Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

Martin A. Grepl; Yvon Maday; Ngoc C. Nguyen; Anthony T. Patera

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 3, page 575-605
  • ISSN: 0764-583X

Abstract

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In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numerical results are presented to assess our approach.

How to cite

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Grepl, Martin A., et al. "Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis 41.3 (2007): 575-605. <http://eudml.org/doc/250042>.

@article{Grepl2007,
abstract = { In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numerical results are presented to assess our approach. },
author = {Grepl, Martin A., Maday, Yvon, Nguyen, Ngoc C., Patera, Anthony T.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Reduced-basis methods; parametrized PDEs; non-affine parameter dependence; offine-online procedures; elliptic PDEs; parabolic PDEs; nonlinear PDEs.; Galerkin method; numerical results; reduced-basis methods; offline-online procedures; nonlinear PDEs},
language = {eng},
month = {8},
number = {3},
pages = {575-605},
publisher = {EDP Sciences},
title = {Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations},
url = {http://eudml.org/doc/250042},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Grepl, Martin A.
AU - Maday, Yvon
AU - Nguyen, Ngoc C.
AU - Patera, Anthony T.
TI - Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/8//
PB - EDP Sciences
VL - 41
IS - 3
SP - 575
EP - 605
AB - In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numerical results are presented to assess our approach.
LA - eng
KW - Reduced-basis methods; parametrized PDEs; non-affine parameter dependence; offine-online procedures; elliptic PDEs; parabolic PDEs; nonlinear PDEs.; Galerkin method; numerical results; reduced-basis methods; offline-online procedures; nonlinear PDEs
UR - http://eudml.org/doc/250042
ER -

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Citations in EuDML Documents

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  1. Yanlai Chen, Jan S. Hesthaven, Yvon Maday, Jerónimo Rodríguez, Improved successive constraint method based error estimate for reduced basis approximation of 2D Maxwell's problem
  2. Karl Kunisch, Stefan Volkwein, Optimal snapshot location for computing POD basis functions
  3. Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
  4. Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
  5. Peng Chen, Alfio Quarteroni, Gianluigi Rozza, A weighted empirical interpolation method: a priori convergence analysis and applications
  6. Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
  7. Alexandre Janon, Maëlle Nodet, Clémentine Prieur, Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values
  8. Jan S. Hesthaven, Benjamin Stamm, Shun Zhang, Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods
  9. Mark Kärcher, Martin A. Grepl, A certified reduced basis method for parametrized elliptic optimal control problems

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