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A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations

Prud'homme, C; Rovas, DV; Veroy, K; Patera, AT; (2002) A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. Mathematical Modelling and Numerical Analysis , 36 (5) pp. 747-771. 10.1051/m2an:2002035. Green open access

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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 SIMTEX and 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.

Type: Article
Title: A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations
Open access status: An open access version is available from UCL Discovery
DOI: 10.1051/m2an:2002035
Publisher version: https://doi.org/10.1051/m2an:2002035
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Mathematical framework / reduced-basis methods / error bounds / computational framework / simulations repository / distributed and parallel computing / CORBA / C++.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of the Built Environment
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of the Built Environment > Bartlett School Env, Energy and Resources
URI: https://discovery.ucl.ac.uk/id/eprint/10063069
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