Eigenvalues Of Integral Operator Research Articles

Approximation tools and decay rates for eigenvalues of integral operators on a general setting

We provide the characterization of the Peetre-type K-functional on a compact two-point homogeneous space in terms of the rate of approximation of a family of multipliers operators. This extends the well known results on the spherical setting. The characterization is employed to prove that an abstract Holder condition or finite order of differentiability assumption on generating positive kernels of integral operators implies a sharp decay rates for their eigenvalues sequences. Consequently, sharp upper bounds for the Kolmogorov n-width of unit balls in reproducing kernel Hilbert space are obtained.

Generalization Analysis for Ranking Using Integral Operator

The study on generalization performance of ranking algorithms is one of the fundamental issues in ranking learning theory. Although several generalization bounds have been proposed based on different measures, the convergence rates of the existing bounds are usually at most O(√1/n), where n is the size of data set. In this paper, we derive novel generalization bounds for the regularized ranking in reproducing kernel Hilbert space via integral operator of kernel function. We prove that the rates of our bounds are much faster than (√1/n). Specifically, we first introduce a notion of local Rademacher complexity for ranking, called local ranking Rademacher complexity, which is used to measure the complexity of the space of loss functions of the ranking. Then, we use the local ranking Rademacher complexity to obtain a basic generalization bound. Finally, we establish the relationship between the local Rademacher complexity and the eigenvalues of integral operator, and further derive sharp generalization bounds of faster convergence rate.

A Note on Asymptotic Expansions to Approximate Eigenvalues of Integral Operators with Green's Kernels using the Iterated Galerkin Method

In this article, we consider approximation of eigenvalues of integral operators with Green's function-type kernels using the iterated Galerkin method. We obtain asymptotic expansions for approximate eigenvalues. The Richardson extrapolation is used to obtain eigenvalue approximations of higher order. A numerical example is considered in order to illustrate our theoretical results.

Sharp estimates for eigenvalues of integral operators generated by dot product kernels on the sphere

We obtain explicit formulas for the eigenvalues of integral operators generated by continuous dot product kernels defined on the sphere via the usual gamma function. Using them, we present both, a procedure to describe sharp bounds for the eigenvalues and their asymptotic behavior near 0. We illustrate our results with examples, among them the integral operator generated by a Gaussian kernel. Finally, we sketch complex versions of our results to cover the cases when the sphere sits in a Hermitian space.

Harnessing GPU Power from High-level Libraries: Eigenvalues of Integral Operators with SLEPc

We consider the approximation of eigenpairs of large-scale matrices arising from the discretization of integral operators with weakly singular function kernels. Efficient and fast solvers to numerically approximate the sought eigenpairs are required. For this, we would like to exploit the computational power of modern graphical processing units (GPUs), and we are interested in doing this from high-level libraries. We show how to use the CUDA add-on in the SLEPc/PETSc libraries to tackle this problem and illustrate our results on a radiative transfer problem in astrophysics. The CUDA-accelerated codes achieve considerable speedups versus the CPU counterparts on the same problem.

Asymptotic Expansions for Approximate Eigenvalues of Integral Operators with Nonsmooth Kernels

We consider approximation of eigenvalues of integral operators with Green's function kernels using the Nyström method and the iterated collocation method and obtain asymptotic expansions for approximate eigenvalues. We show that the Richardson extrapolation is applicable to find eigenvalue approximations of higher order and illustrate our results by numerical examples.

Old and New on the Laplace-Beltrami Derivative

The (strong) Laplace-Beltrami derivative and related operators appear quite frequently in many problems involving Fourier series, spherical harmonics, approximation of functions and smoothness. This article intends to provide an overview with updates on such operators. While most of the results concerning the Laplace-Beltrami derivative found in the literature are stated in the three-dimensional setting, our approach includes higher-dimensional spheres. We present proofs for results which are labeled as known but are just mentioned elsewhere. Spherical moduli of smoothness and decay rates for eigenvalues of integral operators are also among the covered topics.

Eigenvalues of integral operators with positive definite kernels satisfying integrated Hölder conditions over metric compacta

We determine the asymptotic eigenvalue behaviour of integral operators generated by positive definite kernels satisfying an integrated Hölder condition on metric compacta. We also show that this behaviour is the best possible.

Approximation numbers and approximation of the eigenvalues of integral operators

T h e r e f o r e , f o r t h e a p p r o x i m a t i o n o f t h e e i g e n v a l u e s o f H 6 G (X) i t i s n a t u r a l t o u s e t h e eigenvalues of the finite-dimensional operators that approximate H quite well in the operator norm. As we know, the value of the best approximation of a continuous linear operator H, acting from a normal linear space Y into a normed linear space Z, by finite-dimensional operators of rank N in the operator norm is the approximation number [3, p. 146]

Removal of the zero eigenvalues of integral operators in elastostatic boundary value problems

Discretization of various integral equations of the second kind in elastostatic boundary value problems leads to singular or almost singular systems of algebraic equations, i.e. the determinants of the coefficient matrices vanish or are extremely small. The reason for this lies in the fact that the integral operators have isolated zero eigenvalues. In this paper a method of removing the zero eigenvalues is developed which is based mainly on mechanical aspects.

An algebraic system of equations related to the eigenvalues of integral operators with difference kernels

An algebraic system of equations related to the eigenvalues of integral operators with difference kernels

Relation between Chandrasekhar's X-function and the eigenvalues of integral operators with difference kernels

Relation between Chandrasekhar's X-function and the eigenvalues of integral operators with difference kernels

On the eigenvalues of integral operators with difference kernels

On the eigenvalues of integral operators with difference kernels

Asymptotic behaviour of the eigenfunctions and eigenvalues of integral operators with δ-type kernels

Asymptotic behaviour of the eigenfunctions and eigenvalues of integral operators with δ-type kernels