 16

3.2.2 Iterated Arnoldi-Tikhonov Method . . . . . . . . . . . . . . . 17

3.3 Replacing the Original Matrix with a Low Precision Version . . . . . 18

3.3.1 General Case When A is a Real Square Matrix. . . . . . . . . 20

3.3.2 Case When A is Circulant/Block Circulant with Circulant Blocks (BCCB).............................. 22

3.3.3 Case When A is not Square but is Real . . . . . . . . . . . . . 23

4 Numerical Experiments 25

4.1 Spectra Test Problem........................... 25

4.1.1 Results............................... 26

4.1.2 Computation Cost ........................ 28

4.1.3 Other Noise Levels ........................ 29

4.2 Image Deblurring Test Problem ..................... 32

4.2.1 Results............................... 33

4.2.2 Computation Cost ........................ 34

4.2.3 Other Noise Levels ........................ 35

4.3 Impact of matrix size........................... 38

4.4 Sensitivity to the Stopping Criteria ................... 39

4.5 Sensitivity of the Regularization Parameter. . . . . . . . . . . . . . . 40

5 Concluding Remarks 42

Bibliography 45 

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics Applied Mathematics
Keyword	numerical analysis low precision arithmetic iterated Tikhonov
Committee Chair / Thesis Advisor	James Nagy, Emory University
Committee Members	Jinho Choi, Emory University Yuanzhe Xi, Emory University

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