%0 Journal Article %1 H13 %A Haasdonk, Bernard %D 2013 %I EDP Sciences %J ESAIM: Mathematical Modelling and Numerical Analysis %K anm ians imported %N 3 %P 859--873 %R 10.1051/m2an/2012045 %T Convergence Rates of the POD--Greedy Method %U http://dx.doi.org/10.1051/m2an/2012045 %V 47 %X 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.

@article{H13, abstract = {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.}, added-at = {2021-09-29T14:33:27.000+0200}, author = {Haasdonk, Bernard}, biburl = {https://puma.ub.uni-stuttgart.de/bibtex/27027d5273eec5b771aedfb8599d038ce/britsteiner}, doi = {10.1051/m2an/2012045}, file = {:PDF/H13_www_preprint_POD_Greedy_Convergence.pdf:PDF;:PDF/H13_M2AN_POD_Greedy_Convergence_print_version.pdf:PDF}, groups = {haasdonk, haasdonk_all_papers}, interhash = {c2a9404d1cfa8066095b98931efd92e5}, intrahash = {7027d5273eec5b771aedfb8599d038ce}, journal = {{ESAIM}: Mathematical Modelling and Numerical Analysis}, keywords = {anm ians imported}, number = 3, owner = {schmidta}, pages = {859--873}, publisher = {{EDP} Sciences}, timestamp = {2021-09-29T12:35:04.000+0200}, title = {Convergence Rates of the {POD}--{G}reedy Method}, url = {http://dx.doi.org/10.1051/m2an/2012045}, volume = 47, year = 2013 }