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 ...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.展开更多
We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, fin...We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, find n real numbers c1 , c2 , . . . , cn suchthat the matrix diag(c1, c2,..., cn)A has eigenvalues λ1, λ2,..., λn.展开更多
In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed t...In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small,certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.展开更多
This paper deals with the problem of the postbuckling response of a thin cantilever beam ofnon-linear material, subjected to subtangential follower forces. Based on the well-knownBernoulli-Euler bending moment-curvatu...This paper deals with the problem of the postbuckling response of a thin cantilever beam ofnon-linear material, subjected to subtangential follower forces. Based on the well-knownBernoulli-Euler bending moment-curvature relation, the proposed problem is reduced to a specialeigenvalue problem of non-linear differential equation. An approximate solution is achieved byusing a simple and very effective technique, which leads to reliable results even in the case of verylarge deflections. The initial postbuckling path depending on the subtangential follower forces inequilibrium is then obtained. Moreover, the individual and coupling effect of the subtangential fol-lower force, the material non-linearity and the beam slenderness ratio on the initial postbucklingpath are also discussed in detail.展开更多
This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average inter...This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average interpolation operator of nonconforming Crouzeix- Raviart element, and prove a new and optimal error estimate in ｜｜ ｜｜o,δΩ for the eigenfunc- tion of linear conforming finite element and the nonconforming Crouzeix-Raviart element. Finally, we present some numerical results to support the theoretical analysis.展开更多
Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In th...Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method （SLAM）. The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.展开更多
The locally optimal block preconditioned 4-d conjugate gradient method(LOBP4dC G) for the linear response eigenvalue problem was proposed by Bai and Li(2013) and later was extended to the generalized linear response e...The locally optimal block preconditioned 4-d conjugate gradient method(LOBP4dC G) for the linear response eigenvalue problem was proposed by Bai and Li(2013) and later was extended to the generalized linear response eigenvalue problem by Bai and Li(2014). We put forward two improvements to the method: A shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4 dC G(ELOBP4dC G).Numerical results of the ELOBP4 dC G strongly demonstrate the capability of deflation technique and effectiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems.展开更多
This paper deals with multiplicity results for nonlinear elastic equations of thetypewhere e∈L￣2(0,1)g:[0,1]×R×R→R is a bounded continuous function and the pair (α,β)satisfiesand
A Fourier-Chebyshev Petrov-Galerkin spectral method is described for high accuracy computation of linearized dynamics for flow in a circular pipe. The code used here is based on solenoidal velocity variables and is wr...A Fourier-Chebyshev Petrov-Galerkin spectral method is described for high accuracy computation of linearized dynamics for flow in a circular pipe. The code used here is based on solenoidal velocity variables and is written in FORTRAN. Systematic studies are presented of the dependence of eigenval-ues and other quantities on the axial and azimuthal wave numbers;the Reyn-olds’ number of up to 107 and the Weissenberg’s number that is considered lower here. The flow will be considered stable if all the real parts of the ei-genvalues obtained are negative and unstable if only one of these values is positive.展开更多
本文研究如下实对称矩阵广义特征值反问题: 问题IGEP,给定X∈R^(n×m),1=diag(λ_II_k_I,…,λ_pI_k_p)∈R^(n×m),并且λ_I,…,λ_p互异,sum from i=1 to p(k_i=m,求K,M∈SR^(n×n),或K∈SR^(n×n),M∈SR_0^(n×m)...本文研究如下实对称矩阵广义特征值反问题: 问题IGEP,给定X∈R^(n×m),1=diag(λ_II_k_I,…,λ_pI_k_p)∈R^(n×m),并且λ_I,…,λ_p互异,sum from i=1 to p(k_i=m,求K,M∈SR^(n×n),或K∈SR^(n×n),M∈SR_0^(n×m),或K,M∈SR_0^(n×n),或K∈SR^(n×n),M∈SR_+^(n×n),或K∈SR_0^(n×n),M∈SR_+^(n×n),或K,M∈SR_+^(n×m), (Ⅰ)使得 KX=MXA, (Ⅱ)使得 X^TMX=I_m,KX=MXA,其中SR^(n×n)={A∈R^(n×n)|A^T=A},SR_0^(n×n)={A∈SR^(n×n)|X^TAX≥0,X∈R^n},SR_+^(n×n)={A∈SR^(n×n)|X^TAX>0,X∈R^n,X≠0}. 利用矩阵X的奇异值分解和正交三角分解,我们给出了上述问题的解的表达式.展开更多
基金supported by the National Science Foundation of China (11071245)
文摘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.
文摘We present new sufficient conditions on the solvability and numericalmethods for the following multiplicative inverse eigenvalue problem: Given an n × nreal matrix A and n real numbers λ1 , λ2 , . . . ,λn, find n real numbers c1 , c2 , . . . , cn suchthat the matrix diag(c1, c2,..., cn)A has eigenvalues λ1, λ2,..., λn.
基金supported by the Jilin Natural Science Foundation,China(No.20220101040JC)the National Natural Science Foundation of China(No.12271207)+2 种基金supported by the Hong Kong RGC General Research Funds(projects 11311122,12301420 and 11300821)the NSFC/RGC Joint Research Fund(project N-CityU 101/21)the France-Hong Kong ANR/RGC Joint Research Grant,A_CityU203/19.
文摘In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small,certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.
文摘This paper deals with the problem of the postbuckling response of a thin cantilever beam ofnon-linear material, subjected to subtangential follower forces. Based on the well-knownBernoulli-Euler bending moment-curvature relation, the proposed problem is reduced to a specialeigenvalue problem of non-linear differential equation. An approximate solution is achieved byusing a simple and very effective technique, which leads to reliable results even in the case of verylarge deflections. The initial postbuckling path depending on the subtangential follower forces inequilibrium is then obtained. Moreover, the individual and coupling effect of the subtangential fol-lower force, the material non-linearity and the beam slenderness ratio on the initial postbucklingpath are also discussed in detail.
文摘This paper is concerned with the finite elements approximation for the Steklov eigen- value problem on concave polygonal domain. We make full use of the regularity estimate and the characteristic of edge average interpolation operator of nonconforming Crouzeix- Raviart element, and prove a new and optimal error estimate in ｜｜ ｜｜o,δΩ for the eigenfunc- tion of linear conforming finite element and the nonconforming Crouzeix-Raviart element. Finally, we present some numerical results to support the theoretical analysis.
基金supported in part by NSF grants DMS-0611548 and OCI-0749217 and DOE grant DE-FC02-06ER25794supported in part by NSF of China under the contract number 10871049 and Shanghai Down project 200601.
文摘Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method （SLAM）. The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.
基金supported by National Science Foundation of USA(Grant Nos.DMS1522697,CCF-1527091,DMS-1317330 and CCF-1527091)National Natural Science Foundation of China(Grant No.11428104)
文摘The locally optimal block preconditioned 4-d conjugate gradient method(LOBP4dC G) for the linear response eigenvalue problem was proposed by Bai and Li(2013) and later was extended to the generalized linear response eigenvalue problem by Bai and Li(2014). We put forward two improvements to the method: A shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4 dC G(ELOBP4dC G).Numerical results of the ELOBP4 dC G strongly demonstrate the capability of deflation technique and effectiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems.
文摘This paper deals with multiplicity results for nonlinear elastic equations of thetypewhere e∈L￣2(0,1)g:[0,1]×R×R→R is a bounded continuous function and the pair (α,β)satisfiesand
文摘A Fourier-Chebyshev Petrov-Galerkin spectral method is described for high accuracy computation of linearized dynamics for flow in a circular pipe. The code used here is based on solenoidal velocity variables and is written in FORTRAN. Systematic studies are presented of the dependence of eigenval-ues and other quantities on the axial and azimuthal wave numbers;the Reyn-olds’ number of up to 107 and the Weissenberg’s number that is considered lower here. The flow will be considered stable if all the real parts of the ei-genvalues obtained are negative and unstable if only one of these values is positive.
