Bound Of Eigenvalue Research Articles

Lower Bound of Vibration Modes Using the Node-Based Smoothed Finite Element Method (NS-FEM)

The smoothed finite element method (S-FEM) has been recently developed as an effective solver for solid mechanics problems. This paper represents an effective approach to compute the lower bounds of vibration modes or eigenvalues of elasto-dynamic problems, by making use of the important softening effects of node-based S-FEM (NS-FEM). We first use NS-FEM, FEM and the analytic approach to compute the eigenvalues of transverse free vibration in strings and membranes. It is found that eigenvalues by NS-FEM are always smaller than those by FEM and the analytic method. However, NS-FEM produces spurious unphysical modes because of overly soft behavior. A technique is then proposed to remove them by analyzing their vibration shapes (eigenvectors). It is observed that spurious modes with excessively large wave numbers, which are unrelated to the physical deflection shapes but related to the discretization density, therefore can be easily removed. The final results of NS-FEM become the lower bound of eigenvalues and the accuracy can be improved via mesh refinement. And NS-FEM solutions (softer) are more reliable, because the large wave number component can be used as an indicator, which is available in FEM (stiffer), on the quality of the numerical solutions. The proposed NS-FEM procedure offers a viable and practical computational means to effectively compute the lower bounds of eigenvalues for solid mechanics problems.

Convergent Power Series for Boundary Value Problems and Eigenproblems with Application to Atmospheric and Oceanic Tides

It is usually impossible to apply power series to solve boundary value and eigenvalue problems because the radius of convergence does not extend to the boundaries. Hough functions, which solve a second order eigenproblem, are always entire functions or can be computed wholly in terms of entire functions. The eigenvalues ϵ are approximated by the zeros of the highest computed power series coefficient, which is always a polynomial in ϵ. The rate of convergence with series truncation N is proportional to exp(-μN) for some positive constant A brief table contains the entire Maple code for the algorithm. The scheme can also be extended to eigenproblems on an unbounded domain as here illustrated by the quantum harmonic oscillator and quartic oscillator. Computer algebra is very useful; the power series method can also be implemented in a stable, purely numerical fashion by applying trigonometric interpolation on a disk in the complex x-plane and Chebyshev interpolation on a real interval in e. Power series are briefly compared with Chebyshev, Fourier, and spherical harmonic spectral methods, which are mostly superior. The success of power series raises an interesting history-of-science issue: Why was there so little progress in solving Laplace's tidal equations in the nineteenth century?

The eigenvalue problem and infinitely many sign-changing solutions for an elliptic equation with critical Hardy constant

By Li–Yau's idea and a more precise Hardy's inequality, we obtain a new lower bound of eigenvalues for an elliptic equation with critical Hardy constant. Then using the lower bounds, we prove that there are infinitely many sign-changing solutions for a class of elliptic equation with critical Hardy constant which extends the results in Schechter and Zou (2005) [9] to the critical Hardy constant case.

Free Response of a Continuous Vibrational System Using Operational Tau Method

In this paper, the general form of the linear vibrational eigenvalue problems with some boundary conditions is considered. The given problem is then formulated by the operational Tau method with arbitrary polynomial bases. Afterward, considering the most common form of the vibrational eigenvalue problems and boundary conditions, some useful relations are derived from the method to solve such problems easily with a rapid rate of convergence. Finally, some numerical samples are solved with Chebyshev, Legendre and Standard basis and compared with the analytical solutions to clarify the efficiency and accuracy of the method in solving continuous vibrational problems.

A highly efficient spectral-Galerkin method based on tensor product for fourth-order Steklov equation with boundary eigenvalue

In this study, a highly efficient spectral-Galerkin method is posed for the fourth-order Steklov equation with boundary eigenvalue. By making use of the spectral theory of compact operators and the error formulas of projective operators, we first obtain the error estimates of approximative eigenvalues and eigenfunctions. Then we build a suitable set of basis functions included in $H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ and establish the matrix model for the discrete spectral-Galerkin scheme by adopting the tensor product. Finally, we use some numerical experiments to verify the correctness of the theoretical results.

Lower Eigenvalue Bounds on Summation for the Solution of the Lyapunov Matrix Differential Equation

AbstractIn this paper, we offer several lower bounds on eigenvalue summation for the solution of the Lyapunov matrix differential equation applying index matrix eigenvalue inequalities and Hölder inequality. Further, we give a numerical example to illustrate the effectiveness of the derived bounds.

Optimal Spectral Schemes Based on Generalized Prolate Spheroidal Wave Functions of Order $$-1$$

We introduce a family of generalized prolate spheroidal wave functions (PSWFs) of order $$-1,$$ and develop new spectral schemes for second-order boundary value problems. Our technique differs from the differentiation approach based on PSWFs of order zero in Kong and Rokhlin (Appl Comput Harmon Anal 33(2):226–260, 2012); in particular, our orthogonal basis can naturally include homogeneous boundary conditions without the re-orthogonalization of Kong and Rokhlin (2012). More notably, it leads to diagonal systems or direct “explicit” solutions to 1D Helmholtz problems in various situations. Using a rule optimally pairing the bandwidth parameter and the number of basis functions as in Kong and Rokhlin (2012), we demonstrate that the new method significantly outperforms the Legendre spectral method in approximating highly oscillatory solutions. We also conduct a rigorous error analysis of this new scheme. The idea and analysis can be extended to generalized PSWFs of negative integer order for higher-order boundary value and eigenvalue problems.

A full multigrid method for eigenvalue problems

In this paper, a full (nested) multigrid scheme is proposed to solve eigenvalue problems. The idea here is to use a correction method to transform the eigenvalue problem solving to a series of corresponding boundary value problem solving and eigenvalue problems defined on a very low-dimensional finite element space. The boundary value problems which are defined on a sequence of multilevel finite element spaces can be solved by some multigrid iteration steps. The computational work of this new scheme can reach the same optimal order as solving the corresponding boundary value problem by the full multigrid method. Therefore, this type of full multigrid method improves the overfull efficiency of the eigenvalue problem solving.

Bounds for eigenvalues of matrix polynomials with applications to scalar polynomials

We first generalize to complex matrix polynomials an improvement of an upper bound by Cauchy on the zeros of complex scalar polynomials. The bound requires the unique positive root of a real scalar polynomial of the same degree as the given complex scalar or matrix polynomial. We then create a recursive procedure to represent a matrix polynomial by another matrix polynomial of larger size, but of lower degree. We apply this procedure to scalar polynomials, and then apply the generalized improved Cauchy bound to their matrix polynomial representation, often obtaining a better bound by solving a real scalar polynomial equation of significantly reduced degree.

A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids

Abstract This paper is a generalization and improvement of some recent results concerned with the lower bound property of eigenvalues produced by the Morley element of the biharmonic operator. Such an extension is of two fold. First, we generalize those results for the biharmonic operator to general fourth order elliptic operators; second, we extend them from quasi-uniform grids to local quasi-uniform grids. In particular, for the general case, we prove the asymptotic lower bound property of the discrete eigenvalues under a very weak saturation condition in terms of local meshsizes which can in general not be improved; for the special case with B = 0, we show a similar result without any assumption except the fineness of the mesh. These results can be regarded as a substantial improvement of a related result on regular meshes, including adaptive local refined meshes, for general fourth order elliptic operators, due to Yang, Li and Bi [http://arxiv.org/abs/1509.00566], while the analysis herein is completely different and largely simpler than that in the above paper.

A new nonconforming mixed finite element scheme for second order eigenvalue problem

A new nonconforming mixed finite element method (MFEM for short) is established for the Laplace eigenvalue problem. Firstly, the optimal order error estimates for both the eigenvalue and eigenpair (the original variable u and the auxiliary variable p→=∇u) are deduced, the lower bound of eigenvalue is estimated simultaneously. Then, by use of the special property of the nonconforming EQ1rot element (the consistency error is of order O(h2) in broken H1-norm, which is one order higher than its interpolation error), the techniques of integral identity and interpolation postprocessing, we derive the superclose and superconvergence results of order O(h2) for u in broken H1-norm and p→ in L2-norm. Furthermore, with the help of asymptotic expansions, the extrapolation solution of order O(h3) for eigenvalue is obtained. Finally, some numerical results are presented to validate our theoretical analysis.

Two-scale sparse finite element approximations

To reduce computational cost, we study some two-scale finite element approximations on sparse grids for elliptic partial differential equations of second order in a general setting. Over any tensor product domain $\Omega\subset\mathbb{R}^d$ with $d= 2,3$, we construct the two-scale finite element approximations for both boundary value and eigenvalue problems by using a Boolean sum of some existing finite element approximations on a coarse grid and some univariate fine grids and hence they are cheaper approximations. As applications, we obtain some new efficient finite element discretizations for the two classes of problem: The new two-scale finite element approximation on a sparse grid not only has the less degrees of freedom but also achieves a good accuracy of approximation.

Recent developments of constructing adjacency matrix in network analysis

Abstract In this paper, we review recent developments in network analysis using the graphtheory, and introduce ongoing research area with relevant theoretical results. In spe-ci c, we introduce basic notations in graph, and conditional and marginal approach inconstructing the adjacency matrix. Also, we introduce the Marcenko-Pastur law, theTracy-Widom law, the white Wishart distribution, and the spiked distribution. Finally,we mention the relationship between degrees and eigenvalues for the detection of hubsin a network.Keywords: Adjacency matrix, conditional dependency, graph theory, marginal depen-dency, Tracy-Widom law. 1. Introduction Recently Big data is one of the most hot issues in many branches of science, and complexnetwork problem is the central issue in Big data. Lots of interest arose in complex networkssuch as the world-wide web, the internet, biological networks, social networks, and so on. Awonderful scienti c fact is that any complex network can be represented by a graph, and anygraph can be represented by an adjacency matrix. Further, degree matrix and/or Laplacianmatrix are derived by the adjacency matrix.Graph theory has been developed by mathematicians for a long time, and mathematiciansare interested in properties of eigenvalues of adjacency matrix, distribution of eigenvalues(Wigner, 1955; Marcenko and Pastur, 1967) and distribution of the largest eigenvalue (Tracyand Widom, 1996), lower and/or upper bound of eigenvalues, relationship between degreeand eigenvalues, and so on. There are numerous books on the graph theory, and one of therecent and best references is Mieghem (2010).On the other hand, statisticians paid attention to the graph theory quite recently, and theyare mainly interested in graphical models and estimation of adjacency matrix using availableobservations. Recently, statisticians are interested in the case where p(number of variables)is larger than n(sample size) because lots of recent networks such as biological networks and

Two-Sided Bounds for Eigenvalues of Differential Operators with Applications to Friedrichs, Poincaré, Trace, and Similar Constants

We present a general numerical method for computing guaranteed two-sided bounds for principal eigenvalues of symmetric linear elliptic differential operators. The approach is based on the Galerkin method, on the method of a priori--a posteriori inequalities, and on a complementarity technique. The two-sided bounds are formulated in a general Hilbert space setting and as a byproduct we prove an abstract inequality of Friedrichs--Poincaré type. The abstract results are then applied to Friedrichs, Poincaré, and trace inequalities and fully computable two-sided bounds on the optimal constants in these inequalities are obtained. Accuracy of the method is illustrated in numerical examples.

Lowest Eigenvalue Bounds for Markov Processes with Obstacles

We derive lower and upper bound for the first eigenvalue of an infinitesimal generator for symmetric cadlag Markov process via an obstacle enlargement method. Several applications, including α-stable processes on a torus in ℝ d , are presented.

The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension seven

A version of Lichnerowicz’ theorem giving a lower bound of the eigenvalues of the sub-Laplacian on a compact seven dimensional quaternionic contact manifold is proved assuming a lower bound on the Sp(1)Sp(1)-components of the qc-Ricci curvature and the positivity of the P-function of any eigenfunction. The introduced P-function and nonlinearC-operator are motivated by the Paneitz operators defined previously in the Riemannian and CR settings and the P-function used in the theory of elliptic partial differential equations. It is shown that in the case of a seven dimensional compact 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is the round 3-Sasakian sphere.

Interplay between discretization and algebraic computation in adaptive numerical solutionof elliptic PDE problems

AbstractThe Adaptive Finite Element Method (AFEM) for approximating solutions of PDE boundary value and eigenvalue problems is a numerical scheme that automatically and iteratively adapts the finite element space until a sufficiently accurate approximate solution is found. The adaptation process is based on a posteriori error estimators, and at each step of this process an algebraic problem (linear or nonlinear algebraic system or eigenvalue problem) has to be solved. In practical computations the solution of the algebraic problem cannot be obtained exactly. As a consequence, the algebraic error should be incorporated in the context of the AFEM and its a posteriori error estimators. The goal of this paper is to survey some existing approaches in the AFEM context that consider the interplay between the finite element discretization and the algebraic computation. We believe that a better understanding of this interplay is of great importance for the future development in the area of numerically solving large‐scale real‐world motivated problems. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold

The main technical result of the paper is a Bochner type formula for the sub-Laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz’s theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(n)Sp(1) components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a priori estimates for square integrals of horizontal derivatives of smooth compactly supported functions. As an application, we prove a sharp inequality bounding the horizontal Hessian of a function by its sub-Laplacian on the quaternionic Heisenberg group.

New Upper and Lower Eigenvalue Bounds for the Solution of the Continuous AlgebraicRiccati Equation

AbstractIn this paper, applying majorization inequalities, new upper and lower bounds for summation of eigenvalues (including the trace) of the solution of the continuous algebraicRiccati equation are presented. Corresponding numerical examples illustrate that our new bounds extend some of the recent results.

Lyapunov-type inequalities for certain higher order differential equations with anti-periodic boundary conditions

In this work, we will establish several new Lyapunov-type inequalities for certain half-linear higher order differential equations with anti-periodic boundary conditions. A lower bound of eigenvalues will be also given.