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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 goalsindependently, though we consider them here as an ensemble. The first component is a technique forthe rapid and reliable evaluation prediction of linear functional outputs of elliptic (andparabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent globalreduced–basis approximations — Galerkin projection onto a spaceW N spanned by solutions of the governing partial differentialequation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error–residualequation that provide inexpensive yet sharp and rigorous bounds forthe error in the outputs of interest; and (iii) off–line/on–linecomputational procedures — methods which decouple the generationand projection stages of the approximation process. This component is ideally suited — consideringthe operation count of the online stage — for the repeated and rapid evaluation required in thecontext of parameter estimation, design, optimization, andreal–time control. The second component is a framework for distributed simulations. This frameworkcomprises a library providing the necessary abstractions/concepts for distributed simulations and asmall set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of thosesimulations. While the library is the backbone of the framework and is therefore general, thevarious interfaces answer specific needs. We shall describe both components and present how theyinteract.

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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