Smolyak's Algorithm for Integration and L1-Approximation of Multivariate Functions with Bounded Mixed Derivatives of Second Order

We propose and analyze two algorithms for multiple integration and L1-approximation of functions that have bounded mixed derivatives of order 2. The algorithms are obtained by applying Smolyak's construction (see [8]) to one-dimensional composite midpoint rules (for integration) and to one-dime...

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Veröffentlicht in:Numerical algorithms 2004-07, Vol.36 (3), p.229-246
Hauptverfasser: Plaskota, Leszek, Wasilkowski, Grzegorz W
Format: Artikel
Sprache:eng
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Zusammenfassung:We propose and analyze two algorithms for multiple integration and L1-approximation of functions that have bounded mixed derivatives of order 2. The algorithms are obtained by applying Smolyak's construction (see [8]) to one-dimensional composite midpoint rules (for integration) and to one-dimensional piecewise linear interpolation algorithm (for L1-approximation). Denoting by n the number of function evaluations used, the worst case error of the obtained Smolyak's cubature is asymptotically bounded from above by as n→∞. The error of the corresponding algorithm for L1-approximation is bounded by the same expression multiplied by 4s−1.
ISSN:1017-1398
1572-9265
DOI:10.1023/B:NUMA.0000040060.56819.a7