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Ergodic optimization theory for a class of typical maps

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arXiv 2019年

作者： Huang, Wen Lian, Zeng Ma, Xiao Xu, Leiye Zhang, Yiwei Wu Wen-Tsun Key Laboratory of Mathematics USTC Chinese Academy of Sciences Department of Mathematics University of Science and Technology of China Hefei Anhui China College of Mathematical Sciences Sichuan University Chengdu Sichuan610016 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

In this article, we consider the weighted ergodic optimization problem of a class of dynamical systems T: X → X where X is a compact metric space and T is Lipschitz continuous. We show that once T: X → X satisfies both the Anosov shadowing property (ASP) and the Mañé-Conze-Guivarc'h-Bousch property (MCGBP), the minimizing measures of generic Hölder observables are unique and supported on a periodic orbit. Moreover, if T: X → X is a subsystem of a dynamical system f: M → M (i.e. X ⊂ M and f|X= T ) where M is a compact smooth manifold, the above conclusion holds for C1observables. Note that a broad class of classical dynamical systems satisfies both ASP and MCGBP, which includes Axiom A attractors, Anosov diffeomorphisms and uniformly expanding maps. Therefore, the open problem proposed by Yuan and Hunt in [YH] for C1-observables is solved consequentially. Copyright © 2019, The Authors. All rights reserved.

关键词： Dynamical systems

Ergodic optimization theory for axiom a flows

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arXiv 2019年

作者： Huang, Wen Lian, Zeng Ma, Xiao Xu, Leiye Zhang, Yiwei Wu Wen-Tsun Key Laboratory of Mathematics USTC Chinese Academy of Sciences Department of Mathematics University of Science and Technology of China Hefei Anhui China College of Mathematical Sciences Sichuan University Chengdu Sichuan610016 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Hua-Zhong University of Sciences and Technology Wuhan430074 China

In this article, we consider the weighted ergodic optimization problem Axiom A attractors of a C2flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space C0,α(α∈ (0, 1]) or C1the minimizing measure is unique and is supported on a periodic orbit. Copyright © 2019, The Authors. All rights reserved.

关键词： Site selection

A NEW PHYSICS-PRESERVING IMPES SCHEME FOR INCOMPRESSIBLE AND IMMISCIBLE TWO-PHASE FLOW IN HETEROGENEOUS POROUS MEDIA

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arXiv 2019年

作者： Chen, Huangxin Shuyu, S.U.N. School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling High Performance Scientific Computing Xiamen University Fujian361005 China Computational Transport Phenomena Laboratory Division of Physical Science Engineering King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this work we consider a new efficient IMplicit Pressure Explicit Saturation (IMPES) scheme for the simulation of incompressible and immiscible two-phase flow in heterogeneous porous media with capillary pressure. Compared with the conventional IMPES schemes, the new IMPES scheme is inherently physics-preserving, namely, the new algorithm is locally mass conservative for both phases and it also enjoys another appealing feature that the total velocity is continuous in the normal direction. Moreover, the new scheme is unbiased with regard to the two phases and the saturations of both phases are bounds-preserving if the time step size is smaller than a certain value. The key ideas in the new scheme include that the Darcy flows for both phases are rewritten in the formulation based on the total velocity and an auxiliary velocity referring to as the capillary potential gradient, and the total discretized conservation equation is obtained by the summation of the discretized conservation equation for each phase. The upwind strategy is applied to update the saturations explicitly, and the upwind mixed finite element methods are used to solve the pressure-velocity systems which can be decoupled. We also present some interesting examples to demonstrate the efficiency and robustness of the new algorithm. Copyright © 2019, The Authors. All rights reserved.

关键词： Capillarity

VARIATIONAL STRUCTURE AND TWO-DIMENSIONAL SUBSONIC JET FLOWS FOR COMPRESSIBLE EULER SYSTEM WITH GENERAL INCOMING FLOWS

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arXiv 2020年

作者： Li, Yan Shi, Wenhui Tang, Lan Xie, Chunjing School of Mathematical Sciences Shanghai Jiao Tong University 800 Dongchuan Road Shanghai200240 China School of Mathematics Monash University 9 Rainforest Walk ClaytonVIC3800 Australia School of Mathematics and Statistics Central China Normal University Hubei Wuhan430079 China School of mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing CMA-Shanghai Shanghai Jiao Tong University 800 Dongchuan Road Shanghai China

In this paper, we proved the well-posedness theory of compressible subsonic jet flows for two-dimensional steady Euler system with general incoming horizontal velocity as long as the flux is larger than a critical value. One of the key observations is that the stream function formulation for two-dimensional compressible steady Euler system enjoys a variational structure even when the flows have nontrivial vorticity, so that the jet problem can be reformulated as a domain variation problem. This variational structure helps to adapt the framework developed by Alt, Caffarelli, and Friedman to study the jet problem, which is a Bernoulli type free boundary problem. A major technical point to analyze the jet flows is that the inhomogeneous terms in the rescaled equation near the free boundary are always small, even when the vorticity of the flows is big. Copyright © 2020, The Authors. All rights reserved.

关键词： Vorticity

Analysis of the neighborhood pattern similarity measure for the role extraction problem

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arXiv 2020年

作者： Marchand, Melissa Gallivan, Kyle Huang, Wen van Dooren, Paul Department of Mathematics Florida State University 208 Love Building 1017 Academic Way TallahasseeFL32306-4510 United States Department of Mathematics Florida State University 208 Love Building 1017 Academic Way TallahasseeFL32306-4510 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University XiamenFujian China Department of Mathematical Engineering Université catholique de Louvain Louvain-La-Neuve Belgium

In this paper we analyze an indirect approach, called the Neighborhood Pattern Similarity approach, to solve the so-called role extraction problem of a large-scale graph. The method is based on the preliminary construction of a node similarity matrix which allows in a second stage to group together, with an appropriate clustering technique, the nodes that are assigned to have the same role. The analysis builds on the notion of ideal graphs where all nodes with the same role, are also structurally equivalent. Copyright © 2020, The Authors. All rights reserved.

关键词： Extraction

Invariant foliations for stochastic dynamical systems with multiplicative stable Lévy noise

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arXiv 2018年

作者： Chao, Ying Wei, Pingyuan Yuan, Shenglan School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

This work deals with the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Lévy noise. We first establish the existence of stable and unstable foliations for this kind of system via the Lyapunov-Perron method. Then we examine the geometric structure of the invariant foliations, and their relation with invariant manifolds. Finally, we illustrate our results in an example. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

Phase retrieval for sub-Gaussian measurements

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arXiv 2019年

作者： Gao, Bing Liu, Haixia Wang, Yang School of Mathematical Sciences Nankai University Tianjin300071 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong

Generally, phase retrieval problem can be viewed as the reconstruction of a function/signal from only the magnitude of the linear measurements. These measurements can be, for example, the Fourier transform of the density function. Computationally the phase retrieval problem is very challenging. Many algorithms for phase retrieval are based on i.i.d. Gaussian random measurements. However, Gaussian random measurements remain one of the very few classes of measurements. In this paper, we develop an efficient phase retrieval algorithm for sub-gaussian random frames. We provide a general condition for measurements and develop a modified spectral initialization. In the algorithm, we first obtain a good approximation of the solution through the initialization, and from there we use Wirtinger Flow to solve for the solution. We prove that the algorithm converges to the global minimizer linearly. Copyright © 2019, The Authors. All rights reserved.

关键词： Gaussian distribution

Community detection by a riemannian projected proximal gradient method

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arXiv 2020年

作者： Wei, Meng Huang, Wen Gallivan, Kyle A. van Dooren, Paul Department of Mathematics Florida State University 208 Love Building 1017 Academic Way TallahasseeFL32306-4510 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University XiamenFujian361005 China Department of Mathematics Florida State University 208 Love Building 1017 Academic Way TallahasseeFL32306-4510 United States Department of Mathematical Engineering Université catholique de Louvain Louvain-La-Neuve Belgium

Community detection plays an important role in understanding and exploiting the structure of complex systems. Many algorithms have been developed for community detection using modularity maximization or other techniques. In this paper, we formulate the community detection problem as a constrained nonsmooth optimization problem on the compact Stiefel manifold. A Riemannian projected proximal gradient method is proposed and used to solve the problem. To the best of our knowledge, this is the first attempt to use Riemannian optimization for community detection problem. Numerical experimental results on synthetic benchmarks and real-world networks show that our algorithm is effective and outperforms several state-of-art algorithms. Copyright © 2020, The Authors. All rights reserved.

关键词： Population dynamics

A unified framework for asymptotic analysis and computation of finite Hankel transform

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arXiv 2018年

作者： Wang, Haiyong School of Mathematics and Statistics Huazhong University of Science and Technology China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper we present a unified framework for asymptotic analysis and computation of the finite Hankel transform. This framework enables us to derive asymptotic expansions of the transform, including the cases where the oscillator has zeros and stationary points. As a consequence, two efficient and affordable methods for computing the transform numerically are developed and a detailed analysis of their asymptotic error estimate is carried out. Numerical examples are provided to confirm our analysis.65D30, 65D32, 65R10, 41A60 Copyright © 2018, The Authors. All rights reserved.

关键词： Asymptotic analysis