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Sparse time-frequency representation of nonlinear and nonstationary data Dedicated to Professor shi Zhong-Ci on the Occasion of his 80th Birthday

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Science China Mathematics 2013年第12期56卷 2489-2506页

作者： HOU Thomas Yizhao shi zuoqiang Department of Computing and Mathematical Sciences California Institute of Technology Mathematical Sciences Center Tsinghua University

Adaptive data analysis provides an important tool in extracting hidden physical information from multiscale data that arise from various applications. In this paper, we review two data-driven time-frequency analysis methods that we introduced recently to study trend and instantaneous frequency of nonlinear and nonstationary data. These methods are 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（0（t））}, where a is assumed to be less oscillatory than cos（θ（t）） and θ ＇≥ 0. This problem can be formulated as a nonlinear ι0 optimization problem. We have proposed two methods to solve this nonlinear optimization problem. The first one is based on nonlinear basis pursuit and the second one is based on nonlinear matching pursuit. Convergence analysis has been carried out for the nonlinear matching pursuit method. Some numerical experiments are given to demonstrate the effectiveness of the proposed methods.

关键词： sparse representation time-frequency analysis data-driven

CONVERGENCE OF THE WEIGHTED NONLOCAL LAPLACIAN ON RANDOM POINT CLOUD

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Journal of Computational Mathematics 2021年第6期39卷 865-879页

作者： zuoqiang shi Bao Wang Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China Department of Mathematics University of UtahSalt Lake CityUT 84112USA

We analyze the convergence of the weighted nonlocal Laplacian(WNLL)on the high dimensional randomly distributed point *** analysis reveals the importance of the scaling weight,µ∼|P|/|S|with|P|and|S|being the number of entire and labeled data,respectively,in *** established result gives a theoretical foundation of the WNLL for high dimensional data interpolation.

关键词： Weighted nonlocal Laplacian Laplace-Beltrami operator Point cloud Highdimensional interpolation

A Nonlocal Stokes Systemwith Volume Constraints

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Numerical Mathematics(Theory,Methods and Applications) 2022年第4期15卷 903-937页

作者： Qiang Du zuoqiang shi Department of Applied Physics and Applied Mathematics and Data Science InstituteColumbia UniversityNew YorkNY 10027USA Yau Mathematical Sciences Center Tsinghua UniversityBeijing 100084China Yanqi Lake Beijing Institute of Mathematical Sciences and Applications Beijing 101408China

In this paper,we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary *** use the idea of volume constraint to enforce the no-slip boundary condition and prove that the nonlocal model is *** also show that and the solution of the nonlocal system converges to the solution of the original Stokes system as the nonlocality vanishes.

关键词： Nonlocal Stokes system nonlocal operators smoothed particle hydrodynamics incompressible flows well-posedness local limit

A two-level method for sparse time-frequency representation of multiscale data Dedicated to Professor LI TaTsien on the Occasion of His 80th Birthday

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Science China Mathematics 2017年第10期60卷 1733-1752页

作者： LIU ChunGuang shi zuoqiang HOU Thomas Yizhao Department of Mathematics Jinan University Yau Mathematical Sciences Center Tsinghua University Department of Computing and Mathematical Sciences California Institute of Technology

Based on the recently developed data-driven time-frequency analysis(Hou and shi, 2013), we propose a two-level method to look for the sparse time-frequency decomposition of multiscale data. In the two-level method, we first run a local algorithm to get a good approximation of the instantaneous frequency. We then pass this instantaneous frequency to the global algorithm to get an accurate global intrinsic mode function(IMF)and instantaneous frequency. The two-level method alleviates the difficulty of the mode mixing to some *** also present a method to reduce the end effects.

关键词： sparse representation time-frequency analysis matching pursuit two-level method end effects

CONVERGENCE OF LAPLACIAN SPECTR A FROM R ANDOM SA MPLES

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Journal of Computational Mathematics 2020年第6期38卷 952-984页

作者： Wenqi Tao zuoqiang shi Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China

Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality *** Laplacian is one widely used discrete laplacian on point *** was previously proved by Belkin and Niyogithat the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the ***,we introduced Point Integral method(PIM)to solve elliptic equations and corresponding eigenvalue problem on point *** this paper,we prove that the eigenvectors and eigenvalues obtained by PIM converge in the limit of infinitely many random ***,estimation of the convergence rate is also given.

关键词： Graph Laplacian Laplacian spectra Random samples Spectral convergence

Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees

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Communications in Computational Physics 2017年第6期22卷 228-258页

作者： Zhen Li zuoqiang shi Jian Sun Yau Mathematical Sciences Center Tsinghua UniversityBeijing100084China

Partial differential equations(PDE)on manifolds arise in many areas,including mathematics and many applied *** to the complicated geometrical structure of the manifold,it is difficult to get efficient numerical method to solve PDE on *** the paper,we propose a method called point integral method(PIM)to solve the Poisson-type equations from point *** different kinds of PDEs,the Poisson-type equations including the standard Poisson equation and the related eigenproblem of the Laplace-Beltrami operator are one of the most *** PIM,the key idea is to derive the integral equations which approximates the Poisson-type equations and contains no derivatives but only the values of the unknown *** featuremakes the integral equation easy to be discretized frompoint *** the paper,we explain the derivation of the integral equations,describe the point integral method and its implementation,and present the numerical experiments to demonstrate the convergence of PIM.

关键词： Point integral method point cloud Laplace-Beltrami operator convergence

BI-GreenNet:Learning Green’s Functions by Boundary Integral Network

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Communications in Mathematics and Statistics 2023年第1期11卷 103-129页

作者： Guochang Lin Fukai Chen Pipi Hu Xiang Chen Junqing Chen Jun Wang zuoqiang shi Yau Mathematical Sciences Center Tsinghua UniversityBeijing 100084People’s Republic of China Department of Mathematical Sciences Tsinghua UniversityBeijing 100084People’s Republic of China Yanqi Lake Beijing Institute of Mathematical Sciences and Applications Beijing 101408People’s Republic of China Noah’s Ark Lab HuaweiNo.3 Xinxi RoadBeijing 100085People’s Republic of China University College London London WC1E 6EAUK

Green’s function plays a significant role in both theoretical analysis and numerical computing of partial differential equations(PDEs).However,in most cases,Green’s function is difficult to *** troubles arise in the following ***,compared with the original PDE,the dimension of Green’s function is doubled,making it impossible to be handled by traditional mesh-based ***,Green’s function usually contains singularities which increase in the difficulty to get a good ***,the computational domain may be very complex or even *** override these problems,we develop a new framework for computing Green’s function leveraging the fundamental solution,boundary integral method and neural networkswith reasonably high accuracy in this *** focus on Green’s function of Poisson and Helmholtz equations in bounded domains,unbounded *** also consider Poisson equation and Helmholtz equations in domains with *** numerical experiments illustrate the efficiency and accuracy of our method for solving Green’s *** addition,Green’s function provides the operator from the source term and boundary condition to *** applyGreen’s function to solve PDEs with different sources,and obtain reasonably high-precision solutions,which shows the good generalization ability of our ***,the requirements for explicit fundamental solutions to remove the singularity of Green’s function hinder the application of our method in more complex PDEs,such as variable coefficient equations,which will be investigated in our future work.

关键词： Green’s function Partial differential equation Boundary integral Neural network

Non-line-of-sight reconstruction with signal–object collaborative regularization

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Light(Science & Applications) 2021年第10期10卷 1967-1986页

作者： Xintong Liu Jianyu Wang Zhupeng Li zuoqiang shi Xing Fu Lingyun Qiu Yau Mathematical Sciences Center Tsinghua University100084 BeijingChina State Key Laboratory of Precision Measurement Technology and Instruments Department of Precision InstrumentTsinghua University100084 BeijingChina Key Laboratory of Photonic Control Technology(Tsinghua University) Ministry of Education100084 BeijingChina Department of Mathematical Sciences Tsinghua University100084 BeijingChina Yanqi Lake Beijing Institute of Mathematical Sciences and Applications 101408 BeijingChina

Non-line-of-sight imaging aims at recovering obscured objects from multiple scattered *** has recently received widespread attention due to its potential applications,such as autonomous driving,rescue operations,and remote ***,in cases with high measurement noise,obtaining high-quality reconstructions remains a challenging *** this work,we establish a unified regularization framework,which can be tailored for different scenarios,including indoor and outdoor scenes with substantial background noise under both confocal and non-confocal *** proposed regularization framework incorporates sparseness and non-local self-similarity of the hidden objects as well as the smoothness of the *** show that the estimated signals,albedo,and surface normal of the hidden objects can be reconstructed robustly even with high measurement noise under the proposed *** results on synthetic and experimental data show that our approach recovers the hidden objects faithfully and outperforms state-of-the-art reconstruction algorithms in terms of both quantitative criteria and visual quality.

关键词： regularization smoothness operations

An uncertainty investigation for liner impedance eduction methods

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Chinese Journal of Aeronautics 2024年第3期37卷 49-59页

作者： Weihao shi Xiaodong JING School of Energy and Power Engineering Beihang UniversityBeijing 100191China

Impedance eduction methods have been developed for decades to meet the increasing need for high-quality impedance data in the design and optimization of acoustic *** this end,it is important to fully investigate the uncertainty problem,to which only limited attention has been devoted so *** paper considers the possibility of acoustically-induced structural vibration as a nonnegligible uncertainty or error source in impedance eduction *** the frequency moves away from the resonant frequency,with the increase in the value of cavity reactance,the acoustic particle velocity inside liner orifices possibly decreases to the extent comparable to the vibration velocity of liner facing ***,the acoustically-induced vibration,although generally being weak except at the inherent structural frequencies,may considerably affect the impedance eduction results near the anti-resonant frequency where the liner has poor *** demonstrate the effect of structural vibration,the vibration velocity of liner facing sheet is estimated from the experimentally educed admittance of the liner samples whose orifices are sealed with ***,a three-dimensional numerical model is set up,in which normal particle velocity is introduced over the solid portion of liner facing sheet to imitate structural vibration,rather than directly solving the acoustic-structural coupling *** shown by the results,the vibration of liner facing sheet,whose velocity is as small as estimated by the experiment,can result in anomalous deviation of the educed impedance from the impedance model near the anti-resonant *** trend that the anomalous deviation varies with frequency is numerically captured.

关键词： Impedance eduction Uncertainty analysis Acoustically-induced structural vibration Anti-resonant frequency Perforated liner