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Nuclear Science and Techniques 2023年第10期34卷 178-200页

作者： Qi-Hong Yang Yu Yang Yang-Tao Deng Qiao-Lin He He-Lin Gong Shi-Quan Zhang School of Mathdematics Sichuan UniversityChengdu 610065China Paris Elite Institute of Technology Shanghai Jiao Tong UniversityShanghai 200240China

Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years.Despite some progress in one-dimensional problems,there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems.We present two networks,namely the Generalized Inverse Power Method Neural Network(GIPMNN)and Physics-Constrained GIPMNN(PC-GIPIMNN)to solve K-eigenvalue problems in neutron diffusion theory.GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method.The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux.Meanwhile,Deep Ritz Method(DRM)directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form.A comprehensive study was conducted using GIPMNN,PC-GIPMNN,and DRM to solve problems of complex spatial geometry with variant material domains from the fleld of nuclear reactor physics.The methods were compared with the standard flnite element method.The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.

关键词： Neural network Reactor physics Neutron diffusion equation eigenvalue problem Inverse power method

Asymptotic Behavior of Least Energy Solutions for a Fractional Laplacian eigenvalue problem on R^(N)

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Acta Mathematica Sinica,English Series 2023年第4期39卷 707-727页

作者： Yun Bo WANG Xiao Yu ZENG Huan Song ZHOU School of Mathematics and Center for Nonlinear Studies Northwest UniversityXi'an 710127P.R.China Center for Mathematical Sciences Wuhan University of TechnologyWuhan 430070P.R.China

We are interested in the existence and asymptotic behavior for the least energy solutions of the following fractional eigenvalue problem (P)(-△)^(s)u+V(x)u=μu+am(x)|u|^(4s/N)u,∫_(R^(N))|u|^(2)dx=1,u∈H^(s)(R^(N)),where s∈(0,1),μ∈R,a>0,V(x)and m(x)are L^(∞)(R^(N))functions with N≥2.We prove that there is a threshold a^(*)_(s)>0 such that problem(P)has a least energy solution u_(a)(x)for each a∈(0,a^(*)_(s))and u_(a)blows up,as a↗a^(*)_(s),at some point x_(0)∈R^(N),which makes V(x_(0))be the minimum and m(x_(0))be the maximum.Moreover,the precise blowup rates for u_(a)are obtained under suitable conditions on V(x)and m(x).

关键词： Laplacian eigenvalue problem constrained variational problem energ mates

Kinematic analysis of flexible bipedal robotic systems

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Applied Mathematics and Mechanics(English Edition) 2024年第5期45卷 795-818页

作者： R.FAZEL A.M.SHAFEI S.R.NEKOO Department of Mechanical Engineering Shahid Bahonar University of KermanKerman ***Iran Departamento de Ingenieria de Sistemas y Automatica Escuela Técnica Superior de IngenieriaUniversidad de SevillaSeville 41092Spain

In spite of its intrinsic complexities,the passive gait of bipedal robots on a sloping ramp is a subject of interest for numerous researchers.What distinguishes the present research from similar works is the consideration of flexibility in the constituent links of this type of robotic systems.This is not a far-fetched assumption because in the transient(impact)phase,due to the impulsive forces which are applied to the system,the likelihood of exciting the vibration modes increases considerably.Moreover,the human leg bones that are involved in walking are supported by viscoelastic muscles and ligaments.Therefore,for achieving more exact results,it is essential to model the robot links with viscoelastic properties.To this end,the Gibbs-Appell formulation and Newton's kinematic impact law are used to derive the most general form of the system's dynamic equations in the swing and transient phases of motion.The most important issue in the passive walking motion of bipedal robots is the determination of the initial robot configuration with which the system could accomplish a periodic and stable gait solely under the effect of gravitational force.The extremely unstable nature of the system studied in this paper and the vibrations caused by the impulsive forces induced by the impact of robot feet with the inclined surface are some of the very serious challenges encountered for achieving the above-mentioned goal.To overcome such challenges,an innovative method that uses a combination of the linearized equations of motion in the swing phase and the algebraic motion equations in the transition phase is presented in this paper to obtain an eigenvalue problem.By solving this problem,the suitable initial conditions that are necessary for the passive gait of this bipedal robot on a sloping surface are determined.The effects of the characteristic parameters of elastic links including the modulus of elasticity and the Kelvin-Voigt coefficient on the walking stability of this type of robotic systems are also studied.The findings o

关键词： bipedal robot flexible link swing phase transient phase eigenvalue problem Kelvin-Voigt coefficient

A Deep Learning Method for Computing eigenvalues of the Fractional Schrödinger Operator

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Journal of Systems Science & Complexity 2024年第2期37卷 391-412页

作者： GUO Yixiao MING Pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger operator.The proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the problem.These improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded domains.The authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger operators.As an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

关键词： eigenvalue problem deep learning fractional Schrödinger operator isospectral problem

AN INVERSE eigenvalue problem FOR JACOBI MATRICES

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Journal of Computational Mathematics 2007年第5期25卷 620-630页

作者： Haixia Liang Erxiong Jiang Department of Mathematics Shanghai University Shanghai 200444 China

In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.

关键词： Symmetric tridiagonal matrix Jacobi matrix eigenvalue problem Inverse eigenvalue problem.

CONVERGENCE AND SUPERCONVERGENCE OF HERMITE BICUBIC ELEMENT FOR eigenvalue problem OF THE BIHARMONIC EQUATION

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Journal of Computational Mathematics 2001年第2期19卷 139-142页

作者： Dong-sheng Wu (Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China ) 中科院系统科学所北京 100080

Provides information on a study which discussed the convergence and superconvergence for eigenvalue problem of the biharmonic equation using the Hermite bicubic element. Discussion on eigenvalue problem for biharmonic equation; Background on asymptotic error expansions and interpolation postprocessing; Superconvergence approximations to the eigenvalue and eigenfunction.

关键词： Hermite bicubic element biharmonic equation interpolation postprocessing eigenvalue problem

eigenvalue problem of Doubly Stochastic Hamiltonian Systems with Boundary Conditions

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Communications in Mathematical Research 2009年第1期25卷 30-36页

作者： HAN YUE-CAI MA YONG School of Mathematics Jilin University Changchun 130012 School of Mathematics Shandong University Jinan 250100

In this paper, we investigate the eigenvalue problem of forward-backward doubly stochastic dii^erential equations with boundary value conditions. We show that this problem can be represented as an eigenvalue problem of a bounded continuous compact operator. Hence using the famous Hilbert-Schmidt spectrum theory, we can characterize the eigenvalues exactly.

关键词： doubly stochastic Hamiltonian system eigenvalue problem spectrum theory

A BIHARMONIC eigenvalue problem AND ITS APPLICATION

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Acta Mathematica Scientia 2012年第3期32卷 1213-1225页

作者：王江湖张贻民 Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences

In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in R^N：△^2u-αu＋λg（x）u=0 with u ∈H^2（R^N）,u≠0,N≥5Note that there are two parameters α and λ in it, which is different from the usual eigenvalue problems. Here, we consider λ as an eigenvalue and seek Ior a sulble range of parameter α, which ensures that problem （＊） has a maximal eigenvalue. As the loss of strong maximum principle for our problem, we can only get the existence of non-trivial solutions, not positive solutions, in this article. As an application, by using these results, we studied also the existence of non-trivial solutions for an asymptotically linear biharmonic equation in R^N.

关键词： Biharmonic equation potential well eigenvalue problem asymptotically linear

A CASCADIC MULTIGRID METHOD FOR eigenvalue problem

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Journal of Computational Mathematics 2017年第1期35卷 74-90页

作者： Xiaole Han Hehu Xie Fei Xu IAPCM Institute of Applied Physics and Computational Mathematics Beijing 100093 China LSEC NCMIS Institute of Computational Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China Beijing Institute for Scientific and Engineering Computing Beijing University of Technology Beijing 100124 China

A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by linear smoothing steps on a series of multilevel finite element spaces and nonlinear correcting steps on special coarsest spaces. Once the sequence of finite element spaces and the number of smoothing steps are appropriately chosen, the optimal convergence rate with the optimal computational work can be obtained. Some numerical experiments are presented to validate our theoretical analysis.

关键词： eigenvalue problem Cascadic multigrid Multilevel correction scheme Finiteelement method.

AN INVERSE eigenvalue problem FOR JACOBI MATRICES

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Journal of Computational Mathematics 2003年第5期21卷 569-584页

作者： Er-xiong Jiang (Department of Mathematics, Shanghai University, Shanghai 200436, China) Department of Mathematics Shanghai University 上海 200436

Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.

关键词： Symmetric tridiagonal matrix Jacobi matrix eigenvalue problem Inverse eigenvalue problem.