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Smolyak's Algorithm for Integration and L 1-Approximation of Multivariate Functions with Bounded Mixed Derivatives of Second Order

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Abstract

We propose and analyze two algorithms for multiple integration and L 1-approximation of functions \(f:[0,1]^s \to \mathbb{R}\) 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 L 1-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

$$\frac{{16\pi ^2 s}}{{3(s - 1)((s - 2)!)^3 }} \cdot \frac{{(\log _2 n)^{3(s - 1)} }}{{n^2 }} \cdot (1 + o(1))$$

as n→∞. The error of the corresponding algorithm for L 1-approximation is bounded by the same expression multiplied by 4s−1.

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Authors and Affiliations

  1. Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Leszek Plaskota

  2. Department of Computer Science, University of Kentucky, 773 Anderson Hall, Lexington, KY, 40506-0046, USA

    Grzegorz W. Wasilkowski

Authors
  1. Leszek Plaskota
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  2. Grzegorz W. Wasilkowski
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Plaskota, L., Wasilkowski, G.W. Smolyak's Algorithm for Integration and L 1-Approximation of Multivariate Functions with Bounded Mixed Derivatives of Second Order. Numerical Algorithms 36, 229–246 (2004). https://doi.org/10.1023/B:NUMA.0000040060.56819.a7

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