Compact Integral Operator Research Articles

Maximum Smoothed Likelihood Density Estimation for Inverse Problems

We consider the problem of estimating a pdf $f$ from samples $X_1, X_2, \ldots, X_n$ of a random variable with pdf $\mathscr{K}f$, where $\mathscr{K}$ is a compact integral operator. We employ a maximum smoothed likelihood formalism inspired by a nonlinearly smoothed version of the EMS algorithm of Silverman, Jones, Wilson and Nychka. We show that this nonlinearly smoothed algorithm is itself an EM algorithm, which helps explain the strong convergence properties of the algorithm. For the case of (standard) density estimation, that is, the case where $\mathscr{K}$ is the identity, the method yields the standard kernel density estimators. The maximum smoothed likelihood density estimation technique is a regularization technique. We prove an inequality which implies the stability and convergence of the regularization method for the large sample asymptotic problem. Under minimal assumptions it also implies the a.s. convergence of the finite sample density estimate via a uniform version of the strong law of large numbers. Under extra regularity conditions we get a.s. convergence rates via a uniform version of the law of the iterated logarithm (under stronger conditions than usual).

A numerical quadrature for some weakly singular integral operators

We study the behaviour of a class of weakly singular compact integral operators which is approximated by a collectively compact sequence of finite-rank operators defined by truncation followed by a numerical quadrature. Numerical experiments are done on spectral computations.

Analysis of signal optimization problems for dielectric media

The authors have recently presented the results of the numerical implementation of a scheme for designing an incident electromagnetic plane wave which will deposit a maximum amount of energy in any selected portion of an inhomogeneous dielectric slab. This paper presents the mathematical analysis underlying this scheme for solving the signal design problem. The analysis indicates that the maximum energy is achieved by choosing the incident signal to be an appropriately normalized eigenfunction of a compact Fredholm integral operator. A numerical scheme for solving the optimization problem is discussed briefly.

Spectral inequalities for compact integral operators on Banach function spaces

AbstractThis article generalizes some spectral inequalities for non-negative matrices (see [2] and [3]) to compact integral operators with non-negative kernels defined on Banach function spaces. The spectral radius of a sum of such operators is estimated under certain conditions and a generalization of this inequality is given. An inequality for the spectral radius of a compact integral operator with the kernel equal to a weighted geometric mean of non-negative kernels is also proved.

Sampled-data control of continuous-time systems with an H∞ optimality criterion

The synthesis of sampled-data controllers for continuous-time systems is studied in the context of H ∞ control theory. The sampled-data H ∞ optimal control problem consists of finding a sampled-data controller which minimizes the L 2-induced norm from the continuous-time disturbance to the continuous-time output. The problem is here solved via a discrete system representation. This is achieved by describing the closed-loop system as a discrete system with finite state dimension, and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite-dimensional spaces. Via finite-rank approximations of a compact integral operator associated with the system, the sampled-data H ∞ control problem can be solved to any degree of accuracy. From the structure of the problem it follows that irrespective of the order of the approximation, a feasible sampled-data controller can be found which has the same order as the system.

Hammerstein Integral Equations with Nontrivial Solutions

By means of critical point theory, existence theorems for nontrivial solutions to the Hammerstein equation x = KFx are given, where K is a compact linear integral operator and F is a nonlinear superposition operator. To this end, appropriate conditions on the spectrum of the linear parte are combined with growth and representation conditions on the nonlinear part to ensure the applicability of the mountain — pass lemma. The abstract existence theorems are applied to nonlinear elliptic equations and systems subject to Dirichlet boundary conditions.

Rayleigh-schrödinger series for defective spectral elements of compact operators in banach spaces

Numerical computation of Rayleigh-Schrodinger Series for maximal invariant sub-spaces is done for compact integral operators with defective eigenvalues. The series are applied to refine iteratively an approximate starting basis. The integral operators are discretized by different approximation methods which are known to be a strongly stable approximations at any nonzero eigenvalue. Rayleigh-Schrodinger Series are compared with three inexact Newton methods that perform the same goal.

Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy

In this paper the authors propose a numerical method for the computation of the singular system of certain compact integral operators related to the problem of data inversion in confocal microscopy. The method is based on the sampling theorem for functions whose Hankel transform has a bounded support. In this way the original integral equation is transformed into an infinite-dimensional linear system for the values of the unknown function at the sampling point. The singular values of the infinite-dimensional matrix coincide with the singular values of the integral operator. Approximations of the singular systems are obtained by considering finite sections of the infinite-dimensional matrix.

A note on strong stability of finite element approximations

This work deals with the strong stability of finite element approximations in a neighborhood of each isolated eigenvalue of finite multiplicity. We prove the equivalence between a finite element approximation of the variational formulation of a differential eigenelements problem and a projection method on the inverse problem. Conversely, some discretizations of the inverse problem, such as the Sloan approximation of compact integral operators, are shown to be equivalent to a certain finite element method on the differential problem.

Spherical wave operators and the translation formulas

Translational formulas for both scalar and vector spherical wave solutions of the Helmholtz equation are developed in a straightforward manner using differential operator representations for the modal functions and well-known expressions for the scalar and dyadic free-space Green's functions. The expansion coefficients are given in compact integral or differential operator forms useful for analytic investigation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Is the iterative refinement of eigenelements an expensive technique?

Iterative refinement techniques to improve the accuracy of approximate eigenelements of compact integral operators appeared as an alternative to discretization methods to overcome the problem of having to solve large full eigenvalue problems, but other approaches are sometimes possible such as Padé techniques. Here we compare the applicability conditions of these methods and their number of operations for some cases where both approaches are possible.

Identification and Control of Linear Distributed Parameter Systems Through the Use of Experimentally Determined Singular Functions

A new method is presented for the identification of linear Distributed Parameter Systems (DPS) given only limited a priori information about the physical process. Knowledge of the locations of a finite number of spatially varying control actuators and the locations of point measurements is assumed. Basic information from the system inputs and outputs is used to identify approximate spatial characteristics of the DPS. The identification procedure results in a pseudo-modal input/output model for the system, Identification of the spatial character of the process is accomplished by assuming the general model of a linear time-invariant compact unsymmetric integral operator. The kernel of this integral operator is expressed in terms of two sets of orthonormal basis functions, known as singular functions, whose existence is given by singular value theory. Approximations to these singular functions are determined numerically from input/output data using Galerkin's method and the Singular Value Decomposition. Application of this procedure to DPS described by both a self-adjoint and a non-self-adjoint parabolic PDE model is demonstrated. The usefulness of the identified model for an example control system design is shown

Molecular weight distributions from measurements of polymolecular property functions

A general method for the calculation of molecular weight distributions from measurements of polymolecular property functions is given. It is based on the Hilbert-Schmidt theory of compact integral operators. Two variation problems are described, both taking the error of measurement into consideration. The method is applied to light scattering on Gaussian polymer coils (theta state) and to elution curves of Phase Distribution Chromatography (PDC).

The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators

Let K be an eventually compact linear integral operator on L p(Ω, μ), 1 ⩽ p < ∞, with nonnegative kernel k( x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when p = 1. This work extends the previous analysis of the author who characterized the distinguished eigenvalues of K and K ∗, and the support sets for the eigenfunctions and generalized eigenfunctions belonging to the spectral radius of K or K ∗. The characterizations of the support sets for the algebraic eigenspaces of K or K ∗ are phrased in terms of significant k-components which are maximal irreducible subsets of Ω and which yield a positive spectral radius for the integral operator defined by the restriction of k( x, y) to the Cartesian product of such sets. In this paper, we show that a basis for the functions, constituting the algebraic eigenspaces of K and K ∗ belonging to the spectral radius of K, can be chosen to consist of elements which are positive on their sets of support, except possibly on sets of measure less than some arbitrarily specified positive number. In addition, we present necessary and sufficient conditions, in terms of the significant k-components, for both K and K ∗ to possess a positive eigenfunction (a.e. μ) corresponding to the spectral radius, as well as necessary and sufficient conditions for the sequence ∥ γ n K n g ∥ p to converge whenever g⩾ 0, where ∥ − ∥ p denotes the norm in L p(Ω, μ) , and γ 1 the smallest (in modulus) characteristic value of K. This analysis is made possible by introducing the concepts of chains, lengths of chains, height, and depth of a significant k-component as was done by U. Rothblum [ Lin. Alg. Appl. 12 (1975), 281–292] for the matrix setting.

On linear integral operators with nonnegative kernels

Let K be an eventually compact linear integral operator on L p(Ω, μ), 1 ⩽ p < ∞ , with nonnegative kernel k( x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when p = 1. In considering the equation λf = K f + g for given nonnegative g ϵ L p(Ω, μ), λ > 0 , P. Nelson, Jr. provided necessary and sufficient conditions, in terms of the support of g, such that a nonnegative solution f ϵ L p(Ω, μ) was attained. Such conditions led to generalizing some of the graph-theoretic ideas associated with the normal form of a nonnegative reducible matrix. The purpose of this paper is to show that the analysis by Nelson can be enlarged to provide a more complete generalization of the normal form of a nonnegative matrix which can be used to characterize the distinguished eigenvalues of K and K ∗, and to describe sets of support for the eigenfunctions and generalized eigenfunctions of both K and K ∗ belonging to the spectral radius of K.

The Cauchy and the Szegö kernels on multiply connected regions

LetD be a bounded plane region whose boundary ∂D is Dini-smooth. An operatorB:L 2(∂D)→L 2(∂D) that has been considered by Kerzman and Stein is investigated. This operator is a compact self-adjoint integral operator whose kernel β (z, ζ) is bounded on ∂Dx∂D and has a geometric description involving chords inD. With the aid ofB the Szego kernel can be expressed in terms of the Cauchy kernel. Here, the operatorB is recovered from the classical theory of kernel functions resulting in extension of the work of Kerzman and Stein. In particular, the eigenvalue problem associated with the operatorB is studied and its complete equivalence with a very classical problem due to Bergman, Schiffer and Singh is established.

Spectral approximation for compact integral operators by degenerate kernel methods

We establish a new improved error estimate for the solution of the integral equation eigenvalue problem by degenerate kernel methods. In [6] these estimates were proved under the assumption of normality of the original kernel as well as of the approximating degenerate kernel. Now we consider any compact integral operator and a general Banach space situation, in contrast to the Hilbert space setting in [6], This will be done by combining the techniques in [6] with the suitably transformed estimates of [5]. Our results show that degenerate kernel methods have, besides their overall property of furnishing easy approximations to eigenfunctions, for eigenvalues an order of convergence comparable to quadrature methods.

On the compactness of certain integral operators

This paper develops practical methods for deciding whether a given kernel function induces a compact integral operator from certain spaces of functions, defined on a compact subset Ω of R n , into the space of continuous functions over Ω. Necessary and sufficient conditions for compactness are introduced, and several tests for deciding if these conditions are satisfied are developed. The paper concludes with an illustration of the practical use of the theory.

Numerical Computation of the Eigenelements of Linear Integral Operators by Iterations

We are concerned with the numerical solution of the eigenvalue problem $T\varphi = \lambda \varphi ,\varphi \ne 0$, where T is a linear compact integral operator in a Banach space. We present in a unifying manner most of practical numerical methods, i.e. projections and approximate quadratures methods, with a special interest in convergence rates and a posteriors error bounds.The linear operator T and its approximation $T_n$ are defined in the same space. Perturbation theory is then a suitable framework for our problem. It enables us to produce an algorithm which computes $\lambda $ and $\varphi $ by iterations, starting from the eigenelements of $T_n$. The computations require only matrix calculations of a fixed order, which can be kept small relative to the resulting accuracy.The application of the algorithm for projection methods is presented with an emphasis on the Galerkin method and its Sloan variant. The efficiency of the algorithm is illustrated for integral operators, approximated by the projecti...

One dimensional perturbations of compact operators

Let K denote a compact selfadjoint operator acting on a Hilbert space H. L denotes a one dimensional selfadjoint operator also acting on H. It is shown that the eigenvalues of K and K+L interlace on the real axis. The purpose of this article is to prove the following theorem. THEOREM. Let K denote a compact, selfadjoint operator acting on a Hilbert space H. We assume that the nullspace of K is empty. Let L denote a one dimensional, selfadjoint operator, also acting on H. Between every pair of distinct, successive eigenvalues (ti, )i+l) of K there is precisely one eigenvalue of K+L in one of the intervals [2i, Rij?) or (4i, )i+l] or (4 R i+?). Every eigenvalue of multiplicity n> 1 of K is also an eigenvalue of K+L, of multiplicity n or n-1. The above theorem may be viewed as a generalization of certain theorems regarding second order differential equations. Consider, for example, (1) y' + q(x))y = 0 where q(x) is real and continuous on [0, 1], subject to the boundary conditions (a) y(O)=O, y(l)=0, (b) y(O)=0, y'(1)=O. The eigenvalues of (1)(a) and (1)(b) are real and alternate on the real axis. This fact is known. It is however an immediate consequence of the above theorem. Both problems (1)(a) and (1)(b) can be investigated by converting the differential operators into compact integral operators. The latter differ by a one dimensional operator so that the above theorem can be invoked. Received by the editors June 19, 1972. AMS (MOS) subject classifications (1970). Primary 47A55, 47B05; Secondary 34B25. 1 This research was supported by the National Science Foundation under grant GP 27960. ? American Mathematical Society 1973