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

Abstract

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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.


How to cite

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Chen, 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 -

References

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  1. M. Barrault, N.C. Nguyen, Y. Maday and A.T. Patera. An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris Ser. I Math.339 (2004) 667–672.  Zbl1061.65118
  2. A. Barret and G. Reddien, On the reduced basis method. Z. Angew. Math. Mech.75 (1995) 543–549.  Zbl0832.65047
  3. T. Bui-Thanh, K. Willcox and O. Ghattas, Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput.30 (2008) 3270–3288.  Zbl1196.37127
  4. Y. Chen, J.S. Hesthaven, Y. Maday and J. Rodríguez, A monotonic evaluation of lower bounds for Inf-Sup stability constants in the frame of reduced basis approximations. C. R. Acad. Sci. Paris Ser. I Math.346 (2008) 1295–1300.  Zbl1152.65109
  5. M.A. Grepl, Y. Maday, N.C. Nguyen and A.T. Patera. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN41 (2007) 575–605.  Zbl1142.65078
  6. M.D. Gunzburger, Finite element methods for viscous incompressible flows. Academic Press (1989).  Zbl0697.76031
  7. J.S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer Texts in Applied Mathematics54. Springer Verlag, New York (2008).  Zbl1134.65068
  8. D.B.P. Huynh, G. Rozza, S. Sen and A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Acad. Sci. Paris Ser. I Math.345 (2007) 473–478.  Zbl1127.65086
  9. L. Machiels, Y. Maday, I.B. Oliveira, A.T. Patera and D. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Ser. I Math.331 (2000) 153–158.  Zbl0960.65063
  10. Y Maday, Reduced Basis Method for the Rapid and Reliable Solution of Partial Differential Equations, in Proceeding ICM Madrid (2006).  Zbl1100.65079
  11. Y. Maday, A.T. Patera and D.V. Rovas, A blackbox reduced-basis output bound method for noncoercive linear problems, in Nonlinear Partial Differential Equations and Their Applications, D. Cioranescu and J.L. Lions Eds., Collège de France SeminarXIV, Elsevier Science B.V. (2002) 533–569.  Zbl1006.65124
  12. D.A. Nagy, Modal representation of geometrically nonlinear behaviour by the finite element method. Comput. Struct.10 (1979) 683–688.  Zbl0406.73071
  13. N.C. Nguyen, K. Veroy and A.T. Patera. Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling, S. Yip Ed., Springer (2005) 1523–1558.  
  14. A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA Journal18 (1980) 455–462.  
  15. J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput.10 (1989) 777–786.  Zbl0672.76034
  16. C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera and G. Turinici, Reliable realtime solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engineering124 (2002) 70–80.  
  17. G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics. Arch. Comput. Methods Eng.15 (2008) 229–275.  Zbl1304.65251
  18. S. Sen, K. Veroy, D.B.P. Huynh, S. Deparis, N.C. Nguyen and A.T. Patera, “Natural norm” a posteriori error estimators for reduced basis approximations. J. Comput. Phys.217 (2006) 37–62.  Zbl1100.65094

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