Abstract
We introduce an Instantaneous Fourier Analysis method to analyze multiscale nonlinear and non-stationary data. The purpose of this work is to find the sparsest representation of a multiscale signal using basis that is adapted to the data instead of being prescribed a priori. Using a variation approach base on nonlinear L1 optimization, our method defines trends and Instantaneous Frequency of a multiscale signal. One advantage of performing such decomposition is to preserve some intrinsic physical property of the signal without introducing artificial scales or harminics. For multiscale data that have a nonlinear sparse representation, we prove that our nonlinear optimization method converges to the exact signal with the sparse representation. Moreover, we will show that our method is insensitive to noise and robust enough to apply to realistic physical data. For general data that do not have a sparse representation, our method will give an approximate decomposition and the accuracy is controlled by the scale separation property of the original signal.
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.
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