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Quadrature Formulas for the Wiener Measure

1999
*
Journal of Complexity
*

*The*purely deterministic algorithm is a sequence of

*quadrature*

*formulas*

*for*

*the*

*Wiener*

*measure*, where

*the*knots are piecewise linear functions. ... It uses ideas of Smolyak, as well as

*the*multiscale decomposition of

*the*

*Wiener*

*measure*due to Le vy and Ciesielski. ... ACKNOWLEDGMENTS I am grateful to Erich Novak and Klaus Ritter

*for*their advice.

*The*author was supported by

*the*German Research Council (DFG). ...

##
###
Page 756 of Mathematical Reviews Vol. 31, Issue 4
[page]

1966
*
Mathematical Reviews
*

Numerical values

*for**the*weights and abscissae are given*for*n = 2, 3. It should be noted that essentially*the*same class of*quadrature**formulas*has been considered by H. Yanagihara [Bull. ...*The**formula*obtained approximates*the*conditional*Wiener*integral by a k-fold integral with an error of O(1/k?). ...##
###
Numerical evaluation of Wiener integrals

1967
*
Mathematics of Computation
*

A systematic study of

doi:10.1090/s0025-5718-1967-0221753-0
fatcat:j2cmxa7k6jc7rb5eiljnnedwjm
*quadrature**formulae**for**the**Wiener*integral j F[x]w(dx) of*the*type / F[6(u, ■ )]v(du) is presented. ...*The*Cameron and Vladimirov*quadrature**formulae*, which are*the*function space analogues of Simpson's Rule, are shown to fit into this framework. ... In Section 3 we formulate*the*problem*for**quadrature**formulae*which are exact*for*polynomial functionals of degree 2n -f-1. ...##
###
Numerical Evaluation of Wiener Integrals

1967
*
Mathematics of Computation
*

A systematic study of

doi:10.2307/2003470
fatcat:5nyw5puterf6dn4nscytcx4rwa
*quadrature**formulae**for**the**Wiener*integral j F[x]w(dx) of*the*type / F[6(u, ■ )]v(du) is presented. ...*The*Cameron and Vladimirov*quadrature**formulae*, which are*the*function space analogues of Simpson's Rule, are shown to fit into this framework. ... In Section 3 we formulate*the*problem*for**quadrature**formulae*which are exact*for*polynomial functionals of degree 2n -f-1. ...##
###
Accurate evaluation of Wiener integrals

1973
*
Mathematics of Computation
*

A new

doi:10.1090/s0025-5718-1973-0329205-7
fatcat:ioddvskgvvhn5azwtdy7nhoh3m
*quadrature**formula**for*an important class of*Wiener*integrals is presented, in which*the**Wiener*integrals are approximated by «-fold integrals with an error Oin~2). ... Let C be*the*space of continuous real functions x(i) defined on 0 ^ t ^ 1, with x(0) = 0, and endowed with*the**Wiener**measure*W. ...*The*approximation (5) can be derived through*the*use of*the*interpolation*formula**for**Wiener*paths [11] : if xiii -l)/n) = Xj-i , x(i/n) = *,_, + Uj/y/n, then*for*t such that (i -l)/n ¿ t î /n we have ...##
###
Page 5761 of Mathematical Reviews Vol. , Issue 86m
[page]

1986
*
Mathematical Reviews
*

Sh. (2-AOSAZ-C) 86m:65025 Optimal

*quadrature**formulas*and*the**Wiener**measure*. (Russian. English and Azerbaijani summaries) Tzv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.-Tekhn. Mat. ... functions supplied with a*Wiener*probability*measure*. ...##
###
Page 6613 of Mathematical Reviews Vol. , Issue 90K
[page]

1990
*
Mathematical Reviews
*

From

*the*introduction: “Previously,*Wiener**measure*was used to estimate one-dimensional*quadrature**formulas*. ... In this paper we present a method*for*constructing*quadrature**formulas*that is based on*the*theory of divisors of*the*field of algebraic numbers.” ...##
###
Page 1150 of Mathematical Reviews Vol. , Issue 84c
[page]

1984
*
Mathematical Reviews
*

Author’s summary: “It is proved that a Chebyshev

*quadrature**formula*with real nodes*for*two-fold integrals on*the**Wiener**measure*does not exist when*the*number of nodes is larger than four.” 65E Numerical ... /y(f)), with /,(f)*the**quadrature**formula**for*f obtained by*the*process /,. ...##
###
Probabilistic analysis of numerical integration algorithms

1991
*
Journal of Complexity
*

We analyze a type of divided difference test-an error estimate obtained by replacing

doi:10.1016/0885-064x(91)90044-x
fatcat:sqetuwxrk5dzlgoso2jygbrgim
*the*higher derivative of*the*integrand in*the*local error remainder term of a*quadrature**formula*by*the*higher divided ... We study*the*conditional expected error of approximation*for*a class of adaptive numerical integration algorithms. ... ACKNOWLEDGMENT This research was partially supported by*the*Natural Sciences and Engineering Research Council of Canada under Grant OGPO041639. ...##
###
Page 4025 of Mathematical Reviews Vol. , Issue 89G
[page]

1989
*
Mathematical Reviews
*

Finally, some considerations about Turan

*quadrature**formulae*with Chebyshev*measure*are given.” ... (t,s)dtds and p(x) is a*Wiener**measure*(or conditional*Wiener**measure*) on*the*space C2(Q) of continuous functions on*the*unit square Q = [0,1] x [0,1]. ...##
###
Page 2641 of Mathematical Reviews Vol. , Issue 86f
[page]

1986
*
Mathematical Reviews
*

Kronrod [Nodes and weights

*for**quadrature**formulas*. ... (Russian) [Approximation*formulas*with weight*for*path integrals with respect to*the*conditional*Wiener**measure*] With an English summary. Coo6menua O6tequHenHoro Unctutyta Apepuerx VUccneposanun. ...##
###
Page 1308 of Mathematical Reviews Vol. 48, Issue 4
[page]

1974
*
Mathematical Reviews
*

*For*such functionals, explicit evaluation of

*the*

*Wiener*integrals can be made, .and

*quadrature*

*formulae*obtained that yield exact results. ... One is thus able to con- struct

*quadrature*

*formulae*

*for*computing

*the*more general functional F[x]=f,! ...

##
###
Page 42 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 43, Issue 1
[page]

1937
*
American Mathematical Society. Bulletin of the American Mathematical Society
*

Finally

*the*relation of*the*above inversion*formula*to one given recently by Paley and*Wiener*is investigated. ...*For*example, with*the*normal function, constant -exp{ —(x/a)?/2},*the*scale a is usually taken as*the*root-mean-square of*the*given*measurements*.*The*R. A. ...##
###
Page 1710 of Mathematical Reviews Vol. 52, Issue 5
[page]

1976
*
Mathematical Reviews
*

V. 12076

*Quadrature**formulae**for*random processes. (Ukrainian. Eng- lish and Russian summaries) he a Akad. Nauk Ukrain: RSR Ser. ... A*quadrature**formula*L(é, 7, p) is said to be best in*the*class H [best among*quadrature**formulae*with fixed points of inter-, polation] if R,(H, 7, p)=8(H) [R,(H, T, p)=8,(H)}. ...##
###
Page 5334 of Mathematical Reviews Vol. , Issue 95i
[page]

1995
*
Mathematical Reviews
*

*The*set of all g is equipped with

*the*

*Wiener*

*measure*y. ...

*The*authors consider

*the*

*quadrature*

*formula*of

*the*Chebyshev type 1 ie (*) xB | fo) dos) = 5 OIG). 6; EE, j=l where E is a compact set in R? and g is a finite positive

*measure*on E. ...

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