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 |
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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. |
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ISSN: | 1017-1398 1572-9265 |
DOI: | 10.1023/B:NUMA.0000040060.56819.a7 |