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.
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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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