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A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: "Convex inverse" bound conditioners

Veroy, K; Rovas, DV; Patera, AT; (2002) A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: "Convex inverse" bound conditioners. ESAIM - Control, Optimisation and Calculus of Variations , 8 pp. 1007-1028. 10.1051/cocv:2002041. Green open access

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Abstract

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components 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 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. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity , is close to (but always greater than) unity. However, in other cases, the necessary “bound conditioners” – in essence, operator preconditioners that (i ) satisfy an additional spectral “bound” requirement, and (ii ) admit the reduced-basis off-line/on-line computational stratagem – either can not be found, or yield unacceptably large effectivities. In this paper we introduce a new class of improved bound conditioners: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); we thereby accommodate higher-order (e.g., piecewise linear) effectivity constructions while simultaneously preserving on-line efficiency. Simple convex analysis and elementary approximation theory suffice to prove the necessary bounding and convergence properties.

Type: Article
Title: A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: "Convex inverse" bound conditioners
Open access status: An open access version is available from UCL Discovery
DOI: 10.1051/cocv:2002041
Publisher version: https://doi.org/10.1051/cocv:2002041
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: Elliptic partial differential equations, reduced-basis methods, output bounds, Galerkin approximation, a posteriori error estimation, convex analysis.
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/1502808
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