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A Finite-Volume Method for Nonlinear Nonlocal Equations with a gradient flow Structure

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Communications in Computational Physics 2015年第1期17卷 233-258页

作者： Jose A.Carrillo Alina Chertock Yanghong Huang Department of Mathematics Imperial College LondonLondon SW72AZUK Department of Mathematics North Carolina State UniversityRaleighNC 27695USA

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure.These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential.The proposed scheme is able to cope with non-smooth stationary states,different time scales including metastability,as well as concentrations and self-similar behavior induced by singular nonlocal kernels.We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

关键词： Finite-volume method gradient flow nonlinear nonlocal equations

gradient flow of the L_(β)-Functional

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Communications in Mathematical Research 2021年第1期37卷 113-140页

作者： Xiaoli Han Jiayu Li Jun Sun Department of Mathematical Sciences TsinghuaUniversityBeijing 100084China School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026 AMSS CASBeijing 100190China School of Mathematics and Statistics Wuhan UniversityWuhan 430072China Hubei Key Laboratory of Computational Science Wuhan UniversityWuhan 430072China

In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.

关键词： β-symplectic critical surfaces gradient flow monotonicity formula tangent cone

MINIMAL FOLIATIONS FOR THE HIGH-DIMENSIONAL FRENKEL-KONTOROVA MODEL

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Acta Mathematica Scientia 2023年第2期43卷 564-582页

作者：缪雪晴葛建华秦文新王亚南 School of Sciences Nantong UniversityNantong 226019China Department of Mathematics Soochow UniversitySuzhou 215006China School of Mathematical Sciences Nanjing Normal UniversityNanjing 210097China

For the high-dimensional Frenkel-Kontorova(FK)model on lattices,we study the existence of minimal foliations by depinning force.We introduce the tilted gradient flow and define the depinning force as the critical value of the external force under which the average velocity of the system is zero.Then,the depinning force can be used as the criterion for the existence of minimal foliations for the FK model on a Z^(d)lattice for d>1.

关键词： Aubry-Mather theory Frenkel-Kontorova model minimal foliation depinning force gradient flow

Grasping Force Optimization Algorithm of Soft Multi-fingered Hand Based on Safety Margin Detection

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机器人 2017年第6期39卷 844-852页

作者： WANG Xuelin XIAO Yongfei ZHAO Yongguo FAN Xinjian Shandong Provincial Key Laboratory of Robot and Manufacturing Automation Technology Institute of Automation of Shandong Academy of Sciences Qilu University of Technolog (Shandong Academy of Sciences)

The classical gradient flow optimization algorithm requires a valid initial point before starting the recursive algorithm,and the existing methods can’t guarantee that the initial values fully satisfy the friction cone constraints of contact point in the optimization process of gradient flow algorithm.In order to improve safety margin and prevent the finger from slipping at contact point,we present an iterative method of safe initial values with safety margin detection and develop a gradient flow optimization algorithm based on the safe initial values.Firstly,the safety margin is defined which more intuitively reflects the margin of the grasping forces at contact point.The resulting safe initial values can be achieved by the detection of desired safety margin at each iteration.Secondly,the safe initial values are usually not optimal,even with the valid initial values,and it can’t always ensure that the finger contact force always satisfies the friction cone constraints during the optimization.It is an effective way to eliminate the unreliable initial values in the optimization and obtain a safer initial values by increasing the safety margin.By transforming the safe initial values into an initial point of the gradient flow algorithm,the final optimized values of grasping forces can be generated efficiently by gradient flow iteration.Grasp examples of the soft multi-fingered hand indicate the effectiveness of the general solution of the force optimization algorithm based on safety margin detection.The method eliminates the shortcomings of the gradient flow optimization process caused by the initial value problem and provides a more accurate and reliable force optimization result for multi-fingered dexterous manipulation.

关键词： force optimization grasping force multi-fingered hand soft finger gradient flow friction cone

A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets

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Communications in Computational Physics 2019年第1期25卷 155-188页

作者： Amanda E.Diegel Shawn W.Walker Department of Mathematics Louisiana State UniversityBaton RougeLA 70803-4918USA

We develop a novel finite element method for a phase field model of nematic liquid crystal droplets.The continuous model considers a free energy comprised of three components:the Ericksen’s energy for liquid crystals,the Cahn-Hilliard energy representing the interfacial energy of the droplet,and an anisotropic weak anchoring energy that enforces a condition such that the director field is aligned perpendicular to the interface of the droplet.Applications of the model are for finding minimizers of the free energy and exploring gradient flow dynamics.We present a finite element method that utilizes a special discretization of the liquid crystal elastic energy,as well as mass-lumping to discretize the coupling terms for the anisotropic surface tension part.Next,we present a discrete gradient flow method and show that it is monotone energy decreasing.Furthermore,we show that global discrete energy minimizersΓ-converge to global minimizers of the continuous energy.We conclude with numerical experiments illustrating different gradient flow dynamics,including droplet coales-cence and break-up.

关键词： Nematic liquid crystal phase field Ericksen’s energy Γ-convergence gradient flow

Machine learning from a continuous viewpoint,I

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Science China Mathematics 2020年第11期63卷 2233-2266页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

We present a continuous formulation of machine learning,as a problem in the calculus of variations and differential-integral equations,in the spirit of classical numerical analysis.We demonstrate that conventional machine learning models and algorithms,such as the random feature model,the two-layer neural network model and the residual neural network model,can all be recovered(in a scaled form)as particular discretizations of different continuous formulations.We also present examples of new models,such as the flow-based random feature model,and new algorithms,such as the smoothed particle method and spectral method,that arise naturally from this continuous formulation.We discuss how the issues of generalization error and implicit regularization can be studied under this framework.

关键词： machine learning continuous formulation flow-based model gradient flow particle approximation

High-Order Fully Discrete Energy Diminishing Evolving Surface Finite Element Methods for a Class of Geometric Curvature flows

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Annals of Applied Mathematics 2021年第4期37卷 405-436页

作者： Beiping Duan Buyang Li Zhimin Zhang Beijing Computational Science Research Center Beijing 100193China Shenzhen JL Computational Science and Applied Research Institute ShenzhenGuangdong 518129China Department of Applied Mathematics The Hong Kong Polytechnic UniversityHung HornHong KongChina Department of Mathematics Wayne State UniversityDetroitMI 48202USA

This article concerns the construction of high-order energy-decaying numerical methods for gradient flows of evolving surfaces with curvature-dependent energy functionals.The semidiscrete evolving surface finite element method is derived based on the calculus of variation of the semidiscrete surface energy functional.This makes the semidiscrete problem naturally inherit the energy decay structure.With this property,the semidiscrete problem is furthermore formulated as a gradient flow system of ODEs.The averaged vector-field collocation method is used for time discretization of the ODEs to preserve energy decay at the fully discrete level while achieving high-order accuracy in time.Extensive numerical examples are provided to illustrate the accuracy and energy diminishing property of the proposed method,as well as the effectiveness of the method in capturing singularities in the evolution of closed surfaces.

关键词： gradient flow evolving surface curvature energy decay evolving surface finite element method averaged vector-field collocation