On an interpolatory method for high dimensional integration
type: Item
instance of: scholarly article
title: On an interpolatory method for high dimensional integration
zbMATH Open document ID: 0958.65030
DOI: 10.1016/S0377-0427(99)00222-8
author: Klaus Ritter
author: Richard Schmitt
author: Achim Steinbauer
author: Erich Novak
published in: Journal of Computational and Applied Mathematics
publication date: 2000-02-03T00:00:00Z
review text: The numerical integration of functions which are defined on a bounded or unbounded hyperrectangular region is discussed. The integrand must be smooth, possibly very smooth. This means that certain high order derivatives exist and have a small norm. The method is based on a product formula in the sense of \textit{S. A. Smolyak} [Dokl. Akad. Nauk SSSR 148, 1042-1045 (1963; Zbl 0202.39901)]. Only products with a relatively small number of knots are used. Theoretical results are given concerning error bounds and polynomial exactness. Numerical experiments using the classical formulas of \textit{C. W. Clenshaw} and \textit{A. R. Curtis} [Numer. Math. 2, 197-205 (1960; Zbl 0093.140xx)] and a comparison of different methods are presented. To this end the testing package of Genz functions is used. The paper ends with a report on integrands with Gaussian weight functions, because of their utility in a wide range of applications in physics, statistics, and finance.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1398807
zbMATH Keywords: numerical integration
zbMATH Keywords: multidimensional integration
zbMATH Keywords: numerical experiments
zbMATH Keywords: comparison of methods
zbMATH Keywords: error bounds
zbMATH Keywords: polynomial exactness
zbMATH Keywords: Genz functions
zbMATH Keywords: Gaussian weight functions
reviewed by: Heiner Mühlig
describes a project that uses: PATSYM
describes a project that uses: TESTPACK
describes a project that uses: HRMSYM
MaRDI profile type: Publication
cites work: Q4284650
cites work: Q4316803
cites work: Multidimensional quadrature algorithms at higher degree and/or dimension
cites work: An imbedded family of cubature formulae for \(n\)-dimensional product regions
cites work: Boolean Methods for Double Integration
cites work: Sequences, discrepancies and applications
cites work: Q4324796
cites work: Computing discrepancies of Smolyak quadrature rules
cites work: Fully Symmetric Interpolatory Rules for Multiple Integrals
cites work: Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight
cites work: Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region
cites work: Symmetric integration rules for hypercubes. I. Error coefficients
cites work: Cubature Rules of Prescribed Merit
cites work: Construction of fully symmetric numerical integration formulas
cites work: Quasi-Monte Carlo integration
cites work: Q4003879
cites work: High dimensional integration of smooth functions over cubes
cites work: Q4379953
cites work: Average case complexity of multivariate integration for smooth functions
cites work: Q4218396
cites work: Q4284674
cites work: Product-integration with the Clenshaw-Curtis and related points: Convergence properties
cites work: Q4889887
cites work: Q5600030
cites work: ON A WAY OF OBTAINING LOWER ESTIMATES FOR THE ERRORS OF QUADRATURE FORMULAS
cites work: An adaptive algorithm for numerical integration over an \(n\)-dimensional cube
cites work: Explicit cost bounds of algorithms for multivariate tensor product problems
full work available at URL: s0377-0427(99)00222-8
OpenAlex ID: W2046645122
description: scientific article; zbMATH DE number 1398807