Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

Dominique Chapelle; Asven Gariah; Jacques Sainte-Marie

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 4, page 731-757
  • ISSN: 0764-583X

Abstract

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We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

How to cite

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Chapelle, Dominique, Gariah, Asven, and Sainte-Marie, Jacques. "Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 731-757. <http://eudml.org/doc/277849>.

@article{Chapelle2012,
abstract = {We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.},
author = {Chapelle, Dominique, Gariah, Asven, Sainte-Marie, Jacques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {POD; Galerkin approximation error estimates; non-linear parabolic problems; cardiac models; proper orthogonal decomposition; Galerkin approximation; eror estimates; wave-like equations; numerical experiments},
language = {eng},
month = {2},
number = {4},
pages = {731-757},
publisher = {EDP Sciences},
title = {Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples},
url = {http://eudml.org/doc/277849},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Chapelle, Dominique
AU - Gariah, Asven
AU - Sainte-Marie, Jacques
TI - Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 4
SP - 731
EP - 757
AB - We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.
LA - eng
KW - POD; Galerkin approximation error estimates; non-linear parabolic problems; cardiac models; proper orthogonal decomposition; Galerkin approximation; eror estimates; wave-like equations; numerical experiments
UR - http://eudml.org/doc/277849
ER -

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