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.
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {48},
year = {2014},
pages = {259-283},
doi = {10.1051/m2an/2013100},
mrnumber = {3177844},
zbl = {1292.41001},
language = {en},
url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 259-283. doi : 10.1051/m2an/2013100. http://gdmltest.u-ga.fr/item/M2AN_2014__48_1_259_0/
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