NDLI: Reduced-basis output bounds for approximately parametrized elliptic coercive partial differential equations

Content Provider	Springer Nature Link
Author	Prud’homme, C. Patera, A. T.
Copyright Year	2004
Abstract	We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with (approximately) affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations — Galerkin projection onto a spaceW N spanned by solutions of the governing partial differential equation atN selected points in parameter space;(ii) a posteriori error estimation- relaxations of the error-residual equation that provide inexpensive yet sharp 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 TV, typically very small, and the (approximate) 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 rigorousa posteriori error bound framework for the case in which the parametrization of the partial differential equation is exact; in this paper, we address the situation in which our mathematical model is not “complete.” In particular, we permit error in the data that define our partial differential operator: this error may be introduced, for example, by imperfect specification, measurement, calculation, or parametric expansion of a coefficient function. We develop both accurate predictions for the outputs of interestand associated rigorousa posteriori error bounds; and the latter incorporate both numerical discretizationand “model truncation” effects. Numerical results are presented for a particular instantiation in which the model error originates in the (approximately) prescribed velocity field associated with a three-dimensional convection-diffusion problem.
Starting Page	147
Ending Page	162
Page Count	16
File Format	PDF
ISSN	14329360
Journal	Computing and Visualization in Science
Volume Number	6
Issue Number	2-3
e-ISSN	14330369
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2004-03-01
Publisher Place	Berlin, Heidelberg
Access Restriction	Subscribed
Subject Keyword	Computational Mathematics and Numerical Analysis Computer Applications in Chemistry Algorithms Visualization Numerical Analysis Calculus of Variations and Optimal Control; Optimization
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Engineering Computational Theory and Mathematics Computer Vision and Pattern Recognition Modeling and Simulation Software

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