We construct a new algorithm for the numerical integration of functions that are defined on a d -dimensional cube. It is based on the Clenshaw-Curtis rule for d = 1 and on Smolyak's construction. This way we make the best use of the smoothness properties of any (nonperiodic) function. We prove error bounds showing that our algorithm is almost optimal (up to logarithmic factors) for different classes of functions with bounded mixed derivative. Numerical results show that the new method is very competitive, in particular for smooth integrands and d ≥ 8. Classification (1991): 41A55, 41A63, 65D30, 65Y20 Mathematics Subject

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We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As Communicated by Arieh Iserles. 392 Found Comput Math (2009) 9: 391-429 auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.

We analyse a multilevel Monte Carlo method for the approximation of distribution functions of univariate random variables. Since, by assumption, the target distribution is not known explicitly, approximations have to be used. We provide an asymptotic analysis of the error and the cost of the algorithm. Furthermore we construct an adaptive version of the algorithm that does not require any a priori knowledge on weak or strong convergence rates. We apply the adaptive algorithm to smooth path-independent and path-dependent functionals and to stopped exit times of SDEs.

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