We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear ...We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations.Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal,the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.展开更多
This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solve...This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.展开更多
Rotation numbers are used in this paper to study the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic weight which changes sign. The analysis proves that for any nonnegative i...Rotation numbers are used in this paper to study the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic weight which changes sign. The analysis proves that for any nonnegative integer n, ρ -1(n/2) is the union of two closed intervals, one of which lies in [FK(W+3mm?3mm][TPP533A,+3mm?2mm] + and the other in [FK(W+3mm?3mm][TPP533A,+3mm?2mm] -, and the endpoints of these intervals yield the corresponding periodic and anti-periodic eigenvalues.展开更多
In this paper, we determine the infimum and the supremum of the Dirich-let eigenvalues λn(p) (n = 1,2,…)of the problem t∈ ?[0,T], where 1 < p < ∞, and the weights p are nonnegative and are subject to conditi...In this paper, we determine the infimum and the supremum of the Dirich-let eigenvalues λn(p) (n = 1,2,…)of the problem t∈ ?[0,T], where 1 < p < ∞, and the weights p are nonnegative and are subject to conditions p(t)dt = M and max(e[0,T] p(t) = H. It is also explained for whatweights p the infimum and the supremum will be attained.展开更多
We consider a parametric Dirichlet problem driven by the p-Laplacian with a Caratheodory reaction of equidiffusive type. Our hypotheses incorporate as a special case the equidiffusive p-logistic equation. We show that...We consider a parametric Dirichlet problem driven by the p-Laplacian with a Caratheodory reaction of equidiffusive type. Our hypotheses incorporate as a special case the equidiffusive p-logistic equation. We show that if λ1 〉 0 is the principal eigenvalue of the Dirichlet negative p-Laplacian and ）λ 〉 λ1 （/k being the parameter）, the problem has a unique positive solution, while for ）λ ∈ （0, λ1], the problem has no positive solution.展开更多
The properties of the first eigenvalue of a class of (<em>p</em>,<em>q</em>) Laplacian are investigated. A variational formulation for the first eigenvalue of the Laplacian on a closed Riemanni...The properties of the first eigenvalue of a class of (<em>p</em>,<em>q</em>) Laplacian are investigated. A variational formulation for the first eigenvalue of the Laplacian on a closed Riemannian manifold is obtained. This eigenvalue corresponds to a nonlinear, coupled system of <em>p</em>-Laplacian partial differential equations. The main idea is to investigate the evolution of the first eigenvalue of the system under the Ricci harmonic flow. It is also possible to construct monotonic quantities based on them and study their evolution which is done.展开更多
This paper introduces a Takagi-Sugeno(T-S)fuzzy regulator design using the negative absolute eigenvalue(NAE)approach for a class of nonlinear and unstable systems.The open-loop system is initially embodied by the trad...This paper introduces a Takagi-Sugeno(T-S)fuzzy regulator design using the negative absolute eigenvalue(NAE)approach for a class of nonlinear and unstable systems.The open-loop system is initially embodied by the traditional T-S fuzzy model and then,all closed-loop subsystems are combined using the proposed Max-Min operator in place of traditional weighted average operator from the controller side to lessen the coupling virtually and simplify the proposed regulator design.For each virtually decoupled closed-loop subsystem,the composite regulators(i.e.,primary and secondary regulators)are designed by the NAE approach based on the enhanced eigenvalue analysis.The Lyapunov function is utilized to guarantee the asymptotic stability of the overall T-S fuzzy control system.The most popular and widely used nonlinear and unstable systems like the electromagnetic levitation system(EMLS)and the inverted cart pendulum(ICP)are simulated for the wide range of the initial conditions and the enormous variation in the disturbance.The transient and steady-state performance of the considered systems using the proposed design are analyzed in terms of the decay rate,settling time and integral errors as IAE,ISE,ITAE,and ITSE to validate the effectiveness of the proposed approach compared to the most popular and traditional parallel distributed compensation(PDC)approach.展开更多
We propose a finite element method to compute the band structures of dispersive photonic crystals in 3D.The nonlinear Maxwell’s eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator fun...We propose a finite element method to compute the band structures of dispersive photonic crystals in 3D.The nonlinear Maxwell’s eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator function.The N´ed´elec edge elements are employed to discretize the operators,where the divergence free condition for the electric field is realized by a mixed form using a Lagrange multiplier.The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions with the regular approximation of the edge elements.The spectral indicator method is then applied to compute the discrete eigenvalues.Numerical examples are presented demonstrating the effectiveness of the proposed method.展开更多
The major difficulty in achieving good performance of industrial polymerization reactors lies in the lack of understanding of their nonlinear dynamics and the lack of well-developed techniques for the control of nonli...The major difficulty in achieving good performance of industrial polymerization reactors lies in the lack of understanding of their nonlinear dynamics and the lack of well-developed techniques for the control of nonlinear processes, which are usually accompanied with bifurcation phenomenon. This work aims at investigating the nonlinear behavior of the parameterized nonlinear system of vinyl acetate polymerization and further modifying the bifurcation characteristics of this process via a washout filter-aid controller, with all the original steady state equilibria preserved. Advantages and possible extensions of the proposed methodology are discussed to provide scientific guide for further controller design and operation improvement.展开更多
We are concerned with the maximization of tr（V T AV）/tr（V T BV）＋tr（V T CV） over the Stiefel manifold {V ∈ R m×l ｜ V T V = Il} （l 〈 m）, where B is a given symmetric and positive definite matrix, A and...We are concerned with the maximization of tr（V T AV）/tr（V T BV）＋tr（V T CV） over the Stiefel manifold {V ∈ R m×l ｜ V T V = Il} （l 〈 m）, where B is a given symmetric and positive definite matrix, A and C are symmetric matrices, and tr（. ） is the trace of a square matrix. This is a subspace version of the maximization problem studied in Zhang （2013）, which arises from real-world applications in, for example, the downlink of a multi-user MIMO system and the sparse Fisher discriminant analysis in pattern recognition. We establish necessary conditions for both the local and global maximizers and connect the problem with a nonlinear extreme eigenvalue problem. The necessary condition for the global maximizers offers deep insights into the problem, on the one hand, and, on the other hand, naturally leads to a self-consistent-field （SCF） iteration to be presented and analyzed in detail in Part II of this paper.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 91330202, 11371026, 11201501, 11571389, 11001259 and 11031006)National Basic Research Program of China (Grant No. 2011CB309703)the National Center for Mathematics and Interdisciplinary Science, Chinese Academy of Sciences, the President Foundation of Academy of Mathematics and Systems Science, Chinese Academy of Sciences and the Program for Innovation Research in Central University of Finance and Economics
文摘We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations.Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal,the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.
基金the National Science and Tech-nology Council,Taiwan for their financial support(Grant Number NSTC 111-2221-E-019-048).
文摘This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.
基金Supported by the National Basic Research PrioritiesProgram me of China (No.G19990 75 10 8) and theTRAPOYT of the Ministry of Education of China
文摘Rotation numbers are used in this paper to study the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic weight which changes sign. The analysis proves that for any nonnegative integer n, ρ -1(n/2) is the union of two closed intervals, one of which lies in [FK(W+3mm?3mm][TPP533A,+3mm?2mm] + and the other in [FK(W+3mm?3mm][TPP533A,+3mm?2mm] -, and the endpoints of these intervals yield the corresponding periodic and anti-periodic eigenvalues.
基金Project Supported by the National 973 Project(G1999075100)of Chinathe ExcellentPersonnel Supporting Plan of the Ministry of Education of China.
文摘In this paper, we determine the infimum and the supremum of the Dirich-let eigenvalues λn(p) (n = 1,2,…)of the problem t∈ ?[0,T], where 1 < p < ∞, and the weights p are nonnegative and are subject to conditions p(t)dt = M and max(e[0,T] p(t) = H. It is also explained for whatweights p the infimum and the supremum will be attained.
基金supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No.295118the National Science Center of Poland under grant No.N N201 604640+1 种基金the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under grant No.W111/7.PR/2012the National Science Center of Poland under Maestro Advanced Project No.DEC2012/06/A/ST1/00262
文摘We consider a parametric Dirichlet problem driven by the p-Laplacian with a Caratheodory reaction of equidiffusive type. Our hypotheses incorporate as a special case the equidiffusive p-logistic equation. We show that if λ1 〉 0 is the principal eigenvalue of the Dirichlet negative p-Laplacian and ）λ 〉 λ1 （/k being the parameter）, the problem has a unique positive solution, while for ）λ ∈ （0, λ1], the problem has no positive solution.
文摘The properties of the first eigenvalue of a class of (<em>p</em>,<em>q</em>) Laplacian are investigated. A variational formulation for the first eigenvalue of the Laplacian on a closed Riemannian manifold is obtained. This eigenvalue corresponds to a nonlinear, coupled system of <em>p</em>-Laplacian partial differential equations. The main idea is to investigate the evolution of the first eigenvalue of the system under the Ricci harmonic flow. It is also possible to construct monotonic quantities based on them and study their evolution which is done.
文摘This paper introduces a Takagi-Sugeno(T-S)fuzzy regulator design using the negative absolute eigenvalue(NAE)approach for a class of nonlinear and unstable systems.The open-loop system is initially embodied by the traditional T-S fuzzy model and then,all closed-loop subsystems are combined using the proposed Max-Min operator in place of traditional weighted average operator from the controller side to lessen the coupling virtually and simplify the proposed regulator design.For each virtually decoupled closed-loop subsystem,the composite regulators(i.e.,primary and secondary regulators)are designed by the NAE approach based on the enhanced eigenvalue analysis.The Lyapunov function is utilized to guarantee the asymptotic stability of the overall T-S fuzzy control system.The most popular and widely used nonlinear and unstable systems like the electromagnetic levitation system(EMLS)and the inverted cart pendulum(ICP)are simulated for the wide range of the initial conditions and the enormous variation in the disturbance.The transient and steady-state performance of the considered systems using the proposed design are analyzed in terms of the decay rate,settling time and integral errors as IAE,ISE,ITAE,and ITSE to validate the effectiveness of the proposed approach compared to the most popular and traditional parallel distributed compensation(PDC)approach.
基金China Postdoctoral Science Foundation Grant 2019M650460the NSF grant DMS-2011148.The research of J.Sun is supported partially by the Simons Foundation Grant 711922.
文摘We propose a finite element method to compute the band structures of dispersive photonic crystals in 3D.The nonlinear Maxwell’s eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator function.The N´ed´elec edge elements are employed to discretize the operators,where the divergence free condition for the electric field is realized by a mixed form using a Lagrange multiplier.The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions with the regular approximation of the edge elements.The spectral indicator method is then applied to compute the discrete eigenvalues.Numerical examples are presented demonstrating the effectiveness of the proposed method.
基金Supported by the National Basic Research Programme(2012CB720500)the National Natural Science Foundation of China(21306100)
文摘The major difficulty in achieving good performance of industrial polymerization reactors lies in the lack of understanding of their nonlinear dynamics and the lack of well-developed techniques for the control of nonlinear processes, which are usually accompanied with bifurcation phenomenon. This work aims at investigating the nonlinear behavior of the parameterized nonlinear system of vinyl acetate polymerization and further modifying the bifurcation characteristics of this process via a washout filter-aid controller, with all the original steady state equilibria preserved. Advantages and possible extensions of the proposed methodology are discussed to provide scientific guide for further controller design and operation improvement.
基金supported by National Natural Science Foundation of China(Grant Nos.11101257 and 11371102)the Basic Academic Discipline Program+3 种基金the 11th Five Year Plan of 211 Project for Shanghai University of Finance and Economicsa visiting scholar at the Department of Mathematics,University of Texas at Arlington from February 2013 toJanuary 2014supported by National Science Foundation of USA(Grant Nos.1115834and 1317330)a Research Gift Grant from Intel Corporation
文摘We are concerned with the maximization of tr（V T AV）/tr（V T BV）＋tr（V T CV） over the Stiefel manifold {V ∈ R m×l ｜ V T V = Il} （l 〈 m）, where B is a given symmetric and positive definite matrix, A and C are symmetric matrices, and tr（. ） is the trace of a square matrix. This is a subspace version of the maximization problem studied in Zhang （2013）, which arises from real-world applications in, for example, the downlink of a multi-user MIMO system and the sparse Fisher discriminant analysis in pattern recognition. We establish necessary conditions for both the local and global maximizers and connect the problem with a nonlinear extreme eigenvalue problem. The necessary condition for the global maximizers offers deep insights into the problem, on the one hand, and, on the other hand, naturally leads to a self-consistent-field （SCF） iteration to be presented and analyzed in detail in Part II of this paper.
