Solution Of First-kind Integral Equations Research Articles

Linear inverse problems in wave motion: nonsymmetric first-kind integral equations

We present a general framework to study the solution of first-kind integral equations. The integral operator is assumed to be compact and nonself-adjoint and the integral equation can possess a nonempty null space. An approach is presented for adding contributions from the null space to the minimum-energy solution of the integral equation through the introduction of weighted Hilbert spaces. Stability, accuracy, and nonuniqueness of the solution are discussed through the use of model resolution, data fit, and model covariance operators. The application of this study is to inverse problems that exhibit nonuniqueness.

Comments on "On solving first-kind integral equation using wavelets on a bounded interval" [with reply

The author comments that the paper of Goswami, Chan and Chui (see ibid., vol.43, no.6, p.614, 1995) presented an interesting application of semi-orthogonal wavelets on a bounded interval to a numerical solution of first-kind integral equations. The major merit of the wavelet-based methods is to reduce an integral operator into a sparse matrix which is extremely valuable for large-scale problems. Compared with the discussion in the theoretical portion of the paper, the explanation of the numerical results is somewhat short and insufficient. In particular, examples for the demonstration of the sparse matrix lack insight and conviction. Goswami et al. reply that apparently Dr. Wang has not understood the main objective of their paper. The contribution of the paper should be seen not through one specific example, but rather in its totality. The method of moments (MoM) is well known and so is the fact that for a specific example discussed in our paper, namely TM scattering from an infinitely long PEC circular cylinder with small radius, 11 (or even less) basis functions will be sufficient for an accurate representation of the current distribution. The main purpose of our paper is to present wavelet MoM to the electromagnetic community in a simplified way so that readers can apply this technique to their problems which may be more interesting and challenging than the one we discussed. For this purpose, we provided explicit closed-form expressions for scaling functions and wavelets which, to the best of our knowledge, have not appeared anywhere in the literature.

Statistical approach to the solution of first-kind integral equations arising in the study of materials and their properties

There are a number of problems arising when studying the properties of materials, which require for their solution the inversion of a Fredholm first-kind integral equation. Examples include the determination of the distribution of adsorption energies on the surface of a solid and the evaluation of the distribution of pore radii of a solid from diffusion data. Such equations are, in practice, notoriously difficult to solve. This paper describes a general methodology for solving equations of this type. The method combines the ideas of regularization with a quadratic programming algorithm for minimizing quadratic expressions subject to non-negativity constraints. The condition of non-negativity is essential if we are to recover distribution functions for physical attributes of a solid. The method proposed is tested on simulated data for which the true solution to the equation is already known and on real data arising in each of the two situations described above. The method is shown to perform well in recovering the true solution for the simulated data and to produce results in the real data situations that are consistent with the data observed and with observations of related physical quantities.

Confidence Intervals for Discrete Approximations to Ill-Posed Problems

Abstract We consider the linear model Y = Xβ + e that is obtained by discretizing a system of first-kind integral equations describing a set of physical measurements. The n vector β represents the desired quantities, the m x n matrix X represents the instrument response functions, and the m vector Y contains the measurements actually obtained. These measurements are corrupted by random measuring errors e drawn from a distribution with zero mean vector and known variance matrix. Solution of first-kind integral equations is an ill-posed problem, so the least squares solution for the above model is a highly unstable function of the measurements, and the classical confidence intervals for the solution are too wide to be useful. The solution can often be stabilized by imposing physically motivated nonnegativity constraints. In a previous article (O'Leary and Rust 1986) we developed a method for computing sets of nonnegatively constrained simultaneous confidence intervals. In this article we briefly review the ...

The solution of two wave-diffraction problems

A method for obtaining the numerical solution of first-kind integral equations with the Hankel-function kernel H(1)0(k|xt|) is described in relation to two water-wave diffraction problems. The principal feature is the implementation of a new technique for transforming the given equations into second-kind integral equations, which have continuous kernels and from which numerical approximations can readily be determined.

On the approximate solution of first-kind integral equations of Volterra type

The present paper deals with the approximate solution of integral equations of the first kind, ( Open image in new window y)x∈I:=[a, b], where Open image in new window denotes a (linear) integral operator of Volterra (or Abel) type, and whereg∈C(I), withg(a)=0. The given functiong is approximated uniformly onI (or on a finite subsetZ⊂I by using certain weak Chebyshev systems onI which are obtained in a natural way. By the linearity of Open image in new window this yields an approximation to the exact solutiony onI. Questions of uniqueness and characterization of such approximating functions, as well as numerical aspects of the approximation problem are discussed.