Oscillatory data analysis and fast algorithms for integral operators
Abstract/Contents
- 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.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2015 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Yang, Haizhao |
|---|---|
| Associated with | Stanford University, Department of Mathematics. |
| Primary advisor | Ying, Lexing |
| Thesis advisor | Ying, Lexing |
| Thesis advisor | Candès, Emmanuel J. (Emmanuel Jean) |
| Thesis advisor | Ryzhik, Leonid |
| Advisor | Candès, Emmanuel J. (Emmanuel Jean) |
| Advisor | Ryzhik, Leonid |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Haizhao Yang. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2015. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2015 by Haizhao Yang
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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