arXiv subjects

Zuoqiang Shi

Publications and source records attributed to Zuoqiang Shi.

100 records · Page 6Linked to original sources

arXiv2015

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.

cs.IT↗

arXiv2014

Sparse Time-Frequency decomposition by dictionary learning

In this paper, we propose a time-frequency analysis method to obtain instantaneous frequencies and the corresponding decomposition by solving an optimization problem. In this optimization problem, the basis to decompose the signal is not known. Instead, it is adapted to the signal and is determined as part of the optimization problem. In this sense, this optimization problem can be seen as a dictionary learning problem. This dictionary learning problem is solved by using the Augmented Lagrangian Multiplier method (ALM) iteratively. We further accelerate the convergence of the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose several signals, including signals with poor scale separation, signals with outliers and polluted by noise and a real signal. The results show that this method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions.

cs.IT↗

arXiv2013

Convergence of a data-driven time-frequency analysis method

In a recent paper, Hou and Shi introduced a new adaptive data analysis method to analyze nonlinear and non-stationary data. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form $\{a(t) \cos(θ(t))\}$, where $a \in V(θ)$, $V(θ)$ consists of the functions smoother than $\cos(θ(t))$ and $θ'\ge 0$. This problem was formulated as a nonlinear $L^0$ optimization problem and an iterative nonlinear matching pursuit method was proposed to solve this nonlinear optimization problem. In this paper, we prove the convergence of this nonlinear matching pursuit method under some sparsity assumption on the signal. We consider both well-resolved and sparse sampled signals. In the case without noise, we prove that our method gives exact recovery of the original signal.

math.NA↗

arXiv2012

On Singularity Formation of a 3D Model for Incompressible Navier-Stokes Equations

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei in [16] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier-Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.

math.AP↗

arXiv2012

Data-Driven Time-Frequency Analysis

In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and non-stationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form $\{a(t) \cos(θ(t))\}$, where $a \in V(θ)$, $V(θ)$ consists of the functions smoother than $\cos(θ(t))$ and $θ'\ge 0$. This problem can be formulated as a nonlinear $L^0$ optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the $L^0$ optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide a convergence analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the EMD/EEMD method. We also apply our method to study data without scale separation, data with intra-wave frequency modulation, and data with incomplete or under-sampled data.

math.NA↗

arXiv2011

On the Local Well-posedness of a 3D Model for Incompressible Navier-Stokes Equations with Partial Viscosity

In this short note, we study the local well-posedness of a 3D model for incompressible Navier-Stokes equations with partial viscosity. This model was originally proposed by Hou-Lei in \cite{HouLei09a}. In a recent paper, we prove that this 3D model with partial viscosity will develop a finite time singularity for a class of initial condition using a mixed Dirichlet Robin boundary condition. The local well-posedness analysis of this initial boundary value problem is more subtle than the corresponding well-posedness analysis using a standard boundary condition because the Robin boundary condition we consider is non-dissipative. We establish the local well-posedness of this initial boundary value problem by designing a Picard iteration in a Banach space and proving the convergence of the Picard iteration by studying the well-posedness property of the heat equation with the same Dirichlet Robin boundary condition.

math.AP↗

arXiv2010

Dynamic Growth Estimates of Maximum Vorticity for 3D Incompressible Euler Equations and the SQG Model

By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li. Under some additional assumption on the vorticity field, which seems to be consistent with the computational results of Hou-Li, we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fefferman and Deng-Hou-Yu.

math.AP↗

arXiv2010

On Singularity Formation of a Nonlinear Nonlocal System

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove the global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.

math.AP↗

arXiv2008

Removing the Stiffness of Elastic Force from the Immersed Boundary Method for the 2D Stokes Equations

The Immersed Boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization. In this paper, we propose several efficient semi-implicit schemes to remove this stiffness from the Immersed Boundary method for the two-dimensional Stokes flow. First, we obtain a novel unconditionally stable semi-implicit discretization for the immersed boundary problem. Using this unconditionally stable discretization as a building block, we derive several efficient semi-implicit schemes for the immersed boundary problem by applying the Small Scale Decomposition to this unconditionally stable discretization. Our stability analysis and extensive numerical experiments show that our semi-implicit schemes offer much better stability property than the explicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit schemes can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme, but with a much better stability property.

cs.CE↗

arXiv2007

Solitary Waves Bifurcated from Bloch Band Edges in Two-dimensional Periodic Media

Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a periodic potential. Using multi-scale perturbation methods, envelope equations of solitary waves near Bloch bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of second-order dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch modes, the envelope equations lead to a host of solitary wave structures including reduced-symmetry solitons, dipole-array solitons, vortex-cell solitons, and so on -- many of which have never been reported before. It is also shown analytically that the centers of envelope solutions can only be positioned at four possible locations at or between potential peaks. Numerically, families of these solitary waves are directly computed both near and far away from band edges. Near the band edges, the numerical solutions spread over many lattice sites, and they fully agree with the analytical solutions obtained from envelope equations. Far away from the band edges, solitary waves are strongly localized with intensity and phase profiles characteristic of individual families.

nlin.PS↗