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| URI | http://purl.tuc.gr/dl/dias/3BE59C9B-E3CF-455F-92F1-480FD7E6FC0F | - |
| Identifier | https://doi.org/10.1051/cocv:2002041 | - |
| Language | en | - |
| Extent | 21 pages | en |
| Title | A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations:“convex inverse” bound conditioner | en |
| Creator | Rovas Dimitrios | en |
| Creator | Ροβας Δημητριος | el |
| Creator | Patera, Adolf, 1836-1912 | en |
| Creator | Karen Veroy | en |
| Publisher | EDP Sciences | en |
| Content Summary | 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 W N 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. | en |
| Type of Item | Peer-Reviewed Journal Publication | en |
| Type of Item | Δημοσίευση σε Περιοδικό με Κριτές | el |
| License | http://creativecommons.org/licenses/by/4.0/ | en |
| Date of Item | 2015-10-17 | - |
| Date of Publication | 2002 | - |
| Subject | Greek mathematics | en |
| Subject | mathematics greek | en |
| Subject | greek mathematics | en |
| Bibliographic Citation | K. Veroy, D.V. Rovas, A.T. Patera ,"A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations:“convex inverse” bound conditioners," ESAIM: Control, Opt. and Cal. of Vari. ,vol. 8, pp.1007-1028.2002.doi:10.1051/cocv:2002041 | en |
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