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}, publisher = {EDP-Sciences}, volume = {48}, number = {1}, year = {2014}, doi = {10.1051/m2an/2013100}, mrnumber = {3177844}, zbl = {1292.41001}, language = {en}, url = {http://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 - http://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 http://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 , Volume 48 (2014) no. 1, pp. 259-283. doi : 10.1051/m2an/2013100. http://www.numdam.org/articles/10.1051/m2an/2013100/

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