An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

Calabrò, Francesco; Esposito, Antonio Corbo

doi:10.1016/j.cam.2008.10.022

Cited by 4 publications

(1 citation statement)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It seems that a stable and efficient algorithm for the evaluation 3 Clenshaw-Curtis rules have also been studied in this context (see, e.g. [8]), but since Gauss and Clenshaw-Curtis rules typically converge at a similar rate (see, e.g. [27] for the classical case), we shall for brevity restrict our attention to the discussion of Gauss rules here.…”

Section: Gauss Rules In the Case 0 ⊂ Rmentioning

confidence: 99%

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

Gibbs,

Hewett,

Major

2023

Numer Algor

View full text Add to dashboard Cite

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is “disjoint” in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

show abstract

Section: Gauss Rules In the Case 0 ⊂ Rmentioning

confidence: 99%

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

Gibbs,

Hewett,

Major

2023

Numer Algor

View full text Add to dashboard Cite

show abstract

Polynomial Basis Functions and Quadratures

Shizgal

2015

Scientific Computation

View full text Add to dashboard Cite

On the accuracy of interpolation based on single-layer artificial neural networks with a focus on defeating the Runge phenomenon

Auricchio,

Belardo,

Calabrò

et al. 2024

Soft Comput

View full text Add to dashboard Cite

Artificial Neural Networks (ANNs) are a tool in approximation theory widely used to solve interpolation problems. In fact, ANNs can be assimilated to functions since they take an input and return an output. The structure of the specifically adopted network determines the underlying approximation space, while the form of the function is selected by fixing the parameters of the network. In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. Given that the ANN is interpolating, the error incurred occurs outside the sampling interpolation nodes provided by the user. In this study, various choices of nodes are analyzed: equispaced, Chebychev, and randomly selected ones. Then, the focus is on regular target functions, for which it is known that interpolation can lead to spurious oscillations, a phenomenon that in the ANN literature is referred to as overfitting. We obtain good accuracy of the ANN interpolating function in all tested cases using these different types of interpolating nodes and different types of neurons. The following study is conducted starting from the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when the number of interpolation nodes increases, we increase the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge’s function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays, and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training. Then we can conclude that the use of such an ANN defeats the Runge phenomenon. Our results show the power of ANNs to achieve excellent approximations when interpolating regular functions also starting from uniform and random nodes, particularly for Runge’s function.

show abstract

An evaluation of Clenshaw–Curtis quadrature rule for integration w.r.t. singular measures

Cited by 4 publications

References 25 publications

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

Polynomial Basis Functions and Quadratures

On the accuracy of interpolation based on single-layer artificial neural networks with a focus on defeating the Runge phenomenon

Contact Info

Product

Resources

About