Asymptotic Estimates of the Accuracy of Eigenvalues of Fourth Order Elliptic Operator with Mixed Boundary Conditions

We obtain the asymptotic estimates of the accuracy of eigenvalues of the fourth-order operator with mixed boundary conditions on the boundary of a rectangle. Knowledge of the main part of eigenvalues error allows us to reasonably specify eigenvalues on a sequence of grids, obtain discrete analogs of high accuracy, and construct discrete analogs whose eigenvalues give the bilateral approximation to eigenvalues of the original problem.     

A Numerical Method to Verify the Elliptic Eigenvalue Problems Including a Uniqueness Property

We propose a numerical method to enclose the eigenvalues and eigenfunctions of second-order elliptic operators with local uniqueness. We numerically construct a set containing eigenpairs which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space by using a finite element approximation and constructive error estimates. We then prove the local uniqueness separately of eigenvalues and eigenfunctions. This local uniqueness assures the simplicity of the eigenvalue. Numerical examples are presented. Received: November 2, 1998; revised June 5, 1999     

Bounds for theN lowest eigenvalues of fourth-order boundary value problems

We describe a method for the calculation of theN lowest eigenvalues of fourth-order problems inH ₀ ² (Ω). In order to obtain small error bounds, we compute the defects inH ⁻²(Ω) and, to obtain a bound for the rest of the spectrum, we use a boundary homotopy method. As an example, we compute strict error bounds (using interval arithmetic to control rounding errors) for the 100 lowest eigenvalues of the clamped plate problem in the unit square. Applying symmetry properties, we prove the existence of double eigenvalues.     

Attacking a Conjecture in Mathematical Physics by Combining Methods of Computational Analysis and Scientific Computing

We consider a conjecture on the sum of eigenvalues of two integral operators arising in potential and scattering theory for the case that the underlying surface is a triaxial ellipsoid. This concerns computation of Lamé functions which are anyway of great interest in electromagnetics and mechanics. We provide a new effective scheme for the numerical treatment of these special functions. It involves computing the Lamé functions with high accuracy combined with safe error estimates.     

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

The paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The conclusion is that if local approximation properties of nonconforming finite element spaces are better than total errors (sums of global approximation errors and consistency errors) of nonconforming finite element methods, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we analyze abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. We show that this condition hold for most low-order nonconforming finite elements in literature. In addition, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we also propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.     

Numerical calculation of eigenvalues of integral operators for plane elastostatic boundary value problems

For numerical treatment integral equations are discretized and replaced by systems of algebraic equations. The condition properties of these systems depend on the eigenvalues and eigenvectors of the corresponding coefficient matrices. In this paper the eigenvalues of four integral operators for plane elastostatic boundary value problems are calculated numerically and checked against exact data. The results of the evaluations are represented in condensed form as condition numbers, which can be used to estimate the truncation error. The decay behaviour of the error of the numerically calculated eigenvalues permits the precision to be improved by application of Richardson extrapolation.     

Numerical solution of singular ODE eigenvalue problems in electronic structure computations

We put forward a new method for the solution of eigenvalue problems for (systems of) ordinary differential equations, where our main focus is on eigenvalue problems for singular Schrödinger equations arising for example in electronic structure computations. In most established standard methods, the generation of the starting values for the computation of eigenvalues of higher index is a critical issue. Our approach comprises two stages: First we generate rough approximations by a matrix method, which yields several eigenvalues and associated eigenfunctions simultaneously, albeit with moderate accuracy. In a second stage, these approximations are used as starting values for a collocation method which yields approximations of high accuracy efficiently due to an adaptive mesh selection strategy, and additionally provides reliable error estimates. We successfully apply our method to the solution of the quantum mechanical Kepler, Yukawa and the coupled ODE Stark problems.     

Over-iterates of Bernstein-Stancu operators

Abstract In the present note we prove convergence results for over-iterates of certain (generalized) Bernstein-Stancu operators. Similar assertions were obtained in [11]. However, our approach is different in the sense that it uses the spectrum of the operators involved. It is therefore possible to make global statements on [0, 1]. Keywords Bernstein-Stancu operators, eigenvalues, eigenfunctions, iterates. Mathematics Subject Classsification (2000): 41A36, 47A75     

Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction–Subdiffusion Equations

This paper is concerned with unconditionally optimal error estimates of linearized Galerkin finite element methods to numerically solve some multi-dimensional fractional reaction–subdiffusion equations, while the classical analysis for numerical approximation of multi-dimensional nonlinear parabolic problems usually require a restriction on the time-step, which is dependent on the spatial grid size. To obtain the unconditionally optimal error estimates, the key point is to obtain the boundedness of numerical solutions in the \(L^\infty \)-norm. For this, we introduce a time-discrete elliptic equation, construct an energy function for the nonlocal problem, and handle the error summation properly. Compared with integer-order nonlinear problems, the nonlocal convolution in the time fractional derivative causes much difficulties in developing and analyzing numerical schemes. Numerical examples are given to validate our theoretical results.     

An optimal order numerical quadrature approximation of a planar isoparametric eigenvalue problem on triangular finite element meshes

Abstract This paper deals with a finite element numerical quadrature method. It is applied for a class of second-order self-adjoint elliptic operators defined on a bounded domain in the plane. Isoparametric finite element transformations and triangular Lagrange finite elements are used.We establish the rate of convergence for approximate eigenvalues and eigenfunctions of second-order elliptic eigenvalue problems, obtained by a numerical quadrature finite element approximation. Thus the relationship between possible quadrature formulas and the optimal and almost optimal precision of the method is established. The emphasis of the paper is on the error analysis of the approximate eigenpairs. Numerical results confirming the theory are presented.     

On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems

The use of algebraic eigenvalues to approximate the eigenvalues of Sturm-Liouville operators is known to be satisfactory only when approximations to the fundamental and the first few harmonics are required. In this paper, we show how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.     

Precision edge contrast and orientation estimation

The contrast and orientation estimation accuracy of several edge operators that have been proposed in the literature is examined both for the noiseless case and in the presence of additive Gaussian noise. The test image is an ideal step edge that has been sampled with a square-aperture grid. The effects of subpixel translations and rotations of the edge on the performance of the operators are studied. It is shown that the effect of subpixel translations of an edge can generate more error than moderate noise levels. Methods with improved results are presented for Sobel angle estimates and the Nevatia-Babu operator, and theoretical noise performance evaluations are also provided. An edge operator based on two-dimensional spatial moments is presented. All methods are compared according to worst-case and RMS error in an ideal noiseless situation and RMS error under various noise levels     

The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators

This paper is a complement of the work (Hu et al. in arXiv:1112.1145v1[math.NA], 2011), where a general theory is proposed to analyze the lower bound property of discrete eigenvalues of elliptic operators by nonconforming finite element methods. One main purpose of this paper is to propose a novel approach to analyze the lower bound property of discrete eigenvalues produced by the Crouzeix–Raviart element when exact eigenfunctions are smooth. In particular, under some conditions on the triangular mesh, it is proved that the Crouzeix–Raviart element method of the Laplace operator yields eigenvalues below exact ones. Such a theoretical result explains most of numerical results in the literature and also partially answers the problem of Boffi (Acta Numerica 1–120, 2010). This approach can be applied to the Crouzeix–Raviart element of the Stokes eigenvalue problem and the Morley element of the buckling eigenvalue problem of a plate. As a second main purpose, a new identity of the error of eigenvalues is introduced to study the upper bound property of eigenvalues by nonconforming finite element methods, which is successfully used to explain why eigenvalues produced by the rotated $Q_1$ element of second order elliptic operators (when eigenfunctions are smooth), the Adini element (when eigenfunctions are singular) and the new Zienkiewicz-type element of fourth order elliptic operators, are above exact ones.     

Fast numerical approximation for the space-fractional semilinear parabolic equations on surfaces

In this work, a fast numerical method is considered to solve the space-fractional semilinear parabolic equations on closed surfaces. It is a challenge in that how to define the space fractional operator and corresponding semilinear parabolic equation with the energy functional defined on surface. To overcome it, using the local tangential space, we construct the spectral approximation for the space fractional operator on surfaces and apply the matrix transfer technique to avoid the difficulty of fractional nonlocality. The main advantage of the matrix transfer method is the completed diagonal representation of fractional operators from eigenvalue decomposition. Moreover, the time-discrete error estimates are presented as well as the energy stability. Various numerical examples are carried out to verify the theoretical results.

     

Weighted Error Estimates for Finite Element Solutions of Variational Inequalities

In this note the studies begun in Blum and Suttmeier (1999) on adaptive finite element discretisations for nonlinear problems described by variational inequalities are continued. Similar to the concept proposed, e.g., in Becker and Rannacher (1996) for variational equalities, weighted a posteriori estimates for controlling arbitrary functionals of the discretisation error are constructed by using a duality argument. Numerical results for the obstacle problem demonstrate the derived error bounds to be reliable and, used for an adaptive grid refinement strategy, to produce economical meshes. Received September 6, 1999; revised February 8, 2000     

用于故障检测的集成核主分量分析

针对复杂环境下的多变量工业过程在线故障检测问题, 提出基于集成核主分量分析的解决方法. 该方法首先求出样本映射后的无限维空间的多组近似基, 将主分量分析问题特征向量的解空间限定在近似基张成空间求解; 然后集成特征向量和特征值, 并计算Hotelling ??2 统计量和平方预报误差; 最后据此判断检测结果. 该方法对Tennessee Eastman 过程故障检测样本进行测试, 并与其他两种方法进行对比. 测试结果表明了所提出方法的有效性.

     

Investigating Some Spectral Properties of Two-Dimensional Hysteresis Play-Functionals

Spectral properties of two-dimensional generating functionals are considered. They are associated with scalar hysteresis operators. These operators occur as building elements for models of hysteresis within nonlinear analysis. We calculate eigenvalues of the hysteresis playfunctionals and investigate the structure of the corresponding eigenvectors. It turns out that the point spectrum reflects the regularizing property that hysteresis play-operators exhibit in general: Their only possible eigenvalues are attained in the interval thus reflecting the Lipschitz constant less than 1 for the play-operators.     

Un particolare metodo per la determinazione di autovalori di matrici tridiagonali simmetriche

In this paper we given an algorithm of low computational complexity which determines the eigenvalues of a symmetric tridiagonal matrix. The algorithm uses the technique of spectrum slicing together with methods for finding the zeros of polynomials. An application of algorithm for computing Gauss quadrature formulas is given.

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Lavoro svolto nell'ambito del Gruppo Nazionale di Informatica Matematica del C.N.R.     

Spectral characteristics and eigenvalues computation of the harmonic state operators in continuous-time periodic systems

Spectral characteristics of what we call the harmonic state operators in finite-dimensional linear continuous-time periodic (FDLCP) systems are examined by means of the Fredholm theory for the first time. It is shown that the harmonic state operator is a closed, densely defined Fredholm operator on the Hilbert space l₂, and its spectrum contains only eigenvalues of finite type in any bounded neighborhood of the origin of the complex plane. These spectral characteristics lead us to an asymptotic computation algorithm for the eigenvalues of the harmonic state operator via truncations on its Fredholm regularization. Furthermore, the truncation approach also gives us a necessary and sufficient stability criterion in the FDLCP setting, which only involves the Fourier coefficients of the state matrix. Asymptotic stability of the lossy Mathieu differential equation is investigated to illustrate the results.     

A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems

In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented.