Many applications require approximate values of Wiener integrals. A typical approach is to approximate the path integral by a high dimensional integral and apply a Monte Carlo (randomized) method. Here we develop (deterministic) quadrature formulas for the Wiener measure, the `knots' are piecewise linear functions. Our construction is based on polynomial interpolation and uses ideas of Smolyak as well as the multiscale decomposition of the Wiener measure, due to Levy and Ciesielski. Our method works well if the integrand is smooth, as shown by examples.
Approximation Theory IX, Volume 2: Computational Aspects.
C. K. Chui, L. L. Schumaker
(eds.), 251-258, 1998.
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The following paper continues this research:
We present a new method for the approximation of Wiener integrals and provide an explicit error bound for a class F of smooth integrands. 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 Lévy and Ciesielski. For the class F we obtain $n(epsilon) \le \max(1, 2 epsilon^{-4})$, where $n(epsilon)$ is the number of evaluations needed to guarantee an error at most $epsilon$ for $f \in F$.