The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.
Mots-clés : tight surrogates, stable variational formulations, saddle point problems, double greedy schemes, greedy stabilization, rate-optimality, transport equations, convection-diffusion equations
@article{M2AN_2014__48_3_623_0,
author = {Dahmen, Wolfgang and Plesken, Christian and Welper, Gerrit},
title = {Double greedy algorithms: {Reduced} basis methods for transport dominated problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {623--663},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/m2an/2013103},
mrnumber = {3177860},
zbl = {1291.65339},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013103/}
}
TY - JOUR AU - Dahmen, Wolfgang AU - Plesken, Christian AU - Welper, Gerrit TI - Double greedy algorithms: Reduced basis methods for transport dominated problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 623 EP - 663 VL - 48 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013103/ DO - 10.1051/m2an/2013103 LA - en ID - M2AN_2014__48_3_623_0 ER -
%0 Journal Article %A Dahmen, Wolfgang %A Plesken, Christian %A Welper, Gerrit %T Double greedy algorithms: Reduced basis methods for transport dominated problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 623-663 %V 48 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013103/ %R 10.1051/m2an/2013103 %G en %F M2AN_2014__48_3_623_0
Dahmen, Wolfgang; Plesken, Christian; Welper, Gerrit. Double greedy algorithms: Reduced basis methods for transport dominated problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 623-663. doi : 10.1051/m2an/2013103. http://archive.numdam.org/articles/10.1051/m2an/2013103/
[1] , , , , and , Convergence Rates for Greedy Algorithms in Reduced Basis Methods. SIAM J. Math. Anal. 43 (2011) 1457-1472. | MR | Zbl
[2] and , Mixed and Hybrid Finite Element Methods, in vol. 15 of Springer Ser. Comput. Math. Springer-Verlag (1991). | MR | Zbl
[3] , , , and , A Priori convergence of the greedy algorithm for the parameterized reduced basis. ESAIM: M2AN 46 (2012) 595-603. | Numdam | MR | Zbl
[4] , , and , Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524-2550. | MR | Zbl
[5] , and , Adaptivity and Variational Stabilization for Convection-Diffusion Equations. ESAIM: M2AN 46 (2012) 1247-1273. | Numdam | MR | Zbl
[6] , Parameter dependent transport equations, in Workshop J.L.L.-SMP: Reduced Basis Methods in High Dimensions. Available at http://www.ljll.math.upmc.fr/fr/archives/actualites/2011/workshop˙ljll˙smp˙rbihd.html
[7] , , and , Adaptive Petrov−Galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50 (2012) 2420-2445. | MR | Zbl
[8] and , A class of discontinuous Petrov−Galerkin Methods I: The transport equation. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1558-1572. | MR | Zbl
[9] and , A class of discontinuous Petrov−Galerkin methods. Part II: Optimal test functions. Numer. Methods for Partial Differ. Equ. 27 (2011) 70-105. | MR | Zbl
[10] , Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM J. Numer. Anal. 46 (2008) 2039-2067. | MR | Zbl
[11] , and , Greedy algorithms for reduced bases in Banach spaces, Constructive Approximation 37 (2013) 455-466. | MR | Zbl
[12] and , Theory and practice of finite elements. Springer (2004). | MR | Zbl
[13] and , Certified reduced basis methods for parametrized saddle point problems, preprint (2012). To appear in SIAM J. Sci. Comput. | MR | Zbl
[14] and , Reduced basis a posteriori error bounds for the Stokes equations in parameterized domains: A penalty approach. M3AS: Math. Models Methods Appl. Sci. 21 (2011) 2103-2134. | MR
[15] , Certified Reduced Basis Methods for Nonaffine Linear Time-Varying and Nonlinear Parabolic Partial Differential Equations. M3AS: Math. Models Methods Appl. Sci. 22 (2012) 40. | MR | Zbl
[16] and , A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential Equations. ESAIM: M2AN 39 (2005) 157-181. | EuDML | Numdam | MR | Zbl
[17] , Convergence rates for the POD-greedy method. ESAIM: M2AN 47 (2013) 859-873. | EuDML | Numdam | MR | Zbl
[18] and , Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods. SIAM J. Numer. Anal. 45 (2007) 539-557. | MR | Zbl
[19] , , and , Numerical methods in multidimensional radiative transfer, Springer (2009). | MR | Zbl
[20] , and , Constructive approximation: Advanced problems, vol. 304. Springer Grundlehren, Berlin (1996). | MR | Zbl
[21] , and , A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437-446. | MR | Zbl
[22] , , and , First-order system ℒℒ∗ (FOSLL)∗ for general scalar elliptic problems in the plane. SIAM J. Numer. Anal. 43 (2005) 2098-2120. | MR | Zbl
[23] , and , Reduced basis approximation for the time-dependent viscous Burgers' equation. Calcolo 46 (2009) 157-185. | MR | Zbl
[24] and , An improved error bound for reduced basis approximation of linear parabolic problems, submitted to Mathematics of Computation (in press 2013).
[25] and , Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations, Version 1.0, Copyright MIT 2006-2007, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering.
[26] , and , Robust Numerical Methods for Singularly Perturbed Differential Equations, in vol. 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd Edition (2008). | MR | Zbl
[27] G. Rozza and D.B.P. Huynh and A. Manzoni, Reduced basis approximation and a posteriori error estiamtion for Stokes flows in parametrized geometries: roles of the inf-sup stability constants, Numer. Math. DOI: 10.1007/s00211-013-0534-8. | MR
[28] , and , Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15 (2008) 229-275. | MR
[29] and , On the stability of reduced basis techniques for Stokes equations in parametrized domains. Comput. Methods Appl. Mechanics Engrg. 196 (2007) 1244-1260. | MR | Zbl
[30] , A uniform analysis of non-symmetric and coercive linear operators. SIAM J. Math. Anal. 36 (2005) 2033-2048. | MR | Zbl
[31] , On Forward and Inverse Models in Optical Tomography, Ph.D. Thesis. RWTH Aachen (2011).
[32] , , , , and , Natural norm a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217 (2006) 37-62. | MR | Zbl
[33] , Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766-1782. | MR | Zbl
[34] , Infinite dimensional stabilization of convection-dominated problems, Ph.D. Thesis. RWTH Aachen (2012).
[35] , , , , and , A class of discontinuous Petrov−Galerkin methods. Part IV: Wave propagation. J. Comput. Phys. 230 (2011) 2406-2432. | MR
Cité par Sources :
