A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners

Karen Veroy; Dimitrios V. Rovas; Anthony T. Patera

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 1007-1028
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Veroy, Karen, Rovas, Dimitrios V., and Patera, Anthony T.. "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 (2010): 1007-1028. <http://eudml.org/doc/90639>.

@article{Veroy2010,
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. },
author = {Veroy, Karen, Rovas, Dimitrios V., Patera, Anthony T.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Elliptic partial differential equations; reduced-basis methods; output bounds; Galerkin approximation; a posteriori error estimation; convex analysis.; elliptic partial differential equations; output bounds; a posteriori error estimation; convex analysis},
language = {eng},
month = {3},
pages = {1007-1028},
publisher = {EDP Sciences},
title = {A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners},
url = {http://eudml.org/doc/90639},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Veroy, Karen
AU - Rovas, Dimitrios V.
AU - Patera, Anthony T.
TI - A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 1007
EP - 1028
AB - 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.
LA - eng
KW - Elliptic partial differential equations; reduced-basis methods; output bounds; Galerkin approximation; a posteriori error estimation; convex analysis.; elliptic partial differential equations; output bounds; a posteriori error estimation; convex analysis
UR - http://eudml.org/doc/90639
ER -

References

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