# 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

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

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topPrud'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 - Modélisation Mathématique et Analyse Numérique 36.5 (2002): 747-771. <http://eudml.org/doc/245381>.

@article{Prudhomme2002,

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 $W_N$ 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 - Modélisation Mathématique et Analyse Numérique},

keywords = {mathematical framework; reduced-basis methods; error bounds; computational framework; simulations repository; distributed and parallel computing; CORBA; C++; elliptic equations; parabolic equations; convergence; Galerkin projection},

language = {eng},

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/245381},

volume = {36},

year = {2002},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2002

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 $W_N$ 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++; elliptic equations; parabolic equations; convergence; Galerkin projection

UR - http://eudml.org/doc/245381

ER -

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