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A saddle point NUMERICAL METHOD FOR HELMHOLTZ EQUATIONS

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Journal of Computational Mathematics 2017年第1期35卷 19-36页

作者： Russell B. Richins Department of Mathematics and Computer Science Thiel College Pennsylvania USA

In a previous work, the author and D.C. Dobson proposed a numerical method for solving the complex Helmholtz equation based on the minimization variational principles developed by Milton, Seppecher, and Bouchitte. This method results in a system of equations with a symmetric positive definite coefficient matrix, but at the same time requires solving simultaneously for the solution and its gradient. Herein is presented a method based on the saddle point variational principles of Milton, Seppecher, and Bouch- itte, which produces symmetric positive definite systems of equations, but eliminates the necessity of solving for the gradient of the solution. The result is a method for a wide class of Helmholtz problems based completely on the Conjugate Gradient algorithm.

关键词： Helmholtz Conjugate gradient saddle point Finite element.

Globally Proper saddle point in Ic-Cone-Convexlike Set-Valued Optimization Problems

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Acta Mathematica Sinica,English Series 2009年第11期25卷 1921-1928页

作者： Guo Lin YU San Yang LIU Institute of Information and Systems Science The North University for Ethnics Yinchuan 750021 P. R. China Department of Applied Mathematics Xidian University Xi'an 710071 P. R. China

The existence conditions of globally proper efficient points and a useful property of ic- cone-convexlike set-valued maps are obtained. Under the assumption of the ic-cone-convexlikeness, the optimality conditions for globally proper efficient solutions are established in terms of Lagrange multipliers. The new concept of globally proper saddle-point for an appropriate set-valued Lagrange map is introduced and used to characterize the globally proper efficient solutions. The results which are obtained in this paper are proven under the conditions that the ordering cone need not to have a nonempty interior.

关键词： set-valued map ic-cone-convexlikeness globally proper efficiency saddle point

Open Loop saddle point on Linear Quadratic Stochastic Differential Games

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Communications in Mathematical Research 2014年第1期30卷 11-22页

作者： WANG JUN School of Basic Science Changchun University of Technology College of Mathematics Jilin University

In this paper, we deal with one kind of two-player zero-sum linear quadratic stochastic differential game problem. We give the existence of an open loop saddle point if and only if the lower and upper values exist.

关键词： stochastic differential game saddle point open loop strategy

Coincidence point Theorems,Intersection Theorems and saddle point Theorems on FC-spaces

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Communications in Mathematical Research 2009年第2期25卷 115-122页

作者： PIAO YONG-JIE YIN ZHE Department of Mathematics Yanbian University Yanji Jilin 133002

In this paper, we first give the definitions of finitely continuous topological space and FC-subspace generated by some set, and obtain coincidence point theorem, whole intersection theorems and Ky Fan type matching t... 详细信息

关键词： FC-space FC-subspace generated by set saddle point

A New Fixed point Theorem in Noncompact L-Convex Metric Spaces with Applications to Minimax Inequalities and saddle points

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Journal of Mathematical Research and Exposition 2011年第2期31卷 330-336页

作者： Kai Ting WEN Department of Mathematics Bijie University Guizhou 551700 P. R. China

In this paper,a new fixed point theorem is established in noncompact complete Lconvex metric spaces.As applications,a maximal element theorem,a minimax inequality and a saddle point theorem are obtained.

关键词： L-convex metric space fixed point maximal element minimax inequality saddle point

Optimality Conditions for Generalized Convex Nonsmooth Uncertain Multi-objective Fractional Programming

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Journal of the Operations Research Society of China 2023年第4期11卷 809-826页

作者： Xiao Pan Guo-Lin Yu Tian-Tian Gong School of Mathematics and Information Science North Minzu UniversityYinchuan750021NingxiaChina

This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex function pairs,called type-I functions and pseudo-quasi-type-I functions,are introduced in this paper for(NUMFP).Under the assumption that(NUMFP)satisfies the robust constraint qualification with respect to Clarke subdifferential,necessary optimality conditions of the robust weak efficient solution are given.Sufficient optimality conditions are obtained under pseudo-quasi-type-I generalized convexity assumption.Furthermore,we introduce the concept of robust weak saddle points to(NUMFP),and prove two theorems about robust weak saddle points.The main results in the present paper are verified by concrete examples.

关键词： Multi-objective fractional programming Robust weak efficient solution Generalized convex function Optimality condition saddle point

Consecutive topological transitions of helical Fermi arcs at saddle points in CoSi

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Science China(Physics,Mechanics & Astronomy) 2022年第5期65卷 97-103页

作者： Zhicheng Rao Shangjie Tian Shunye Gao Quanxin Hu Wenhui Fan Jierui Huang Cenyao Tang Yaobo Huang Hechang Lei Yujie Sun Tian Qian Hong Ding Beijing National Laboratory for Condensed Matter Physics and Institute of Physics Chinese Academy of SciencesBeijing 100190China University of Chinese Academy of Sciences Beijing 100049China Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials&Micro-nano Devices Renmin University of ChinaBeijing 100872China Shanghai Synchrotron Radiation Facility Shanghai Advanced Research InstituteChinese Academy of SciencesShanghai 201210China Songshan Lake Materials Laboratory Dongguan 523808China CAS Center for Excellence in Topological Quantum Computation University of Chinese Academy of SciencesBeijing 100190China Department of Physics South University of Science and TechnologyShenzhen 518055China

The CoSi family hosts unconventional topological nodes with nonzero Chern numbers.The nontrivial topology is manifested by conspicuous surface Fermi arcs connecting surface projections of the nodes.Here,using angle-resolved photoemission spectroscopy,we have systematically investigated the(001)surface states of pristine and Ni-doped Co Si.The surface states form saddle-like band structures at/near the time-reversal invariant point near the Fermi level.The Fermi arcs undergo consecutive Lifshitz transitions at the saddle points X,leading to changes of the Fermi arc configuration.As the density of states has a van Hove singularity at the saddle points,exotic many-body physical phenomena may emerge accompanied by the topological transitions of surface Fermi arcs.

关键词： topological transition Fermi arcs saddle point

Optimality Conditions and saddle-point Theory for Nonsmooth Generalized Convexity Multiobjective Programming

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Optimality Conditions and Saddle-Point Theory for Nonsmooth ...

Error Estimate for Semi-implicit Method of Sphere-Constrained High-Index saddle Dynamics

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作者： Cailing Wang~1,Wenjuan Sun~2,You Zhang~3,Zhongfan Li~4 1.School of Mathematics,Jilin University,Changchun 130012,China 2.School of Science,Shenyang Ligong University,Shenyang 110168,China 3.School of Mathematics & Statistics,Northeast Normal University,Changchun 130024,China 4.School of Mathematics,Jilin University,Changchun 130012,China

＜正＞In this paper,we introduce generalized essentially pseudoconvex function and generalized essentially quasiconvex function,and give sufficient optimality conditions of the nonsmooth generalized convex multi-objec... 详细信息

关键词： Multiobjective programming optimality conditions saddle point

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Chinese Annals of Mathematics,Series B 2023年第5期44卷 765-780页

作者： Lei ZHANG Pingwen ZHANG Xiangcheng ZHENG Beijing International Center for Mathematical Research Center for Machine Learning ResearchCenterfor Quantitative BiologyPeking UniversityBeijing 100871China School of Mathematics and Statistics Wuhan UniversityWuhan 430072China School of MathematicalSciences Laboratory of Mathematics and Applied MathematicsPeking UniversityBeijing 100871China School of Mathematics Shandong UniversityJinan 250100China

The authors prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics,which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere.Due to the semi-implicit treatment and the novel computational procedure,the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations,and the computations of the state variable and directional vectors are coupled with the retraction,the vector transport and the orthonormalization procedure,which significantly complicates the analysis.They address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings.

关键词： saddle point Constrained saddle dynamics Solution landscape Semiimplicit Numerical analysis