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

Abstract

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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.

How to cite

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Grepl, 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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Citations in EuDML Documents

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  1. Karl Kunisch, Stefan Volkwein, Proper orthogonal decomposition for optimality systems
  2. Martin A. Grepl, Yvon Maday, Ngoc C. Nguyen, Anthony T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations
  3. Ekkehard W. Sachs, Matthias Schu, A priori error estimates for reduced order models in finance
  4. Alexandre Janon, Maëlle Nodet, Clémentine Prieur, Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values
  5. Wolfgang Dahmen, Christian Plesken, Gerrit Welper, Double greedy algorithms: Reduced basis methods for transport dominated problems
  6. Jan S. Hesthaven, Benjamin Stamm, Shun Zhang, Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods
  7. Mark Kärcher, Martin A. Grepl, A certified reduced basis method for parametrized elliptic optimal control problems
  8. Mark Kärcher, Martin A. Grepl, A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems
  9. Bernard Haasdonk, Mario Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations

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