## 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 ^{2}s/3(s-1)((s- 2)!) ^{3}.(log _{2}n) ^{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 language | English |
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Pages (from-to) | 229-246 |

Number of pages | 18 |

Journal | Numerical Algorithms |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - 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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