A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations

Christophe Prud'homme; Dimitrios V. Rovas; Karen Veroy; Anthony T. Patera

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 5, page 747-771
  • ISSN: 0764-583X

Abstract

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We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error–residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off–line/on–line computational procedures — methods which decouple the generation and projection stages of the approximation process. This component is ideally suited — considering the operation count of the online stage — for the repeated and rapid evaluation required in the context of parameter estimation, design, optimization, and real–time control. The second component is a framework for distributed simulations. This framework comprises a library providing the necessary abstractions/concepts for distributed simulations and a small set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of those simulations. While the library is the backbone of the framework and is therefore general, the various interfaces answer specific needs. We shall describe both components and present how they interact.

How to cite

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Prud'homme, Christophe, et al. "A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations." ESAIM: Mathematical Modelling and Numerical Analysis 36.5 (2010): 747-771. <http://eudml.org/doc/194124>.

@article{Prudhomme2010,
abstract = { We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error–residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off–line/on–line computational procedures — methods which decouple the generation and projection stages of the approximation process. This component is ideally suited — considering the operation count of the online stage — for the repeated and rapid evaluation required in the context of parameter estimation, design, optimization, and real–time control. The second component is a framework for distributed simulations. This framework comprises a library providing the necessary abstractions/concepts for distributed simulations and a small set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of those simulations. While the library is the backbone of the framework and is therefore general, the various interfaces answer specific needs. We shall describe both components and present how they interact. },
author = {Prud'homme, Christophe, Rovas, Dimitrios V., Veroy, Karen, Patera, Anthony T.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Mathematical framework; reduced-basis methods; error bounds; computational framework; simulations repository; distributed and parallel computing; CORBA; C++.; mathematical framework; C++; elliptic equations; parabolic equations; convergence; Galerkin projection},
language = {eng},
month = {3},
number = {5},
pages = {747-771},
publisher = {EDP Sciences},
title = {A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations},
url = {http://eudml.org/doc/194124},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Prud'homme, Christophe
AU - Rovas, Dimitrios V.
AU - Veroy, Karen
AU - Patera, Anthony T.
TI - A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 5
SP - 747
EP - 771
AB - We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error–residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off–line/on–line computational procedures — methods which decouple the generation and projection stages of the approximation process. This component is ideally suited — considering the operation count of the online stage — for the repeated and rapid evaluation required in the context of parameter estimation, design, optimization, and real–time control. The second component is a framework for distributed simulations. This framework comprises a library providing the necessary abstractions/concepts for distributed simulations and a small set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of those simulations. While the library is the backbone of the framework and is therefore general, the various interfaces answer specific needs. We shall describe both components and present how they interact.
LA - eng
KW - Mathematical framework; reduced-basis methods; error bounds; computational framework; simulations repository; distributed and parallel computing; CORBA; C++.; mathematical framework; C++; elliptic equations; parabolic equations; convergence; Galerkin projection
UR - http://eudml.org/doc/194124
ER -

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Citations in EuDML Documents

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  1. Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
  2. Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
  3. Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
  4. Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie, Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
  5. Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie, Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
  6. Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie, Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
  7. Bernard Haasdonk, Mario Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations

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