Double greedy algorithms: Reduced basis methods for transport dominated problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 3, p. 623-663

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.

DOI : https://doi.org/10.1051/m2an/2013103
Classification:  65J10,  65N12,  65N15,  35B30
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {3},
     year = {2014},
     pages = {623-663},
     doi = {10.1051/m2an/2013103},
     zbl = {1291.65339},
     mrnumber = {3177860},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 3, pp. 623-663. doi : 10.1051/m2an/2013103. http://www.numdam.org/item/M2AN_2014__48_3_623_0/

[1] 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. | MR 2821591 | Zbl 1229.65193

[2] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, in vol. 15 of Springer Ser. Comput. Math. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[3] A. Buffa, Y. Maday, A.T. Patera, C. Prud'Homme and G. Turinici, A Priori convergence of the greedy algorithm for the parameterized reduced basis. ESAIM: M2AN 46 (2012) 595-603. | Numdam | MR 2877366 | Zbl 1272.65084

[4] J.M. Cascon, C. Kreuzer, R.H. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524-2550. | MR 2421046 | Zbl 1176.65122

[5] A. Cohen, W. Dahmen and G. Welper, Adaptivity and Variational Stabilization for Convection-Diffusion Equations. ESAIM: M2AN 46 (2012) 1247-1273. | Numdam | MR 2916380 | Zbl 1270.65065

[6] W. Dahmen, 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] W. Dahmen, C. Huang, C. Schwab and G. Welper, Adaptive Petrov−Galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50 (2012) 2420-2445. | MR 3022225 | Zbl 1260.65091

[8] L.F. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov−Galerkin Methods I: The transport equation. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1558-1572. | MR 2630162 | Zbl 1231.76142

[9] L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov−Galerkin methods. Part II: Optimal test functions. Numer. Methods for Partial Differ. Equ. 27 (2011) 70-105. | MR 2743600 | Zbl 1208.65164

[10] S. Deparis, 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 2399407 | Zbl 1177.35148

[11] R. Devore, G. Petrova and P. Wojtaszczyk, Greedy algorithms for reduced bases in Banach spaces, Constructive Approximation 37 (2013) 455-466. | MR 3054611 | Zbl 1276.41021

[12] A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer (2004). | MR 2050138 | Zbl 1059.65103

[13] A. Gerner and K. Veroy-Grepl, Certified reduced basis methods for parametrized saddle point problems, preprint (2012). To appear in SIAM J. Sci. Comput. | MR 3023727 | Zbl 1255.76024

[14] A.-L. Gerner and K. Veroy, 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 2851708 | Zbl pre06045578

[15] M.A. Grepl, 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 2890453 | Zbl 1241.35116

[16] M.A. Grepl and A.T. Patera, A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential Equations. ESAIM: M2AN 39 (2005) 157-181. | Numdam | MR 2136204 | Zbl 1079.65096

[17] B. Haasdonk, Convergence rates for the POD-greedy method. ESAIM: M2AN 47 (2013) 859-873. | Numdam | MR 3056412 | Zbl 1277.65074

[18] T. Hughes and G. Sangalli, Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods. SIAM J. Numer. Anal. 45 (2007) 539-557. | MR 2300286 | Zbl 1152.65111

[19] G. Kanschat, E. Meinköhn, R. Rannacher and R. Wehrse, Numerical methods in multidimensional radiative transfer, Springer (2009). | MR 2530844 | Zbl 1155.65004

[20] G.G. Lorentz, M. Von Golitschek and Yu. Makovoz, Constructive approximation: Advanced problems, vol. 304. Springer Grundlehren, Berlin (1996). | MR 1393437 | Zbl 0910.41001

[21] Y. Maday, A.T. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437-446. | MR 1910581 | Zbl 1014.65115

[22] T. Manteuffel, S. Mccormick, J. Ruge and J.G. Schmidt, First-order system ℒℒ∗ (FOSLL)∗ for general scalar elliptic problems in the plane. SIAM J. Numer. Anal. 43 (2005) 2098-2120. | MR 2192333 | Zbl 1103.65117

[23] N.-C. Nguyen, G. Rozza and A.T. Patera, Reduced basis approximation for the time-dependent viscous Burgers' equation. Calcolo 46 (2009) 157-185. | MR 2533748 | Zbl 1178.65109

[24] T. Patera and K. Urban, An improved error bound for reduced basis approximation of linear parabolic problems, submitted to Mathematics of Computation (in press 2013). | Zbl pre06300438

[25] A.T. Patera and G. Rozza, 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. | Zbl pre05344486

[26] H.-J. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, in vol. 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd Edition (2008). | MR 2454024 | Zbl 1155.65087

[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 3090657 | Zbl pre06210474

[28] 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. Arch. Comput. Methods Eng. 15 (2008) 229-275. | MR 2430350 | Zbl pre05344486

[29] G. Rozza and K. Veroy, On the stability of reduced basis techniques for Stokes equations in parametrized domains. Comput. Methods Appl. Mechanics Engrg. 196 (2007) 1244-1260. | MR 2281777 | Zbl 1173.76352

[30] G. Sangalli, A uniform analysis of non-symmetric and coercive linear operators. SIAM J. Math. Anal. 36 (2005) 2033-2048. | MR 2178232 | Zbl 1114.35060

[31] M. Schlottbom, On Forward and Inverse Models in Optical Tomography, Ph.D. Thesis. RWTH Aachen (2011).

[32] S. Sen, K. Veroy, D.B.P. Huynh, S. Deparis, N.C. Nguyn and A.T. Patera, Natural norm a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217 (2006) 37-62. | MR 2250524 | Zbl 1100.65094

[33] R. Verfürth, Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005) 1766-1782. | MR 2182149 | Zbl 1099.65100

[34] G. Welper, Infinite dimensional stabilization of convection-dominated problems, Ph.D. Thesis. RWTH Aachen (2012).

[35] J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo and V. Calo, A class of discontinuous Petrov−Galerkin methods. Part IV: Wave propagation. J. Comput. Phys. 230 (2011) 2406-2432. | MR 2772923 | Zbl pre05909482