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Quadrature Formulas for the Wiener Measure

Achim Steinbauer
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).  ... 
doi:10.1006/jcom.1999.0518 fatcat:kewuq5p5jfcpfmgnieirnlsvr4

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

Alan G. Konheim, Willard L. Miranker
1967 Mathematics of Computation  
A systematic study of 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.  ... 
doi:10.1090/s0025-5718-1967-0221753-0 fatcat:j2cmxa7k6jc7rb5eiljnnedwjm

Numerical Evaluation of Wiener Integrals

Alan G. Konheim, Willard L. Miranker
1967 Mathematics of Computation  
A systematic study of 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.  ... 
doi:10.2307/2003470 fatcat:5nyw5puterf6dn4nscytcx4rwa

Accurate evaluation of Wiener integrals

Alexandre Joel Chorin
1973 Mathematics of Computation  
A new 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  ... 
doi:10.1090/s0025-5718-1973-0329205-7 fatcat:ioddvskgvvhn5azwtdy7nhoh3m

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

Feng Gao
1991 Journal of Complexity  
We analyze a type of divided difference test-an error estimate obtained by replacing 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.  ... 
doi:10.1016/0885-064x(91)90044-x fatcat:sqetuwxrk5dzlgoso2jygbrgim

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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