Smolyak's algorithm for integration and L 1-approximation of multivariate functions with bounded mixed derivatives of second order

Leszek Plaskota, Grzegorz W. Wasilkowski

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

We propose and analyze two algorithms for multiple integration and L 1-approximation of functions f:[0,1] s → ℝ that have bounded mixed derivatives of order 2. The algorithms are obtained by applying Smolyak's construction (see [16]) 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 16π 2s/3(s-1)((s- 2)!) 3.(log 2n) 3(s-1)/n 2.(1+o(1)) as n→∞. The error of the corresponding algorithm for L 1-approximation is bounded by the same expression multiplied by 4 s-1.

Original languageEnglish
Pages (from-to)229-246
Number of pages18
JournalNumerical Algorithms
Volume36
Issue number3
DOIs
StatePublished - Jul 2004

Bibliographical note

Funding Information:
The research of the first author has been partially supported by the State Committee for Scientific Research of Poland under Grant 5 P03A 007 21, and of the second author by the National Science Foundation under Grant CCR-0095709.

Keywords

  • Smolyak's algorithm
  • multivariate approximation
  • multivariate integration

ASJC Scopus subject areas

  • Applied Mathematics

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