Stochastic properties of quadrature formulas
type: Item
instance of: scholarly article
title: Stochastic properties of quadrature formulas
zbMATH Open document ID: 0655.65041
DOI: 10.1007/BF01397555
author: Erich Novak
published in: Numerische Mathematik
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133294
review text: Stochastic properties of several quadrature formulas are discussed. Let \(F\subset L_ 1([0,1]\beta d,\lambda \beta d)\) be a given set of functions and let \(S(f)=\int_{[0,1]\beta d} f(x)dx\) and \(\tilde S(f)=\phi(a_ 1,a_ 2,...,a_ n)\) be a quadrature method only usingthe function values at n knots \(a_ 1,a_ 2,...,a_ n\in [0,1]\beta d\). The maximal error of a method \(\tilde S\) is given by \(\Delta_{\max}(S)=\sup \| S(f)-S(f)\|,\) while the average error is given by \(\Delta_{\mu}(S)=\int_{F}| S(f)-\tilde S(f)| d\mu(f),\) where \(\mu\) is a Borel probability measure on \((F,\| \|_{\infty})\). It is shown that stochastic error bounds, i.e., the error bounds of the average error, for quadratic formulas are much smaller than deterministic ones (bound for the maximal error) in many cases, though it dependson the class of F on functions to be considered. Moreover, nonlinear methods, adaptive methods or even methods with varying cardinality are shown not be significantly better than the simple linear method with respect to certain stochastic error bounds.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 4070162
zbMATH Keywords: stochastic error bounds
zbMATH Keywords: quadrature formulas
reviewed by: Katsuji Uosaki
MaRDI profile type: Publication
cites work: Q3892702
cites work: Multidimensional Spline Approximation
cites work: Q5525016
cites work: Q4064914
cites work: Approximation of linear operators on a Wiener space
cites work: Approximation of linear functionals on a Banach space with a Gaussian measure
cites work: A decomposition of measures in euclidean space yielding error bounds for quadrature formulas
cites work: Sequences of functionals with a boundary layer
cites work: Unbiased Monte Carlo Integration Methods with Exactness for Low Order Polynomials
cites work: Information of varying cardinality
cites work: Randomization for continuous problems
OpenAlex ID: W1978505256
description: scientific article; zbMATH DE number 4070162