Issue |
ESAIM: M2AN
Volume 48, Number 1, January-February 2014
|
|
---|---|---|
Page(s) | 259 - 283 | |
DOI | https://doi.org/10.1051/m2an/2013100 | |
Published online | 10 January 2014 |
Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods∗
1
Division of Applied Mathematics, Box F, Brown
University, 182 George
St., Providence,
RI
02912,
USA
Jan.Hesthaven@Brown.edu
2
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions,
F-75005,
Paris,
France
stamm@ann.jussieu.fr
3
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions,
F-75005,
Paris,
France
4
Department of Mathematics, City University of Hong
Kong, Kowloon Tong,
Hong Kong,
China
shun.zhang@cityu.edu.hk
Received: 4 August 2011
We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy algorithm. In a further improved approach, this is combined with an algorithm in which the train set for the greedy approach is adaptively sparsified and enriched. A safety check step is added at the end of the algorithm to certify the quality of the sampling. Both these techniques are applicable to high-dimensional problems and we shall demonstrate their performance on a number of numerical examples.
Mathematics Subject Classification: 41A05 / 41A46 / 65N15 / 65N30
Key words: Greedy algorithm / reduced basis method / empirical interpolation method
© EDP Sciences, SMAI, 2014
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