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Convergence Rates of the POD–Greedy Method

Published online by Cambridge University Press:  17 April 2013

Bernard Haasdonk
Bernard Haasdonk*
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Germany. haasdonk@mathematik.uni-stuttgart.de

Abstract

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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.

Keywords

Greedy approximationproper orthogonal decompositionconvergence ratesreduced Basis methods

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013, pp. 859 - 873
Copyright
© EDP Sciences, SMAI, 2013

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