# 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

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topVeroy, 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 -

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