The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539-542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the empirical interpolation method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.
Mots-clés : reduced basis method, a posteriori error bound, round-off errors, boundary element method, empirical interpolation method, acoustics
@article{M2AN_2014__48_1_207_0, author = {Casenave, Fabien and Ern, Alexandre and Leli\`evre, Tony}, title = {Accurate and online-efficient evaluation of the \protect\emph{a posteriori }error bound in the reduced basis method}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {207--229}, publisher = {EDP-Sciences}, volume = {48}, number = {1}, year = {2014}, doi = {10.1051/m2an/2013097}, zbl = {1288.65157}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2013097/} }
TY - JOUR AU - Casenave, Fabien AU - Ern, Alexandre AU - Lelièvre, Tony TI - Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 207 EP - 229 VL - 48 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013097/ DO - 10.1051/m2an/2013097 LA - en ID - M2AN_2014__48_1_207_0 ER -
%0 Journal Article %A Casenave, Fabien %A Ern, Alexandre %A Lelièvre, Tony %T Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 207-229 %V 48 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013097/ %R 10.1051/m2an/2013097 %G en %F M2AN_2014__48_1_207_0
Casenave, Fabien; Ern, Alexandre; Lelièvre, Tony. Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 1, pp. 207-229. doi : 10.1051/m2an/2013097. http://archive.numdam.org/articles/10.1051/m2an/2013097/
[1] Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43 (2002) 9-44. | MR | Zbl
and ,[2] Sur le problème de Helmholtz. Rendiconti del Seminario matematico della Università e Politecnico di Torino (2007) 427-450. | MR | Zbl
,[3] An ‘empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004) 667-672. | MR | Zbl
, , and ,[4] Loss and recapture of orthogonality in the modified Gram-Schmidt algorithm. SIAM J. Matrix Anal. Appl. 13 (1992) 176-190. | MR | Zbl
and ,[5] Mathematical modelling and numerical simulation in materials science. Ph.D. thesis, Université Paris-Est (2009).
,[6] Regularized combined field integral equations. Numer. Math. 100 (2005) 1-19. | MR | Zbl
and ,[7] Numerical Analysis. PWS Publishing Company (1993). | Zbl
and ,[8] Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math. Models Methods Appl. Sci. 21 (2011) 2433-2467. | MR | Zbl
, and ,[9] Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539-542. | MR | Zbl
,[10] Ph.D. thesis, in preparation (2013).
,[11] A multiscale problem in thermal science. ESAIM: Proceedings 38 (2012) 202-219.
, and ,[12] An introduction to the proper orthogonal decomposition. Curr. Sci. 78 (2000) 808-817.
,[13] Certified reduced basis method for electromagnetic scattering and radar cross section estimation. Technical Report 2011-28, Scientific Computing Group, Brown University, Providence, RI, USA (2011). | MR | Zbl
, , , and ,[14] Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell's problem. ESAIM: M2AN 43 (2009) 1099-1116. | Numdam | MR | Zbl
, , and ,[15] A short review on model order reduction based on proper generalized decomposition. Arch. Comput. Methods Eng. 18 (2011) 395-404.
, and ,[16] 3S harmonique : Justifications Mathématiques : Partie I. Technical report, EADS CCR (2001).
, ,[17] 3S, Justifications Mathématiques : Partie II, présence d'un écoulement uniforme. Technical report, EADS CCR (2002).
, ,[18] Theory and Practice of Finite Elements, in vol. 159 of Applied Mathematical Sciences. Springer (2004). | MR | Zbl
and ,[19] The reduced basis method for the electric field integral equation. J. Comput. Phys. 230 (2011) 5532-5555. | MR | Zbl
, , and ,[20] When modified Gram-Schmidt generates a well-conditioned set of vectors. IMA J. Numer. Anal. 22 (2002) 521-528. | MR | Zbl
and ,[21] What every computer scientist should know about floating point arithmetic. ACM Computing Surveys 23 (1991) 5-48.
,[22] Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press (1996). | MR | Zbl
and ,[23] Reduction of stiffness and mass matrices. AIAA J. 3 (1965) 380.
,[24] Coercive combined field integral equations. J. Numer. Math. 11 (2003) 115-134. | MR | Zbl
,[25] Stable FEM-BEM Coupling for Helmholtz Transmission Problems. ETH, Seminar für Angewandte Mathematik (2005). | MR | Zbl
and ,[26] Boundary Element Methods: Foundation and Error Analysis. John Wiley & Sons, Ltd (2004).
and ,[27] A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. Acad. Sci. Paris 345 (2007) 473-478. | MR | Zbl
, , and ,[28] Compensated Horner scheme. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2006).
, and ,[29] Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Math. Acad. Sci. Paris 331 (2000) 153-158. | MR | Zbl
, , , and ,[30] A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2008) 383-404. | Zbl
, , and ,[31] Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000). | MR | Zbl
,[32] Generalized spectral decomposition for stochastic nonlinear problems. J. Comput. Phys. 228 (2009) 202-235. | MR | Zbl
and ,[33] Private communication (2012).
,[34] Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering (2007).
and ,[35] Dynamic condensation. AIAA J. 22 (1984) 724-727.
,[36] Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Eng. 124 (2002) 70-80.
, , , , , and ,[37] Boundary Element Methods. Springer Series in Computational Mathematics. Springer (2010). | MR | Zbl
and ,[38] Sparse polynomial approximation in finite fields. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, STOC '01. ACM, New York, USA (2001) 209-215. | MR
,[39] Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47 (2005) 773-788. | MR | Zbl
and ,[40] Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. C. R. Math. Acad. Sci. Paris 337 (2003) 619-624. | MR | Zbl
, and ,Cité par Sources :