Many high dimensional problems are difficult to solve for any numerical method. This curse of dimension means that the computational cost must increase exponentially with the dimension of the problem. A high dimension, however, can be compensated by a high degree of smoothness. We study numerical integration and prove that such a compensation is possible by a recently invented method. The method is shown to be universal, i.e., simultaneously optimal up to logarithmic factors, on two different smoothness scales. The first scale is defined by isotropic smoothness conditions, while the second scale involves anisotropic smoothness and is related to partially separable functions.
Multivariate Approximation and Splines,
G. Nürnberger, J. W. Schmidt, G. Walz (eds.),
ISNM, 1997, pp. 177-188.
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