# A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations

Martin A. Grepl; Anthony T. Patera

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 39, Issue: 1, page 157-181
- ISSN: 0764-583X

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topGrepl, Martin A., and Patera, Anthony T.. "A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis 39.1 (2010): 157-181. <http://eudml.org/doc/194255>.

@article{Grepl2010,

abstract = {
In this paper, we extend the reduced-basis methods and associated a
posteriori error estimators developed earlier for elliptic partial
differential equations to parabolic problems with affine parameter
dependence. The essential new ingredient is the presence of time in the
formulation and solution of the problem – we shall “simply” treat
time as an additional, albeit special, parameter. First, we introduce
the reduced-basis recipe – Galerkin projection onto a space WN
spanned by solutions of the governing partial differential equation at
N selected points in parameter-time space – and develop a new greedy
adaptive procedure to “optimally” construct the parameter-time sample
set. Second, we propose error estimation and adjoint procedures that
provide rigorous and sharp bounds for the error in specific outputs of
interest: the estimates serve a priori to construct our samples,
and a posteriori to confirm fidelity. Third, based on the
assumption of affine parameter dependence, we develop offline-online
computational procedures: in the offline stage, we generate the
reduced-basis space; in the online stage, given a new parameter value, we
calculate the reduced-basis output and associated error bound. The
operation count for the online stage depends only on N (typically
small) and the parametric complexity of the problem; the method is thus
ideally suited for repeated, rapid, reliable evaluation of input-output
relationships in the many-query or real-time contexts.
},

author = {Grepl, Martin A., Patera, Anthony T.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Parabolic partial differential equations; diffusion equation;
parameter-dependent systems; reduced-basis methods; output bounds;
Galerkin approximation; a posteriori error estimation.; parabolic equations; parameter-dependent systems; Galerkin approximation; a posteriori error estimation; numerical results},

language = {eng},

month = {3},

number = {1},

pages = {157-181},

publisher = {EDP Sciences},

title = {A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations},

url = {http://eudml.org/doc/194255},

volume = {39},

year = {2010},

}

TY - JOUR

AU - Grepl, Martin A.

AU - Patera, Anthony T.

TI - A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 39

IS - 1

SP - 157

EP - 181

AB -
In this paper, we extend the reduced-basis methods and associated a
posteriori error estimators developed earlier for elliptic partial
differential equations to parabolic problems with affine parameter
dependence. The essential new ingredient is the presence of time in the
formulation and solution of the problem – we shall “simply” treat
time as an additional, albeit special, parameter. First, we introduce
the reduced-basis recipe – Galerkin projection onto a space WN
spanned by solutions of the governing partial differential equation at
N selected points in parameter-time space – and develop a new greedy
adaptive procedure to “optimally” construct the parameter-time sample
set. Second, we propose error estimation and adjoint procedures that
provide rigorous and sharp bounds for the error in specific outputs of
interest: the estimates serve a priori to construct our samples,
and a posteriori to confirm fidelity. Third, based on the
assumption of affine parameter dependence, we develop offline-online
computational procedures: in the offline stage, we generate the
reduced-basis space; in the online stage, given a new parameter value, we
calculate the reduced-basis output and associated error bound. The
operation count for the online stage depends only on N (typically
small) and the parametric complexity of the problem; the method is thus
ideally suited for repeated, rapid, reliable evaluation of input-output
relationships in the many-query or real-time contexts.

LA - eng

KW - Parabolic partial differential equations; diffusion equation;
parameter-dependent systems; reduced-basis methods; output bounds;
Galerkin approximation; a posteriori error estimation.; parabolic equations; parameter-dependent systems; Galerkin approximation; a posteriori error estimation; numerical results

UR - http://eudml.org/doc/194255

ER -

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