Abstract

Data are one of the most important links that we have with the physical world. Developing effective data analysis methods is an important path through which we can understand the underlying processes of natural phenomena. So far, most data analysis methods use a pre-determined basis to process data. These methods often assume linearity and stationarity of data. In real-world experimental and theoretical studies, we often deal with signals that have  ever-changing frequency. Chirp signal is one class of the signals used by bats as well as in radar. For any nonlinear system, the frequency is modulating not only among different oscillation periods, but also within one period such as in the nonlinear pendulum problem. To understand the underlying mechanisms of these processes, we can not use the traditional Fourier analysis with components of constant frequency. We need to examine the true physical processes using a truly adaptive data analysis methods.

In this short course, we will review the traditional Time-Frequecy analysis methods such as Fourier transform, Windowed Fourier transform; Winger-Ville transform; and Wavelet transform. We then review some recent progress on the nonlinear Time-Frequency analysis methods, including the Empirical Mode Decomposition method, and Compressed Sensing for recovery of sparse data. Finally, we will discuss some recently developed adaptive data analysis method based on nonlinear optimization and a sparse representation of multiscale data, and introduce some open problems for future research.

There is no prerequisite for this short interdisciplinary course. We will start from the very basic principle and gradually move to the frontier of some of the most exciting research topics. We will provide some matlab codes so that the stduents can have some hand-on experience on various data analysis methods. Most of the problems we consider have a strong connection to physical and/or bio-medical applications. Innovative and efficient numerical methods will be introduced with guidance from analysis.

About the speaker

Thomas Y. Hou is the Charles Lee Powell professor of applied and computational mathematics at Caltech, and is one of the leading experts in vortex dynamics and multiscale problems. His research interests are centered around developing analytical tools and effective numerical methods for vortex dynamics, interfacial flows, and multiscale problems. He received his Ph.D. from UCLA in 1987. Upon graduating from UCLA, he joined the Courant Institute as a postdoc and then became a faculty member in 1989. He moved to the applied math department at Caltech in 1993, and is currently the executive director of applied and computational mathematics. Dr. Hou has received a number of honors and awards, including the SIAM Fellow in 2009, the Computational and Applied Sciences Award from USACM in 2005, the Morningside Gold Medal in Applied Mathematics in 2004, the SIAM Wilkinson Prize in Numerical Analysis and Scientific Computing in 2001, the Francois N. Frenkiel Award from the Division of Fluid Mechanics of APS in 1998, the Feng Kang Prize in Scientific Computing in 1997, a Sloan fellow from 1990 to 1992. He was an invited plenary speaker at the International Congress of Industrial and Applied Mathematics in 2003, and an invited speaker of the International Congress of Mathematicians in 1998. He was also the founding Editor-in-Chief of the SIAM Journal on Multiscale Modeling and Simulation from 2002 to 2007.