Volume 48, Number 1, January-February 2014
|Page(s)||259 - 283|
|Published online||10 January 2014|
Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods∗
Division of Applied Mathematics, Box F, Brown
University, 182 George
2 CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
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
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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