Efficient algorithms for multivariate and \(\infty\)-variate integration with exponential weight
type: Item
instance of: scholarly article
title: Efficient algorithms for multivariate and \(\infty\)-variate integration with exponential weight
zbMATH Open document ID: 1303.65009
published in: Numerical Algorithms
publication date: 2014-10-31T00:00:00Z
review text: The authors present an efficient algorithm for \(\infty\)-variate integration with exponential weight function \[ I_{\infty}(f) = \lim_{d\to \infty} \int_{{\mathbb R}_+^d} f(x_1,\dots,x_d,0,0,\dots)\, \exp(-\sum_{j=1}^d x_j)\,dx\,, \] where the function \(f\) is differentiable with respect to each variable and vanishes if there is at least one \(x_j =0\). The \(d\)-variate problem is solved by the \(d\)-variate integration method of \textit{S. A. Smolyak} [Dokl. Akad. Nauk SSSR 148, 1042--1045 (1963; Zbl 0202.39901)] which requires efficient algorithms for the univariate integral \[ I(f) = \int_0^{\infty} f(x)\, \exp(-x)\, dx. \] Using weighted trapezoidal rules, (nearly) optimal quadratures for \(I(f)\) are presented in detail.
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 6363122
zbMATH Keywords: multivariate integration
zbMATH Keywords: exponential weight
zbMATH Keywords: \(\infty\)-variate integration
zbMATH Keywords: Smolyak's algorithm
zbMATH Keywords: univariate integral with exponential weight
zbMATH Keywords: weighted trapezoidal rule
zbMATH Keywords: nearly optimal quadrature
author: Leszek Plaskota
author: Grzegorz W. Wasilkowski
MaRDI profile type: Publication
full work available at URL: s11075-013-9798-4
OpenAlex ID: W2009612317
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DOI: 10.1007/S11075-013-9798-4
description: scientific article; zbMATH DE number 6363122