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Spectral gap of sparse bistochastic matrices with exchangeable rows

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arXiv 2018年

作者： Bordenave, Charles Qiu, Yanqi Zhang, Yiwei Institut de Mathématiques de Marseille CNRS Aix-Marseille University 39 Rue Frédéric Joliot Curie Marseille13013 France Institute of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics AMSS Chinese Academy of Sciences Beijing100190 China CNRS Institut de Mathématiques de Toulouse University of Toulouse III 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

We consider a random bistochastic matrix of size n of the form MQ where M is a uniformly distributed permutation matrix and Q is a given bistochastic matrix. Under sparsity and regularity assumptions on Q, we prove that the second largest eigenvalue of MQ is essentially bounded by the normalized Hilbert-Schmidt norm of Q when n grows large. We apply this result to random walks on random regular digraphs.60B20, 60C05, 05C80, 05C38 Copyright © 2018, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

Corrigendum to: A convergent adaptive finite element method for elliptic Dirichlet boundary control problems

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IMA Journal of Numerical Analysis 2020年第1期40卷 800-800页

作者： Gong, Wei Liu, Wenbin Tan, Zhiyu Yan, Ningning National Center for Mathematics and Interdisciplinary Sciences Chinese Academy of Sciences Beijing China The State Key Laboratory of Scientific and Engineering Computing Chinese Academy of Sciences Beijing China Institute of Computational Mathematics Chinese Academy of Sciences Beijing China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Kent Business School University of Kent Kent UK Department of Mathematics Hong Kong Baptist University Kowloon Tong Hong Kong SAR NCMIS Chinese Academy of Sciences Beijing China LSEC Chinese Academy of Sciences Beijing China Institute of Systems Science Chinese Academy of Sciences Beijing China

Analysis of the Mean Field Free Energy Functional of Electrolyte Solution with Non-zero Boundary Conditions and the Generalized PB/PNP Equations with Inhomogeneous Dielectric Permittivity

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arXiv 2017年

作者： Liu, Xuejiao Qiao, Yu Lu, Benzhuo State Key Laboratory of Scientific and Engineering Computing National Center for Mathematics and Interdisciplinary Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

The energy functional, the governing partial differential equation(s) (PDE), and the boundary conditions need to be consistent with each other in a modeling system. In electrolyte solution study, people usually use a free energy form of an infinite domain system (with vanishing potential boundary condition) and the derived PDE(s) for analysis and computing. However, in many real systems and/or numerical computing, the objective domain is finite, and people still use the similar energy form, PDE(s) but with different boundary conditions, which may cause inconsistency. In this work, (1) we present a mean field free energy functional for electrolyte solution within a finite domain with either physical or numerically required artificial boundary. Apart from the conventional energy components (electrostatic potential energy, ideal gas entropy term and chemical potential term), new boundary interaction terms are added for both Neumann and Dirichlet boundary conditions. These new terms count for physical interactions with the boundary (for real boundary) or the environment influence on the computational domain system (for non-physical but numerically designed boundary). (2) The traditional physical-based Poisson-Boltzmann (PB) equation and Poisson-Nernst-Planck (PNP) equations are proved to be consistent with the new free energy form, and different boundary conditions can be applied. (3) In particular, for inhomogeneous electrolyte with ionic concentration-dependent dielectric permittivity, we derive the generalized Boltzmann distribution (thereby the generalized PB equation) for equilibrium case, and the generalized PNP equations for non-equilibrium case, under different boundary conditions. Numerical tests are performed to demonstrate the different consequences resulted from different energy forms and their derived PDE(s). Copyright © 2017, The Authors. All rights reserved.

关键词： Poisson equation

An hp-adaptive strategy for elliptic problems

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arXiv 2017年

作者： Liu, Hui Cui, Tao Leng, Wei Zhang, Linbo State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences P.O. Box 2719 Beijing100190 China

In this paper a new hp-adaptive strategy for elliptic problems based on refinement history is proposed, which chooses h-, p- or hp-refinement on individual elements according to a posteriori error estimate, as well as smoothness estimate of the solution obtained by comparing the actual and expected error reduction rate. Numerical experiments show that exponential convergence can be achieved with this strategy. Copyright © 2017, The Authors. All rights reserved.

关键词： Finite element method

A wong-zakai approximation for random slow manifolds with application to parameter estimation

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arXiv 2018年

作者： He, Ziying Zhang, Xinyong Jiang, Tao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Mathematical Sciences Tsinghua University Beijing100084 China Collaborative Innovation Center of China Pilot Reform Exploration and Assessment Hubei Sub-Center Hubei University of Economics Wuhan430205 China

We study a Wong-Zakai approximation for the random slow manifold of a slow-fast stochastic dynamical system. We first deduce the existence of the random slow manifold about an approximation system driven by an integrated Ornstein-Uhlenbeck (O-U) process. Then we compute the slow manifold of the approximation system, in order to gain insights of the long time dynamics of the original stochastic system. By restricting this approximation system to its slow manifold, we thus get a reduced slow random system. This reduced slow random system is used to accurately estimate a system parameter of the original system. An example is presented to illustrate this *** Codes Primary: 37L55, 35R60, Secondary: 60H15, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Parameter estimation

Anomalous time-scaling of extreme events in infinite systems and birkhoff sums of infinite observables

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arXiv 2018年

作者： Galatolo, Stefano Holland, Mark Persson, Tomas Zhang, Yiwei Dipartimento di Matematica Via Buonattoti 1 Pisa56127 Italy North Park Road EXETEREX4 4QF United Kingdom Centre for Mathematical Sciences Lund University Box 118 Lund221 00 Sweden Scool 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

We establish quantitative results for the statistical behaviour of infinite systems. We consider two kinds of infinite system: i) a conservative dynamical system (f, X, µ) preserving a σ-finite measure µ such that µ(X) = ∞;ii) the case where µ is a probability measure but we consider the statistical behaviour of an observable φ: X → [0, ∞) which is non-integrable: R φ dµ = ∞. In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove a general relation between the behavior of φ, the local dimension of µ, and the scaling rate of the growth of Birkhoff sums of φ as time tends to infinity. We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings in the case of logarithm laws of entrance times, dynamical Borel-Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.37A40, 37A50, 37A25, 60G75, 37D25 Copyright © 2018, The Authors. All rights reserved.

关键词： Chaotic systems

Adaptive regularized newton method for riemannian optimization

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arXiv 2017年

作者： Hu, Jiang Milzarek, Andre Wen, Zaiwen Yuan, Yaxiang Beijing International Center for Mathematical Research Peking University China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences China

Optimization on Riemannian manifolds widely arises in eigenvalue computation, density functional theory, Bose-Einstein condensates, low rank nearest correlation, image registration, and signal processing, etc. We propose an adaptive regularized Newton method which approximates the original objective function by the second-order Taylor expansion in Euclidean space but keeps the Riemannian manifold constraints. The regularization term in the objective function of the subproblem enables us to establish a Cauchy-point like condition as the standard trust-region method for proving global convergence. The subproblem can be solved inexactly either by first-order methods or a modified Riemannian Newton method. In the later case, it can further take advantage of negative curvature directions. Both global convergence and superlinear local convergence are guaranteed under mild conditions. Extensive computational experiments and comparisons with other state-of-the-art methods indicate that the proposed algorithm is very promising. Copyright © 2017, The Authors. All rights reserved.

关键词： Newton-Raphson method

Incorporating Born solvation energy into the three-dimensional Poisson-Nernst-Planck model to study ion selectivity in KcsA K+ channels

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Physical Review E 2017年第6期96卷 062416-062416页

作者： Xuejiao Liu Benzhuo Lu State Key Laboratory of Scientific and Engineering Computing National Center for Mathematics and Interdisciplinary Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China and School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China

Potassium channels are much more permeable to potassium than sodium ions, although potassium ions are larger and both carry the same positive charge. This puzzle cannot be solved based on the traditional Poisson-Nernst-Planck (PNP) theory of electrodiffusion because the PNP model treats all ions as point charges, does not incorporate ion size information, and therefore cannot discriminate potassium from sodium ions. The PNP model can qualitatively capture some macroscopic properties of certain channel systems such as current-voltage characteristics, conductance rectification, and inverse membrane potential. However, the traditional PNP model is a continuum mean-field model and has no or underestimates the discrete ion effects, in particular the ion solvation or self-energy (which can be described by Born model). It is known that the dehydration effect (closely related to ion size) is crucial to selective permeation in potassium channels. Therefore, we incorporated Born solvation energy into the PNP model to account for ion hydration and dehydration effects when passing through inhomogeneous dielectric channel environments. A variational approach was adopted to derive a Born-energy-modified PNP (BPNP) model. The model was applied to study a cylindrical nanopore and a realistic KcsA channel, and three-dimensional finite element simulations were performed. The BPNP model can distinguish different ion species by ion radius and predict selectivity for K+ over Na+ in KcsA channels. Furthermore, ion current rectification in the KcsA channel was observed by both the PNP and BPNP models. The I−V curve of the BPNP model for the KcsA channel indicated an inward rectifier effect for K+ (rectification ratio of ∼3/2) but indicated an outward rectifier effect for Na+ (rectification ratio of ∼1/6).

关键词： Electrostatic interactions Biomolecules Finite-element method

Anisotropic adaptive finite element method for magnetohydrodynamic flow at high Hartmann numbers

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Applied Mathematics and Mechanics(English Edition) 2016年第11期37卷 1479-1500页

作者： Jikun ZHAO Shipeng MAO Weiying ZHENG School of Mathematics and Statistics Zhengzhou University State Key Laboratory of Scientific and Engineering Computing (LSEC) and Institute of Computational Mathematics Academy of Mathematics and Systems Science (AMSS) Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences

This paper presents an anisotropic adaptive finite element method （FEM） to solve the governing equations of steady magnetohydrodynamic （MHD） duct flow. A resid- ual error estimator is presented for the standard FEM, and two-sided bounds on the error independent of the aspect ratio of meshes are provided. Based on the Zienkiewicz-Zhu es- timates, a computable anisotropic error indicator and an implement anisotropic adaptive refinement for the MHD problem are derived at different values of the Hartmann number. The most distinguishing feature of the method is that the layer information from some directions is captured well such that the number of mesh vertices is dramatically reduced for a given level of accuracy. Thus, this approach is more suitable for approximating the layer problem at high Hartmann numbers. Numerical results show efficiency of the algorithm.

关键词： magnetohydrodynamic （MHD） flow posteriori error estimate anisotropicadaptive finite element method （FEM）