On the quantum and randomized approximation of linear functionals on function spaces

A modified approach to numerical solution of Fredholm integral equations of the second kind

An adaptive method for the numerical solution of Fredholm integral equations of the second kind. I. regular kernels

Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets

article Free Access Share on On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature Author: S. Twomey U. S. Weather Bureau, Washington, D. C. U. S. Weather Bureau, Washington, D. C.View Profile Authors Info & Claims Journal of the ACMVolume 10Issue 1Jan. 1963 pp 97–101https://doi.org/10.1145/321150.321157Published:01 January 1963Publication History 721citation2,290DownloadsMetricsTotal Citations721Total Downloads2,290Last 12 Months190Last 6 weeks25 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

This paper provides a new representation formula for the solution of Fredholm integral equations of the second kind. The formula has certain computational advantages over the usual representation involving the resolvent or the representation formula of Krein. As an application, the Bellman-Krein partial differential equation for the resolvent is derived.

The use of Sherman–Morrison formula in the solution of Fredholm integral equation of second kind

Closed-form algebraic formulae are derived for the local error incurred in the numerical solution of integral equations by iterated collocation methods, the analysis being illustrated by application to Fredholm integral equations of the second kind. The novel error analysis uses an asymptotic approach, in the small parameter of the numerical mesh size, applied to a finite-rank degenerate-kernel orthogonalpolynomial approximation of the exact kernel. It is proved that, under suitable conditions, the discrepancy between our theoretically predicted error and the actual numerical error tends to zero at the rate ‖K − KM‖, where M is the rank of the degenerate-kernel approximation. A leading-order error analysis is validated on three increasingly accurate projection methods applied to both smooth and sharply peaked kernels, our error predictions being demonstrated to be exponentially convergent to experimentally obtained global numerical errors. The new method is demonstrated to be cost-effective relative to standard extrapolation.

Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem

Complexity of solving nonlinear equations in the deterministic, randomized and quantum settings

The basic problem addressed here is that of obtaining the unknown retardation time spectrum from experimental creep response curves. The spectrum may be a set of discrete times or it may be a spectrum characterized by a continuous distribution function. The present work employs a general model which assumes that the observed creep compliance is due to the summed effect of an arbitrary distribution of mechanisms. The analysis requires the solution of Fredholm integral equations of the first kind. It is well known that this problem is ill-conditioned so that any numerical scheme will have to involve some smoothing to obtain accurate solutions. The present work employs Butler’s method of constrained regularization which takes advantage of the fact that the solution is positive and uses data-dependent smoothing. This work indicates that the imposition of the positivity constraint makes the computation of the solution much better conditioned. Computations with the method of constrained regularization employing near-optimal smoothing demonstrate its superiority over the method of Schapery for obtaining accurate solutions when the data are very noisy.

Localized adsorption on heterogeneous surfaces

In this paper multiple grid methods are applied for the fast solution of the large nonsparse systems of equations that arise from the discretization of Fredholm integral equations of the second kind.Various multiple grid schemes, both with Nystrm and with direct interpolation, are considered.For these iterative methods, the rates of convergence are derived using the collectively compact operator theory by Anselone and Atkinson.Estimates for the asymptotic computational complexity are given, which show that the multiple grid schemes result in B(N2) arithmetic operations.

Information Complexity of Multivariate Fredholm Integral Equations in Sobolev Classes

Two previously given methods for the numerical solution of Fredholm integral equations of the first kind are investigated by the use of polynomial spline functions. As a byproduct a new method is presented for obtaining the eigensolutions of the kernel function. From numerical experiments zero order regularization appears to give more accurate results than higher orders. A comparison is made as to the relative accuracy and speed of the two methods. A scheme is presented to enable a choice to be made, from the set of truncated solutions, as to which member is closest to the solution of the integral equation.