The focus is on a model reduction framework for parameterized elliptic eigenvalue problems by a reduced basis method. In contrast to the standard single output case, one is interested in approximating several outputs simultaneously, namely a certain number of the smallest eigenvalues. For a fast and reliable evaluation of these input-output relations, we analyze a posteriori error estimators for eigenvalues. Moreover, we present different greedy strategies and study systematically their performance. Special attention needs to be paid to multiple eigenvalues whose appearance is parameter-dependent. Our methods are of particular interest for applications in vibro-acoustics.
Accepté le :
DOI : 10.1051/m2an/2016025
Mots-clés : A posteriori error estimation, eigenvalue problem, finite element method, model reduction, multiple eigenvalues, parameter-dependent partial differential equation, reduced basis method
@article{M2AN_2017__51_2_443_0,
author = {Horger, Thomas and Wohlmuth, Barbara and Dickopf, Thomas},
title = {Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {443--465},
publisher = {EDP-Sciences},
volume = {51},
number = {2},
year = {2017},
doi = {10.1051/m2an/2016025},
mrnumber = {3626406},
zbl = {1362.65121},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2016025/}
}
TY - JOUR AU - Horger, Thomas AU - Wohlmuth, Barbara AU - Dickopf, Thomas TI - Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 443 EP - 465 VL - 51 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016025/ DO - 10.1051/m2an/2016025 LA - en ID - M2AN_2017__51_2_443_0 ER -
%0 Journal Article %A Horger, Thomas %A Wohlmuth, Barbara %A Dickopf, Thomas %T Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 443-465 %V 51 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016025/ %R 10.1051/m2an/2016025 %G en %F M2AN_2017__51_2_443_0
Horger, Thomas; Wohlmuth, Barbara; Dickopf, Thomas. Simultaneous reduced basis approximation of parameterized elliptic eigenvalue problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 443-465. doi : 10.1051/m2an/2016025. http://www.numdam.org/articles/10.1051/m2an/2016025/
and , Finite Element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52 (1989) 275–297. | DOI | MR | Zbl
I. Babuška and J. Osborn, Eigenvalue problems, in vol. 2 of Handbook of Numerical Analysis, edited by P. Ciarlet and J. Lions. (1991) 641–787. | MR | Zbl
, and , Regularity and numerical solution of eigenvalue problems with piecewise analytic data. SIAM J. Numer. Anal. 26 (1989) 1534–1560. | DOI | MR | Zbl
, , , , and , Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. | DOI | MR | Zbl
, , , , and , A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: M2AN 46 (2012) 595–603. | DOI | Numdam | MR | Zbl
, and , Greedy algorithms for high-dimensional eigenvalue problems. Constructive Approximation 40 (2014) 387–423. | DOI | MR | Zbl
, and , Double greedy algorithms: Reduced basis methods for transport dominated problems. ESAIM: M2AN 48 (2014) 623–663. | DOI | Numdam | MR | Zbl
, and ,Greedy algorithms for reduced bases in banach spaces. Constructive Approximation 37 (2013) 455–466. | DOI | MR | Zbl
, and , Adaptive reduced basis methods for nonlinear convection–diffusion equations. Finite Volumes for Complex Applications VI Problems & Perspectives 4 (2011) 369–377. | DOI | MR | Zbl
M. Drohmann, B. Haasdonk, S. Kaulmann and M. Ohlberger, A software framework for reduced basis methods using DUNE-RB and RBmatlab. Advances in DUNE (2012) 77–88.
and , Reduced basis collocation methods for partial differential equations with random coefficients. SIAM/ASA J. Uncertainty Quantification 1 (2013) 192–217. | DOI | MR | Zbl
, , and , Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems. ESAIM: M2AN 50 (2016) 1857–1885. | DOI | Numdam | MR | Zbl
and , On non-coercive variational inequalities. SIAM J. Numer. Anal. 52 (2014) 2250–2271. | DOI | MR | Zbl
N. Gräbner, S. Quraishi, C. Schröder, V. Mehrmann and U. von Wagner, New numerical methods for the complex eigenvalue analysis of disk brake squeal, in EuroBrake 2014 Conference Proceedings (2014).
and , A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN 39 (2005) 157–181. | DOI | Numdam | MR | Zbl
and , Reduced basis method for finite volume approximations of parametrized linear evolution equations. M2AN 42 (2008) 277–302. | DOI | Numdam | MR | Zbl
, and , A reduced basis method for parametrized variational inequalities. SIAM J. Math. Anal. 50 (2012) 2656–2676. | MR | Zbl
, , , and , A new mortar formulation for modeling elastomer bedded structures with modal-analysis in 3D. Adv. Model. Simul. Eng. Sci. 2 (2014) 18. | DOI
, , and , A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Acad. Sci., Paris, Sér. I 345 (2007) 473–478. | DOI | MR | Zbl
, , and , Reduced basis method for the stokes equations in decomposable domains using greedy optimization. ECMI 2014 (2014) 1–7.
and , Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems. Discussiones Mathematicae: Differential Inclusions, Control and Optimization 27 (2007) 95–117. | MR | Zbl
R. Lehoucq, D. Sorensen and C. Yang, Arpack users guide: Solution of large-scale eigenvalue problems by implicitely restarted arnoldi methods. SIAM (1998). | MR | Zbl
, and , A reduced basis element method for the steady stokes problem. ESAIM: M2AN 40 (2006) 529-552. | DOI | Numdam | MR | Zbl
, , , and , Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci., Paris, Sér. I 331 (2000) 153–158. | DOI | MR | Zbl
, and , A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem. C. R. Acad. Sci., Paris, Sér. I 328 (1999) 823–828. | DOI | MR | Zbl
, and , Global a priori convergence theory for reduced basis approximations of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci., Paris, Ser. I 335 (2002) 289–294. | DOI | MR | Zbl
, and , A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437–446. | DOI | MR | Zbl
and , Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455–462. | DOI
A. Patera and G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Version 1.0, Copyright MIT 2006–2007.
, Reduced-basis method for band structure calculations. Phys. Rev. E 76 (2007) 046–704.
G. Pau, Reduced Basis Method for Quantum Models of Crystalline Solids. Ph. D. thesis, Massachusetts Institute of Technology (2007).
, Reduced basis method for simulation of nanodevices. Phys. Rev. B 78 (2008) 155–425.
A. Quarteroni, Numerical Models for Differential Problems, Vol. 8 of MS&A. Springer, Milan, 2nd edition (2014). | MR | Zbl
A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations. An Introduction. Springer (2015). | MR | Zbl
, and , Reduced basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423–445. | DOI | MR | Zbl
and , On the stability of the reduced basis method for stokes equations in parametrized domains. Comput. Methods. Appl. Mech. Engrg. 196 (2007) 1244–1260. | DOI | MR | Zbl
, and , 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. | DOI | MR | Zbl
, and , Reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries: roles of the inf-sup stability constants. Numer. Math. 125 (2013) 115–152. | DOI | MR | Zbl
and , An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comput. 83 (2014) 1599–1615. | DOI | MR | Zbl
K. Urban and B. Wieland, Affine decompositions of parametric stochastic processes for application within reduced basis methods, in 7th Vienna International Conference on Mathematical Modelling, MATHMOD 2012, Vienna (2012) 15–17.
K. Urban, S. Volkwein and O. Zeeb, Greedy sampling using nonlinear optimization, in Vol. 9 of Reduced Order Methods for Modeling and Computational Reductions (2014) 137–157. | MR
, , , and , Component-based reduced basis for parametrized symmetric eigenproblems. Adv. Model. Simul. Eng. Sci. 2 (2015). | DOI
K. Veroy, Reduced-Basis Methods Applied to Problems in Elasticity: Analysis and Applications. Ph. D. thesis, Massachusetts Institute of Technology (2003).
, A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement. ESAIM: M2AN 50 (2016) 163–185. | DOI | Numdam | MR | Zbl
and , The reduced basis method for an elastic buckling problem. PAMM, Proc. Appl. Math. Mech. 13 (2013) 439–440. | DOI
Cité par Sources :
