Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods∗. (10th January 2014)

• Record Type: Journal Article
• Authors: Hesthaven, Jan S.; Stamm, Benjamin; Zhang, Shun
• Is Part Of: Mathematical modelling and numerical analysis. Volume 48:Part 1(2013)
• Page Start: 259
• Page End: 283
• Publication Date: 2014-01-10
• Subjects: Numerical analysis -- Periodicals; Mathematical models -- Periodicals; 510
• DOI: 10.1051/m2an/2013100
• Languages: English
• ISSNs: 0764-583X

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>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.</p> </abstract>