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

Published online by Cambridge University Press:  15 October 2002

Christophe Prud'homme ,
Dimitrios V. Rovas ,
Karen Veroy  and
Anthony T. Patera
Christophe Prud'homme
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Massachusetts Ave., Cambridge, MA 02139, USA. prudhomm@MIT.EDU.
Dimitrios V. Rovas
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Massachusetts Ave., Cambridge, MA 02139, USA. prudhomm@MIT.EDU.
Karen Veroy
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Massachusetts Ave., Cambridge, MA 02139, USA. prudhomm@MIT.EDU.
Anthony T. Patera
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Massachusetts Ave., Cambridge, MA 02139, USA. prudhomm@MIT.EDU.
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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.

Keywords

Mathematical frameworkreduced-basis methodserror boundscomputational frameworksimulations repositorydistributed and parallel computingCORBAC++.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 5: Special issue on Programming , September 2002 , pp. 747 - 771
Copyright
© EDP Sciences, SMAI, 2002

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