Theory Of Compact Operators Research Articles

Finite element analysis for the Navier-Lamé eigenvalue problem

The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree k≥1 is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.

Idempotent vector spaces and their linear transformations

Abstract This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n × n n\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.

Spectral approximation and error analysis for the transmission eigenvalue problem with an isotropic inhomogeneous medium

In this paper, we propose an effective Legendre-Fourier spectral method for the transmission eigenvalue problem in polar geometry with an isotropic inhomogeneous medium. The basic idea of this methodology is to rewrite the initial problem into its equivalent form by using polar coordinates and some specially designed polar conditions. A variational method and its discrete version (i.e., Legendre-Fourier spectral method) are then presented within a class of weighted Sobolev spaces. With the help of the spectral theory of compact operators and the approximation properties of some specially designed projections in non-uniformly weighted Sobolev spaces, error estimates with spectral accuracy of the Legendre-Fourier spectral method for both the eigenvalue and eigenfunction approximations are established. Numerical experiments are presented to confirm the theoretical findings and the efficiency of our algorithm.

Numerical algorithm with fifth‐order accuracy for axisymmetric Laplace equation with linear boundary value problem

AbstractIn order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Green's formula. Secondly, A quadrature method is constructed by combing the mid‐rectangle formula with a singular integral formula to solve the integral equation, which has the accuracy of and low computational complexity. Thirdly, the convergence of the numerical solutions is proved based on the theory of compact operators and the single parameter asymptotic expansion of errors with odd power is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. After one extrapolation, the accuracy of the numerical solutions can reach the accuracy of . Finally, two numerical examples are presented to demonstrate the efficiency of the method.

An Efficient Finite Element Method and Error Analysis Based on Dimension Reduction Scheme for the Fourth-Order Elliptic Eigenvalue Problems in a Circular Domain

In this paper, we propose an effective finite element method for the fourth order elliptic eigenvalue problems in a circular domain. First, by using polar coordinates transformation and the orthogonal property of Fourier basis functions, the original problem is turned into a series of equivalent one-dimensional eigenvalue problems. Second, according to the properties of Laplace operator in polar coordinate, we deduce the polar conditions and introduce suitable weighted Sobolev space. Based on the polar conditions and weighted Sobolev space, we establish the weak form and the corresponding discrete form. Third, we prove the error estimates of approximation eigenvalues and eigenvectors by means of the spectral theory of compact operators for each one-dimensional eigenvalue system. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results.

Virtual element method for the modified transmission eigenvalue problem in inverse scattering theory

In this paper, we present the virtual element approximation for the modified transmission eigenvalue problem. The eigenvalue problem arises in inverse scattering theory for artificial inhomogeneous media characterized as a metamaterial. By the shifting argument and spectral approximation theory of compact operators, we prove the correct spectral approximation of the virtual element scheme, and derive the optimal order error estimates for the eigenfunctions and the double order for the eigenvalues. Finally, we implement numerical experiments to illustrate behaviors of the standard and serendipity virtual element schemes, and the finite element scheme.

Collocation method for solving two-dimensional nonlinear Volterra–Fredholm integral equations with convergence analysis

This paper presents a method for solving the two-dimensional nonlinear Volterra–Fredholm integral equation. The main idea of the method is to use the Lagrange interpolation function to approximate the unknown solution and the Legendre–Gauss quadrature formula to approximate the integral. The advantage of the method is that it requires relatively few collocation points to obtain a relatively small error and does not require the calculation of integrals. Under certain sufficient conditions, the existence and uniqueness of the original equation are given. In addition, the existence and uniqueness of the solutions of the discrete equations are given using the theory of compact operators. The convergence analysis and error estimates of the method are also derived. Finally, several numerical examples are used to demonstrate its efficiency and accuracy.

Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

Abstract This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear Riemann–Liouville fractional differential equations with mixed boundary value conditions. An exhaustive study of the sign of the related Green’s function is carried out. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed point theory of compact operators defined in suitable cones, it is proved that there exists at least one solution of the considered problem. Moreover, the method of lower and upper solutions is developed and the existence of solutions is deduced by a combination of both techniques. In particular cases, the Banach contraction principle is used to ensure the uniqueness of solutions.

Existence of periodic solutions and bifurcation points for generalized ordinary differential equations

The generalized ordinary differential equations (shortly GODEs), introduced by J. Kurzweil in 1957, encompass other types of equations. The first main result of this paper extends to GODEs some classical conditions on the existence of a periodic solution of a nonautonomous ODE. By means of the correspondence between impulse differential equations (shortly IDEs) and GODEs, we translate the result to IDEs. Instead of the classical hypotheses that the functions on the righthand side of an IDE are piecewise continuous, it is enough to require that they are integrable in the sense of Lebesgue, allowing such functions to have many discontinuities. Our second main result provides conditions for the existence of a bifurcation point with respect to the trivial solution of a periodic boundary value problem for a GODE depending upon a parameter, and, again, we apply such result to IDEs. The machinery employed to obtain the main results are the topological degree theory, tools from the theory of compact operators and an Arzelà-Ascoli-type theorem for regulated functions.

Exact Boundary Synchronization by Groups for a Coupled System of Wave Equations with Coupled Robin Boundary Controls on a General Bounded Domain

In this paper, by means of a suitable projection of the system, we introduce the concept of a presynchronized system and propose a new method based on Fredholm's theory of compact operators to char...

An efficient spectral-Galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries

In this paper, we put forward an efficient spectral-Galerkin approximation in view of dimension reduction scheme for transmission eigenvalue problem in polar geometries. Firstly, we turn the original problem into an equivalent fourth order nonlinear eigenvalue problem. Then the fourth order nonlinear eigenvalue problem is transformed into a coupled fourth order linear eigenvalue system by introducing an auxiliary Poisson equation. Secondly, based on polar coordinate transformation, we further reduce the coupled fourth order linear eigenvalue system to a series of equivalent one-dimensional eigenvalue systems. Thirdly, we derive the essential polar condition and introduce the appropriate weighted Sobolev space according to the polar condition, and establish the weak form and the corresponding discrete form. In addition, by utilizing spectral theory of compact operators, we prove the error estimates of approximation eigenvalues and eigenvectors for each one-dimensional eigenvalue system. Finally, we provide ample numerical experiments, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.

Discontinuous Galerkin methods of the non-selfadjoint Steklov eigenvalue problem in inverse scattering

In this paper, we apply discontinuous Galerkin methods to the non-selfadjoint Steklov eigenvalue problem arising in inverse scattering. The variational formulation of the problem is non-selfadjoint and does not satisfy H1-elliptic condition. By using the spectral approximation theory of compact operators, we prove the spectral approximation and optimal convergence order for the eigenvalues. Finally, some numerical experiments are reported to show that the proposed numerical schemes are efficient for real and complex Steklov eigenvalues.

A virtual element method for the Laplacian eigenvalue problem in mixed form

In this paper, the virtual element method for the approximation of Laplacian eigenvalue problem in mixed form is studied. We show that the discrete form satisfies the hypotheses required by the Brezzi-Babǔska theory. Under some assumptions on polygonal meshes, we employ the spectral theory of compact operators to prove the spectral approximation and the optimal order for the eigenvalues. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order.

The modified quadrature method for Laplace equation with nonlinear boundary conditions

Here, the numerical solutions for Laplace equation with nonlinear boundary conditions is studied. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equation. The modified quadrature method is presented for solving the nonlinear equation, which possesses high accuracy order $O(h^3)$ and low computing complexities. A nonlinear system is obtained by discretizing the nonlinear equation and the convergence of numerical solutions is proved by the theory of compact operators. Moreover, in order to solve the nonlinear system, the Newton iteration is provided by using the Ostrowski fixed point theorem. Finally, numerical examples support the theoretical results.

A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem

The goal of this paper is to develop numerical methods computing a few smallest elastic interior transmission eigenvalues, which are of practical importance in inverse elastic scattering theory. The problem is challenging since it is nonlinear, nonselfadjoint, and of fourth order. In this paper, we construct a lowest order mixed finite element method which is close to the Ciarlet-Raviart mixed finite element method. This scheme is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. Due to the nonselfadjointness, the discretization of elastic transmission eigenvalue problem leads to a non-classical mixed method which does not fit into the framework of classical theoretical analysis. In stead, we obtain the convergence analysis based on the spectral approximation theory of compact operators. Numerical examples are presented to verify the theory. Both real and complex eigenvalues can be obtained.

An efficient spectral Galerkin approximation to a Helmholtz transmission eigenvalue problem in spherical geometries

In this paper, we present an efficient spectral method based on Legendre-Galerkin approximation for the Helmholtz transmission eigenvalue problem in spherical geometries. By means of the spherical coordinate transformation and spherical harmonic expansion, we decompose the original problem into a sequence of equivalent one-dimensional generalized eigenvalue problems. Then we establish corresponding weak forms and discrete schemes for each one-dimensional generalized eigenvalue problem. Especially for the case of solid spherical domain, we need to deal with singularities introduced by spherical coordinate transformation. In order to overcome the difficulty, we derive the pole conditions and introduce the weighted Sobolev space according to pole conditions. The corresponding weak forms and discrete schemes are also established according to the weighted Sobolev space. In addition, we prove the error estimates of eigenvalues and eigenfunctions by using the spectral theory of compact operators for each one-dimensional generalized eigenvalue problems.We also provide some numerical experiments to demonstrate the validity of the algorithm and the correctness of the theoretical results.

A mixed Legendre-Galerkin spectral method for the buckling problem of simply supported Kirchhoff plates

In this paper, we develop a mixed Legendre-Galerkin spectral method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. By the spectral theory of compact operators, the rigorous error estimates for the approximate eigenvalues and eigenfunctions are provided. Finally, we present some numerical experiments which support our theoretical results.

Spectral Theory for Random Poincaré Maps

We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting $N$ asymptotically stable periodic orbits. We construct a discrete-time, continuous-space Markov chain, called a random Poincaré map, which encodes the metastable behavior of the system. We show that this process admits exactly $N$ eigenvalues which are exponentially close to $1$, and we provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighborhoods of periodic orbits. The eigenvalues and eigenfunctions are well-approximated by principal eigenvalues and quasi-stationary distributions of processes killed upon hitting some of these neighborhoods. The proofs rely on Feynman--Kac-type representation formulas for eigenfunctions, Doob's $h$-transform, spectral theory of compact operators, and a recently discovered detailed-balance property satisfied by committor functions.

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.

Kernel canonical correlation analysis via gradient descent

Kernel canonical correlation analysis (CCA) is a powerful statistical tool characterizing nonlinear relations between two sets of multidimensional variables. It has been widely used in many branches of science and technology, e.g. bioinfomatics, multi-media information retrieval, cross-language document retrieval, fMRI (functional magnetic resonance imaging). Previous algorithms focus on sparsity analysis of kernel CCA. In this paper, from another viewpoint, we address a new gradient descent kernel CCA algorithm, which is based on the relation between kernel CCA and linear systems of equations. Meanwhile, stability analysis of the algorithm is addressed by means of suitable error decomposition formula and compact operator theory. Theoretical analysis is elegantly investigated in terms of choices of regularization parameter and step size. Experimental results on real-world datasets demonstrate the effectiveness of the algorithm for content-based image retrieval task. The results indicate that the proposed algorithm is stable and the performance is comparable with several state-of-the-art CCA algorithms.