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Oscillatory Data Analysis and Fast Algorithms for Integral Operators OSCILLATORY DATA ANALYSIS AND FAST ALGORITHMS FOR INTEGRAL OPERATORS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Haizhao Yang July 2015 c Copyright by Haizhao Yang 2015 All Rights Reserved Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/fq061ny3299 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Lexing Ying) Principal Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Emmanuel Cand`es) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. (Lenya Ryzhik) Approved for the University Committee on Graduate Studies iii Abstract This dissertation consists of two independent parts: oscillatory data analysis (Part I) and fast algorithms for integral operators in computational harmonic analysis (Part II). The first part concentrates on developing theory and efficient tools in applied and computational harmonic analysis for oscillatory data analysis. In modern data science, oscillatory data analysis aims at identifying and extracting principle wave-like components, which might be nonlinear and non-stationary, underlying a complex physical phenomenon. Estimating instantaneous properties of one-dimensional components or local properties of multi-dimensional components has been an important topic in various science and engineering problems in resent three decades. This thesis introduces several novel synchrosqueezed transforms (SSTs) with rigorous mathematical, statistical analysis, and efficient implementation to tackle challenging problems in oscillatory data analysis. Several real applications show that these transforms provide an elegant tool for oscillatory data analysis. In many applications, the SST-based algorithms are significantly faster than the existing state-of-art algorithms and obtain better results. The second part of this thesis proposes several fast algorithms for the numerical implementation of several integral operators in harmonic analysis including Fourier integral operators (including pseudo differential operators, the generalized Radon transform, the nonuniform Fourier transform, etc.) and special function transforms (including the Fourier-Bessel transform, the spherical harmonic transform, etc.). These are useful mathematical tools in a wide range of science and engineering problems, e.g., imaging science, weather and climate modeling, electromagnetics, quantum chemistry, and phenomena modeled by wave equations. Via hierarchical domain decomposition, randomized low-rank approximations, interpolative low-rank approximations, the fast Fourier transform, and the butterfly algorithm, I propose several novel fast algorithms for applying or recovering these operators. iv Acknowledgments I would like to express my deepest gratitude to my advisor, Prof. Lexing Ying, for his support, guidance and encouragement in my graduate study. His creative and critical thinking has been inspiring me throughout my study at Stanford. He leads me by example and teaches me how to ask questions and solve problems in the way towards a pure scientist. I also extend my gratitude to people in Prof. Ying's group for the happy time together. I am also grateful to Prof. Ingrid Daubechies and Prof. Jianfeng Lu at Duke University for their tremendous support and fruitful discussions in the research of oscillatory data analysis and its applications. They opened my eyes to the joy and beauty of mathematics in materials science and art investigation. I feel privileged to have many fantastic professors who taught me and inspired me a lot at Stanford. I express my special thanks to my thesis committee members, Prof. Biondo Biondi, Prof. Emmanuel Cand`es,Prof. Lenya Ryzhik, and Prof. Andr`asVasy for their help and insights in the mathematical science. I would like to thank Prof. Kui Ren at the University of Texas at Austin, Prof. Weinan E at Princeton University, Prof. David Cai in Courant Institute of Mathematical Science at New York University, and Prof. Jingfang Huang at the University of North Carolina, Chapel Hill for their help, care and encouragement. I would also like to thank my collaborators and friends for their help and discussions in my research: William P. Brown, Prof. Xiaoling Cheng, Prof. Charles K. Chui, Prof. Sergey Fomel, Dr. Kenneth L. Ho, Prof. Jingwei Hu, Yingzhou Li, Eileen R. Martin, Prof. Benedikt Wirth, Prof. Hau-Tieng Wu, Prof. Zhenli Xu. Most of all, I would like to thank my parents Zhenqiang Yang, Guiying Yang and my wife Tian Yang for their endless source of love and support. I dedicate this thesis to them. v Contents Abstract iv Acknowledgments v I Oscillatory Data Analysis 1 1 Introduction 2 1.1 Time-Frequency Geometry . .3 1.2 Mode Decomposition . .5 1.3 Contributions . .7 2 Theory of Synchrosqueezed Transforms 10 2.1 1D Synchrosqueezed Wave Packet Transform . 11 2.1.1 Motivation . 11 2.1.2 Definition of 1D SSWPT . 12 2.1.3 Analysis . 14 2.2 Multi-Dimensional Synchrosqueezed Wave Packet Transform . 22 2.2.1 Motivation . 22 2.2.2 Definition . 22 2.2.3 Analysis . 25 2.3 2D Synchrosqueezed Curvelet Transform . 32 2.3.1 Motivation . 32 2.3.2 Definition . 33 2.3.3 Analysis . 36 3 Robustness of Synchrosqueezed Transforms 49 3.1 Introduction . 49 3.1.1 Motivation . 49 vi 3.1.2 Significance . 51 3.2 1D Synchrosqueezed Wave Packet Transform (SSWPT) . 52 3.3 2D Synchrosqueezed Wave Packet Transform (SSWPT) . 72 3.4 2D Synchrosqueezed Curvelet Transform (SSCT) . 74 4 Discrete Synchrosqueezed Transforms 79 4.1 Fast Discrete SSWPT and Mode Decomposition . 79 4.1.1 Implementation . 79 4.1.2 Numerical Examples . 86 4.2 Fast Discrete SSCT and Mode Decomposition . 89 4.2.1 Implementation . 89 4.2.2 Numerical Examples . 94 4.2.3 Intrinsic Mode Decomposition for Synthetic Data . 96 4.3 Numerical Robustness Analysis . 98 4.3.1 Robustness Tests for 1D SST . 99 4.3.2 Robustness Tests for 2D SST . 102 4.3.3 Component Test . 103 4.3.4 Real Examples . 104 5 Diffeomorphism Based Spectral Analysis 107 5.1 Introduction . 107 5.2 Diffeomorphism Based Spectral Analysis (DSA) . 109 5.2.1 Implementation of the DSA . 109 5.2.2 Analysis of the DSA . 112 5.3 Numerical Examples . 116 5.3.1 Synthetic Examples . 117 5.3.2 Real Applications . 121 5.4 Conclusion . 124 6 Applications 126 6.1 Atomic Crystal Analysis . 126 6.1.1 Introduction . 126 6.1.2 Crystal Image Models and Theory . 130 6.1.3 Crystal Defect Analysis Algorithms and Implementations . 137 6.1.4 Variational Model to Retrieve Deformation Gradient . 147 6.1.5 Examples and Discussions . 155 6.1.6 Conclusion . 160 6.2 Canvas Weave Analysis in Art Forensics . 162 vii 6.2.1 Introduction . 162 6.2.2 Model of the Canvas Weave Pattern in X-Radiography . 163 6.2.3 Fourier-Space Based Canvas Analysis . 164 6.2.4 Applications to Art Investigations . 166 6.2.5 Conclusion . 171 7 Conclusions of Part I 176 7.1 Summary . 176 7.2 Future Work . 176 II Fast Algorithms for Integral Operators in Harmonic Analysis 178 8 Introduction 179 9 Multiscale Butterfly Algorithm 182 9.1 Introduction . 182 9.1.1 Previous Work . 183 9.1.2 Motivation . 184 9.1.3 Our Contribution . 185 9.1.4 Organization . 186 9.2 The Butterfly Algorithm . 186 9.3 Low-Rank Approximations . 189 9.4 Multiscale Butterfly Algorithm . 194 9.4.1 Cartesian Butterfly Algorithm . 194 9.4.2 Complexity Analysis . 196 9.5 Numerical Results . 197 9.6 Conclusion . 200 10 One-Dimensional Butterfly Factorization 201 10.1 Introduction . 201 10.1.1 Complementary Low-Rank Matrices and Butterfly Algorithm . 201 10.1.2 Motivations and Significance . 203 10.1.3 Content . 204 10.2 Preliminaries . 205 10.2.1 SVD via Random Matrix-Vector Multiplication . 206 10.2.2 SVD via Random Sampling . 206 10.3 Butterfly Factorization . 208 10.3.1 Middle Level Factorization . 209 viii 10.3.2 Recursive Factorization . ..
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