Sparse Time Frequency Representations and Dynamical Systems

In this paper, we establish a connection between the recently developed data-driven time-frequency analysis [T.Y. Hou and Z. Shi, Advances in Adaptive Data Analysis, 3, 1–28, 2011], [T.Y. Hou and Z. Shi, Applied and Comput. Harmonic Analysis, 35, 284–308, 2013] and the classical second order differential equations. The main idea of the data-driven time-frequency analysis is to decompose a multiscale signal into the sparsest collection of Intrinsic Mode Functions (IMFs) over the largest possible dictionary via nonlinear optimization. These IMFs are of the form a(t)cos(θ(t)), where the amplitude a(t) is positive and slowly varying. The non-decreasing phase function θ(t) is determined by the data and in general depends on the signal in a nonlinear fashion. One of the main results of this paper is that we show that each IMF can be associated with a solution of a second order ordinary differential equation of the form x+p(x,t)x+q(x,t)=0. Further, we propose a localized variational formulation for this problem and develop an effective l1-based optimization method to recover p(x,t) and q(x,t) by looking for a sparse representation of p and q in terms of the polynomial basis. Depending on the form of nonlinearity in p(x,t) and q(x,t), we can define the order of nonlinearity for the associated IMF. This generalizes a concept recently introduced by Prof. N. E. Huang et al. [N.E. Huang, M.-T. Lo, Z. Wu, and Xianyao Chen, US Patent filling number 12/241.565, Sept. 2011]. Numerical examples will be provided to illustrate the robustness and stability of the proposed method for data with or without noise. This manuscript should be considered as a proof of concept

Extracting a shape function for a signal with intra-wave frequency modulation

In this paper, we consider signals with intra-wave frequency modulation. To handle this kind of signals effectively, we generalize our data-driven time-frequency analysis by using a shape function to describe the intra-wave frequency modulation. The idea of using a shape function in time-frequency analysis was first proposed by Wu. A shape function could be any periodic function. Based on this model, we propose to solve an optimization problem to extract the shape function. By exploring the fact that s is a periodic function, we can identify certain low rank structure of the signal. This structure enables us to extract the shape function from the signal. To test the robustness of our method, we apply our method on several synthetic and real signals. The results are very encouraging

An L1 Penalty Method for General Obstacle Problems

We construct an efficient numerical scheme for solving obstacle problems in divergence form. The numerical method is based on a reformulation of the obstacle in terms of an L1-like penalty on the variational problem. The reformulation is an exact regularizer in the sense that for large (but finite) penalty parameter, we recover the exact solution. Our formulation is applied to classical elliptic obstacle problems as well as some related free boundary problems, for example the two-phase membrane problem and the Hele-Shaw model. One advantage of the proposed method is that the free boundary inherent in the obstacle problem arises naturally in our energy minimization without any need for problem specific or complicated discretization. In addition, our scheme also works for nonlinear variational inequalities arising from convex minimization problems.Comment: 20 pages, 18 figure

On the Compressive Spectral Method

The authors of [Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 6634--6639] proposed sparse Fourier domain approximation of solutions to multiscale PDE problems by soft thresholding. We show here that the method enjoys a number of desirable numerical and analytic properties, including convergence for linear PDEs and a modified equation resulting from the sparse approximation. We also extend the method to solve elliptic equations and introduce sparse approximation of differential operators in the Fourier domain. The effectiveness of the method is demonstrated on homogenization examples, where its complexity is dependent only on the sparsity of the problem and constant in many cases

The Hilbert-Huang Transform: A Theoretical Framework and Applications to Leak Identification in Pressurized Space Modules

Any manned space mission must provide breathable air to its crew. For this reason, air leaks in spacecraft pose a danger to the mission and any astronauts on board. The purpose of this work is twofold: the first is to address the issue of air pressure loss from leaks in spacecraft. Air leaks present a danger to spacecraft crew, and so a method of finding air leaks when they occur is needed. Most leak detection systems localize the leak in some way. Instead, we address the identification of air leaks in a pressurized space module, we aim to determine the material in which the leak occurs. This is done with methods centered on statistics and machine learning. In addition to these findings, we investigate one of the methods used in the leak identification section, the Hilbert-Huang Transform. This method has seen many demonstrations of its effectiveness, however this method lacks a solid theoretical framework. We make some contributions to the background of the Hilbert-Huang Transform