Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1857-1885.

We develop a new reduced basis (RB) method for the rapid and reliable approximation of parametrized elliptic eigenvalue problems. The method hinges upon dual weighted residual type a posteriori error indicators which estimate, for any value of the parameters, the error between the high-fidelity finite element approximation of the first eigenpair and the corresponding reduced basis approximation. The proposed error estimators are exploited not only to certify the RB approximation with respect to the high-fidelity one, but also to set up a greedy algorithm for the offline construction of a reduced basis space. Several numerical experiments show the overall validity of the proposed RB approach.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016009
Classification : 65M15, 65N25, 65N30, 78M34
Mots clés : Parametrized eigenvalue problems, reduced basis method, a posteriori error estimation, greedy algorithm, dual weighted residual
Fumagalli, Ivan 1 ; Manzoni, Andrea 2 ; Parolini, Nicola 1 ; Verani, Marco 1

1 MOX – Modellistica e Calcolo Scientifico, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy.
2 CMCS-MATHICSE-SB, Ecole Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland.
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     title = {Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems},
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Fumagalli, Ivan; Manzoni, Andrea; Parolini, Nicola; Verani, Marco. Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1857-1885. doi : 10.1051/m2an/2016009. http://archive.numdam.org/articles/10.1051/m2an/2016009/

A. Ammar and F. Chinesta, Circumventing curse of dimensionality in the solution of highly multidimensional models encountered in quantum mechanics using meshfree finite sums decomposition. In Meshfree Methods for Partial Differential Equations IV, edited by M. Griebel and M. Schweitzer. Vol. 65 of Lect. Notes Comput. Sci. Eng. Springer, Berlin, Heidelberg (2008) 1–17. | Zbl

M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004) 667–672. | DOI | MR | Zbl

R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1–102. | DOI | MR | Zbl

L. Beirão Da Veiga and M. Verani, A posteriori boundary control for FEM approximation of elliptic eigenvalue problems. Numer. Methods Partial Differential Equations 28 (2012) 369–388. | DOI | MR | Zbl

P. Binev, A. Cohen, W. Dahmen, R. Devore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. | DOI | MR | Zbl

A. Buffa, Y. Maday, A.T. Patera, C. Prud’Homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: M2AN 46 (2012) 595–603. | DOI | Numdam | MR | Zbl

E. Cancès, V. Ehrlacher and T. Lelièvre, Greedy algorithms for high-dimensional eigenvalue problems. Constructive Approximation 40 (2013) 387–423. | DOI | MR | Zbl

A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs. Acta Numerica, 1–159 (2015). | MR

L. Dedè, Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems. SIAM J. Sci. Comput. 32 (2010) 997–1019. | DOI | MR | Zbl

T. Dickopf, T. Horger and B. Wohlmuth, Simultaneous reduced basis approximation of parameterized eigenvalue problems. Preprint (2015). | arXiv | Numdam | MR

D.C. Dobson and F. Santosa, Optimal localization of eigenfunctions in an inhomogeneous medium. SIAM J. Appl. Math. 64 (2004) 762–774. | DOI | MR | Zbl

L. Evans, Partial differential equations. Vol. 19 of Graduate Studies in Mathematics, 2nd edition. American Mathematical Society, Providence, RI (2010). | MR | Zbl

M. Fares, J. Hesthaven, Y. Maday and B. Stamm, The reduced basis method for the electric field integral equation. J. Comp. Phys. 230 (2011) 5532–5555. | DOI | MR | Zbl

G.H. Golub and C.F. Van Loan, Matrix Computations, 4th edition. The John Hopkins University Press, Baltimore (2013). | MR | Zbl

J.S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics. Springer (2016). | MR

V. Heuveline and R. Rannacher, A posteriori error control for finite approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15 (2001) 107–138. | DOI | MR | Zbl

M. Hintermüller, C.-Y. Kao and A. Laurain, Principal eigenvalue minimization for an elliptic problem with indefinite weight and Robin boundary conditions. Appl. Math. Optim. 65 (2012) 111–146. | DOI | MR | Zbl

D.B.P. Huynh, D.J. Knezevic and A.T. Patera. A static condensation reduced basis element method: approximation and a posteriori error estimation. ESAIM: M2AN 47 (2013) 213–251. | DOI | Numdam | MR | Zbl

T. Lassila and G. Rozza, Parametric free-form shape design with PDE models and reduced basis method. Comput. Meth. Appl. Mech. Engrg. 199 (2010) 1583–1592. | DOI | MR | Zbl

T. Lassila, A. Manzoni, A. Quarteroni and G. Rozza, Model order reduction in fluid dynamics: challenges and perspectives. In Reduced order methods for modeling and computational reduction, edited by A. Quarteroni and G. Rozza. Vol. 9. Springer, MS&A Series (2013) 235–274. | MR

L. Machiels, Y. Maday, I. Oliveira, A.T. Patera and D. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 153–158. | DOI | MR | Zbl

Y. Maday, A.T. Patera and J. Peraire, A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem. C. R. Acad. Sci. Paris, Série I 327 (1998) 823–828. | MR | Zbl

A. Manzoni, An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows. ESAIM: M2AN 48 (2014) 1199–1226. | DOI | Numdam | MR | Zbl

A. Manzoni, A. Quarteroni and G. Rozza, Shape optimization of cardiovascular geometries by reduced basis methods and free-form deformation techniques. Int. J. Numer. Meth. Fluids 70 (2012) 646–670. | DOI | MR | Zbl

A. Manzoni and F. Negri, Heuristic strategies for the approximation of stability factors in quadratically nonlinear parametrized PDEs. Adv. Comput. Math. 41 (2015) 1255–1288. | DOI | MR | Zbl

J.A. Méndez-Bermùdez and F.M. Izrailev, Transverse localization in quasi-1d corrugated waveguides (2008) 1376–1378.

J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Paris; Academia, Prague (1967). | MR | Zbl

F. Negri, G. Rozza, A. Manzoni and A. Quarteroni, Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35 (2013) A2316–A2340. | DOI | MR | Zbl

N.C. Nguyen, K. Veroy and A.T. Patera, Certified real-time solution of parametrized partial differential equations. In Handbook of Materials Modeling, edited by S. Yip (2005) 1523–1558.

S.J. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints i. frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2001) 272–288. | DOI | MR | Zbl

G.S.H. Pau, Reduced Basis Method for Quantum Models of Crystalline Solids. Ph.D. thesis, Massachusetts Institute of Technology (2007).

C. Prud’Homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: reduced-basis output bounds methods. J. Fluids. Engng. 124 (2002) 70–80. | DOI

A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations in industrial applications. J. Math. Ind. 1 (2011). | MR | Zbl

A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations. An Introduction. Vol. 92 of Unitext Series. Springer (2016). | MR | Zbl

D.V. Rovas, Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. Ph.D. thesis, Massachusetts Institute of Technology (2003).

G. Rozza, D.B.P. Huynh and A. Manzoni, 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

B. Sapoval, O. Haeberlé and S. Russ. Acoustical properties of irregular and fractal cavities. Acoust. Soc. Am. J. 102 (1997) 2014–2019. | DOI

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