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 language | English |
|---|---|
| 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.
Funding
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.
| Funders | Funder number |
|---|---|
| U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China | CCR-0095709 |
| State Committee for Scientific Research of Poland | 5 P03A 007 21 |
Keywords
- Smolyak's algorithm
- multivariate approximation
- multivariate integration
ASJC Scopus subject areas
- Applied Mathematics