Optimal integration of Lipschitz functions with a Gaussian weight
type: Item
instance of: scholarly article
title: Optimal integration of Lipschitz functions with a Gaussian weight
zbMATH Open document ID: 0908.65013
author: Francisco Curbera
published in: Journal of Complexity
publication date: 1999-03-18T00:00:00Z
review text: The aim of this work is to study numerical integration of scalar functions over the infinite interval \((-\infty,+\infty)\) with a Gaussian weight. Such integrals, and their multivariate analogs, have numerous applications. The author assumes that the functions satisfy a Lipschitz condition with known constant \(L\). For this class of functions he studies the (worst case) complexity of integration with the Gaussian weight \(\exp(- x^2/(2\sigma))/\sqrt{2\pi\sigma}\), for a positive variance \(\sigma\). First, he studies the problem of finding optimal information. That is, for a given number \(n\), he finds optimal sample points \(x_1,\dots, x_n\) that yield the minimal error \(r(n)\) of the optimal algorithm. The first major result is to show that (asymptotically) the optimal information is given by the quantiles of the Gaussian weight with double variance, i.e., \[ {1\over\sqrt{2\pi(2\sigma)}} \int^{x_i}_0 \exp(- t^2/(2(2\sigma)))dt\sim{i\over n+1}. \] The second major result of this paper is to show that the worst case error \(r(n)\) of the optimal information is \[ r(n)= \sqrt{{\pi\sigma\over 2}}\cdot{L\over n} (1- O(1))\quad\text{as }n\to+\infty. \] Numerical examples are given.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 1206671
zbMATH Keywords: optimal integration
zbMATH Keywords: Lipschitz functions
zbMATH Keywords: worst case complexity
zbMATH Keywords: numerical examples
zbMATH Keywords: infinite interval
zbMATH Keywords: Gaussian weight
zbMATH Keywords: optimal algorithm
zbMATH Keywords: optimal information
reviewed by: Dumitru Acu
MaRDI profile type: Publication
full work available at URL: jcom.1997.0464
OpenAlex ID: W2057055143
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DOI: 10.1006/JCOM.1997.0464
description: scientific article; zbMATH DE number 1206671