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.
Keywords: greedy algorithm, reduced basis method, empirical interpolation method
@article{M2AN_2014__48_1_259_0,
author = {Hesthaven, Jan S. and Stamm, Benjamin and Zhang, Shun},
title = {Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {259--283},
year = {2014},
publisher = {EDP Sciences},
volume = {48},
number = {1},
doi = {10.1051/m2an/2013100},
mrnumber = {3177844},
zbl = {1292.41001},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2013100/}
}
TY - JOUR AU - Hesthaven, Jan S. AU - Stamm, Benjamin AU - Zhang, Shun TI - Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 259 EP - 283 VL - 48 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2013100/ DO - 10.1051/m2an/2013100 LA - en ID - M2AN_2014__48_1_259_0 ER -
%0 Journal Article %A Hesthaven, Jan S. %A Stamm, Benjamin %A Zhang, Shun %T Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 259-283 %V 48 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2013100/ %R 10.1051/m2an/2013100 %G en %F M2AN_2014__48_1_259_0
Hesthaven, Jan S.; Stamm, Benjamin; Zhang, Shun. Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 1, pp. 259-283. doi: 10.1051/m2an/2013100
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