Reduced basis method for finite volume approximations of parametrized linear evolution equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 2, p. 277-302

The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests. We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.

DOI : https://doi.org/10.1051/m2an:2008001
Classification:  76M12,  65M15,  35L90,  35K90,  76R99
Keywords: model reduction, reduced basis methods, finite volume methods, a-posteriori error estimates
@article{M2AN_2008__42_2_277_0,
     author = {Haasdonk, Bernard and Ohlberger, Mario},
     title = {Reduced basis method for finite volume approximations of parametrized linear evolution equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {2},
     year = {2008},
     pages = {277-302},
     doi = {10.1051/m2an:2008001},
     zbl = {pre05262088},
     mrnumber = {2405149},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2008__42_2_277_0}
}
Haasdonk, Bernard; Ohlberger, Mario. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 2, pp. 277-302. doi : 10.1051/m2an:2008001. http://www.numdam.org/item/M2AN_2008__42_2_277_0/

[1] B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA J. 16 (1978) 525-528.

[2] D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749-1779. | MR 1885715 | Zbl 1008.65080

[3] C. Bardos, A.Y. Leroux and J.C. Nedelec, First order quasilinear equations with boundary conditions. Comm. Partial Diff. Eq. 4 (1979) 1017-1034. | MR 542510 | Zbl 0418.35024

[4] 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. Acad. Sci. Paris Ser. I Math. 339 (2004) 667-672. | MR 2103208 | Zbl 1061.65118

[5] T. Barth and M. Ohlberger, Finite volume methods: Foundation and analysis, in Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes Eds., John Wiley & Sons (2004).

[6] J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147 (1999) 269-361. | MR 1709116 | Zbl 0935.35056

[7] B. Cockburn, Discontinuous Galerkin methods for computational fluid dynamics, in Encyclopedia of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes Eds., John Wiley & Sons (2004). | MR 2015264

[8] B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001) 173-261. | MR 1873283 | Zbl 1065.76135

[9] Y. Coudiere, J.P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493-516. | Numdam | MR 1713235 | Zbl 0937.65116

[10] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, volume VII, North-Holland, Amsterdam (2000) 713-1020. | MR 1804748 | Zbl 0981.65095

[11] R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41-82. | MR 1917365 | Zbl 1005.65099

[12] R. Eymard, T. Gallouët and R. Herbin, A cell-centred finite volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal. 26 (2006) 326-353. | MR 2218636 | Zbl 1093.65110

[13] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996). | MR 1410987 | Zbl 0860.65075

[14] M.A. Grepl, Reduced-basis Approximations and a Posteriori Error Estimation for Parabolic Partial Differential Equations. Ph.D. thesis, Massachusetts Institute of Technology, USA (2005).

[15] 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

[16] P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées 22 [Research in Applied Mathematics]. Masson, Paris (1992). | MR 1173209 | Zbl 0766.35001

[17] R. Herbin and M. Ohlberger, A posteriori error estimate for finite volume approximations of convection diffusion problems, in Proc. 3rd Int. Symp. on Finite Volumes for Complex Applications - Problems and Perspectives (2002) 753-760. | MR 2009001 | Zbl 1060.65103

[18] R.L. Higdon, Initial-boundary value problems for linear hyperbolic systems. SIAM Rev. 28 (1986) 177-217. | MR 839822 | Zbl 0603.35061

[19] M.-J. Jasor and L. Lévi, Singular perturbations for a class of degenerate parabolic equations with mixed Dirichlet-Neumann boundary conditions. Ann. Math. Blaise Pascal 10 (2003) 269-296. | Numdam | MR 2031272 | Zbl 1065.35158

[20] D. Kröner, Numerical Schemes for Conservation Laws. John Wiley & Sons and Teubner (1997). | MR 1437144 | Zbl 0872.76001

[21] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). | MR 1925043 | Zbl 1010.65040

[22] L. Machiels, Y. Maday, I.B. Oliveira, A. Patera and D.V. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Ser. I Math. 331 (2000) 153-158. | MR 1781533 | Zbl 0960.65063

[23] M. Mangold and M. Sheng, Nonlinear model reduction of a 2D MCFC model with internal reforming. Fuel Cells 4 (2004) 68-77.

[24] B.C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Automat. Control AC-26 (1981) 17-32. | MR 609248 | Zbl 0464.93022

[25] N.C. Nguyen, K. Veroy and A.T. Patera, Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling, S. Yip Ed., Springer (2005) 1523-1558.

[26] A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455-462.

[27] M. Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. ESAIM: M2AN 35 (2001) 355-387. | Numdam | MR 1825703 | Zbl 0992.65100

[28] M. Ohlberger, A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Numer. Math. 87 (2001) 737-761. | MR 1815733 | Zbl 0973.65076

[29] M. Ohlberger and J. Vovelle, Error estimate for the approximation of non-linear conservation laws on bounded domains by the finite volume method. Math. Comp. 75 (2006) 113-150. | MR 2176392 | Zbl 1082.65112

[30] 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, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering.

[31] T.A. Porsching and M.L. Lee, The reduced basis method for initial value problems. SIAM J. Numer. Anal. 24 (1987) 1277-1287. | MR 917452 | Zbl 0639.65039

[32] C. Prud'Homme, D. Rovas, K. Veroy and A.T. Patera, A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. ESAIM: M2AN 36 (2002) 747-771. | Numdam | MR 1955536 | Zbl 1024.65104

[33] C. Prud'Homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engineering 124 (2002) 70-80.

[34] A. Quarteroni, G. Rozza, L. Dede and A. Quaini, Numerical approximation of a control problem for advection-diffusion processes, in System Modeling and Optimization, Proceedings of 22nd IFIP TC7 Conference (2006). | MR 2249340

[35] D.V. Rovas, L. Machiels and Y. Maday, Reduced basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423-445. | MR 2241309 | Zbl 1101.65099

[36] C.W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcat. Chaos 15 (2005) 997-1013. | MR 2136757 | Zbl 1140.76443

[37] G. Rozza, Shape design by optimal flow control and reduced basis techniques: Applications to bypass configurations in haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Switzerland (2005).

[38] B. Schölkopf and A.J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond. MIT Press (2002).

[39] T. Tonn and K. Urban, A reduced-basis method for solving parameter-dependent convection-diffusion problems around rigid bodies. Technical Report 2006-03, Institute for Numerical Mathematics, Ulm University, ECCOMAS CFD (2006).

[40] K. Veroy and A.T. Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Meth. Fluids 47 (2005) 773-788. | MR 2123791 | Zbl 1134.76326

[41] K. Veroy, C. Prud'Homme and A.T. Patera, Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds. C. R. Acad. Sci. Paris Ser. I Math. 337 (2003) 619-624. | MR 2017737 | Zbl 1036.65075