Journal
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
Volume 47, Issue 3, Pages 859-873Publisher
EDP SCIENCES S A
DOI: 10.1051/m2an/2012045
Keywords
Greedy approximation; proper orthogonal decomposition; convergence rates; reduced Basis methods
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Funding
- Baden-Wurttemberg Stiftung gGmbH
- German Research Foundation (DFG) within the Cluster of Excellence in Simulation Technology at the University of Stuttgart [EXC 310/1]
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Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD-Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD-Greedy algorithm, we show that the existing convergence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD-Greedy algorithm.
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