Quadrature formulas for the Wiener measure
type: Item
instance of: scholarly article
title: Quadrature formulas for the Wiener measure
zbMATH Open document ID: 0965.41017
author: Achim Steinbauer
published in: Journal of Complexity
publication date: 2000-09-03T00:00:00Z
review text: This paper is concerned with path integrals \(I_\infty (f) =\int _Cf(x) dw(x)\) with respect to the Wiener measure on \(C=C[0,1]\). The author presents a new method for the approximation of \(I_\infty (f)\) by means of a sequence \((A_s)_{s=1,2,\dots }\) of quadrature formulas with the error bound \[ \sup _{f\in {\mathcal F} }|I_\infty (f) -A_s(f) |<2^{-1/4} n_s ^{-1/4}, \] where \({\mathcal F}\) is a class of smooth functions. The quadrature formulas \(A_s \) use \(n_s \) knots and have the form \[ A_s(f)= \sum _{i=1}^{n_s}a_i F(x_i), \] where the \(x_i \) are piecewise linear functions. For the class \({\mathcal F}\) it is proved that \(n(\varepsilon) \leq \max (1,2\varepsilon ^{-4})\) where \(n(\varepsilon)\) is the number of integrand evaluations needed to guarantee an error at most \(\varepsilon \) for \( f\in {\mathcal F}\). The effectiveness of the proposed method is illustrated with test examples.
reviewed by: Luigi Gatteschi
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 65D32
zbMATH DE Number: 1499175
zbMATH Keywords: Wiener measure
MaRDI profile type: Publication
full work available at URL: jcom.1999.0518
OpenAlex ID: W2010728253
cites work: A ``Simpson's rule'' for the numerical evaluation of Wiener's integrals in function space
cites work: Accurate Evaluation of Wiener Integrals
cites work: Q5288943
cites work: Approximation of Wiener integrals
cites work: Q3231529
cites work: Q4039796
cites work: The path integral approach to financial modeling and options pricing
cites work: Deterministic computation of functional integrals
cites work: Generating Quasi-Random Paths for Stochastic Processes
cites work: On an interpolatory method for high dimensional integration
cites work: Q4240374
cites work: A new algorithm and worst case complexity for Feynman-Kac path integration.
cites work: Approximation and optimization on the Wiener space
cites work: Q4226447
cites work: Q4226567
cites work: Explicit cost bounds of algorithms for multivariate tensor product problems
cites work: On tractability of path integration
DOI: 10.1006/JCOM.1999.0518
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description: scientific article; zbMATH DE number 1499175