# Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell's problem

Yanlai Chen; Jan S. Hesthaven; Yvon Maday; Jerónimo Rodríguez

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 6, page 1099-1116
- ISSN: 0764-583X

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topChen, Yanlai, et al. "Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell's problem." ESAIM: Mathematical Modelling and Numerical Analysis 43.6 (2009): 1099-1116. <http://eudml.org/doc/250581>.

@article{Chen2009,

abstract = {
In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-sup
stability constants is essential. In [Huynh et al., C. R. Acad.
Sci. Paris Ser. I Math.345 (2007) 473–478], the authors presented an efficient
method, compatible with an off-line/on-line strategy, where the on-line computation is reduced to
minimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient and robust due to two related properties: (i) the lower bound is
obtained by a monotonic process with respect to the size of the nested sets; (ii) less eigen-problems need to be solved. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of
the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.
},

author = {Chen, Yanlai, Hesthaven, Jan S., Maday, Yvon, Rodríguez, Jerónimo},

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

keywords = {Reduced basis method; successive constraint method; inf-sup constant; a posteriori error estimate; Maxwell's equation; discontinuous Galerkin method.; reduced basis method; a posteriori error estimate; discontinuous Galerkin method},

language = {eng},

month = {8},

number = {6},

pages = {1099-1116},

publisher = {EDP Sciences},

title = {Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell's problem},

url = {http://eudml.org/doc/250581},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Chen, Yanlai

AU - Hesthaven, Jan S.

AU - Maday, Yvon

AU - Rodríguez, Jerónimo

TI - Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell's problem

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2009/8//

PB - EDP Sciences

VL - 43

IS - 6

SP - 1099

EP - 1116

AB -
In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-sup
stability constants is essential. In [Huynh et al., C. R. Acad.
Sci. Paris Ser. I Math.345 (2007) 473–478], the authors presented an efficient
method, compatible with an off-line/on-line strategy, where the on-line computation is reduced to
minimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient and robust due to two related properties: (i) the lower bound is
obtained by a monotonic process with respect to the size of the nested sets; (ii) less eigen-problems need to be solved. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of
the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.

LA - eng

KW - Reduced basis method; successive constraint method; inf-sup constant; a posteriori error estimate; Maxwell's equation; discontinuous Galerkin method.; reduced basis method; a posteriori error estimate; discontinuous Galerkin method

UR - http://eudml.org/doc/250581

ER -

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## Citations in EuDML Documents

top- Fabien Casenave, Alexandre Ern, Tony Lelièvre, Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method
- Jan S. Hesthaven, Benjamin Stamm, Shun Zhang, Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods

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