Double greedy algorithms: Reduced basis methods for transport dominated problems

Wolfgang Dahmen; Christian Plesken; Gerrit Welper

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 3, page 623-663
  • ISSN: 0764-583X

Abstract

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The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

How to cite

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Dahmen, Wolfgang, Plesken, Christian, and Welper, Gerrit. "Double greedy algorithms: Reduced basis methods for transport dominated problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.3 (2014): 623-663. <http://eudml.org/doc/273103>.

@article{Dahmen2014,
abstract = {The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.},
author = {Dahmen, Wolfgang, Plesken, Christian, Welper, Gerrit},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {tight surrogates; stable variational formulations; saddle point problems; double greedy schemes; greedy stabilization; rate-optimality; transport equations; convection-diffusion equations},
language = {eng},
number = {3},
pages = {623-663},
publisher = {EDP-Sciences},
title = {Double greedy algorithms: Reduced basis methods for transport dominated problems},
url = {http://eudml.org/doc/273103},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Dahmen, Wolfgang
AU - Plesken, Christian
AU - Welper, Gerrit
TI - Double greedy algorithms: Reduced basis methods for transport dominated problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 623
EP - 663
AB - The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.
LA - eng
KW - tight surrogates; stable variational formulations; saddle point problems; double greedy schemes; greedy stabilization; rate-optimality; transport equations; convection-diffusion equations
UR - http://eudml.org/doc/273103
ER -

References

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