Reduced basis method for finite volume approximations of parametrized linear evolution equations

Bernard Haasdonk; Mario Ohlberger

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 2, page 277-302
  • ISSN: 0764-583X

Abstract

top
The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) 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.

How to cite

top

Haasdonk, Bernard, and Ohlberger, Mario. "Reduced basis method for finite volume approximations of parametrized linear evolution equations." ESAIM: Mathematical Modelling and Numerical Analysis 42.2 (2008): 277-302. <http://eudml.org/doc/250341>.

@article{Haasdonk2008,
abstract = { The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) 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. },
author = {Haasdonk, Bernard, Ohlberger, Mario},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Model reduction; reduced basis methods; finite volume methods; a-posteriori error estimates.; model reduction; a-posteriori error estimates},
language = {eng},
month = {3},
number = {2},
pages = {277-302},
publisher = {EDP Sciences},
title = {Reduced basis method for finite volume approximations of parametrized linear evolution equations},
url = {http://eudml.org/doc/250341},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Haasdonk, Bernard
AU - Ohlberger, Mario
TI - Reduced basis method for finite volume approximations of parametrized linear evolution equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/3//
PB - EDP Sciences
VL - 42
IS - 2
SP - 277
EP - 302
AB - The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) 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.
LA - eng
KW - Model reduction; reduced basis methods; finite volume methods; a-posteriori error estimates.; model reduction; a-posteriori error estimates
UR - http://eudml.org/doc/250341
ER -

References

top
  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.  
  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.  
  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.  
  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.  
  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).  
  8. B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput.16 (2001) 173–261.  
  9. Y. Coudiere, J.P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN33 (1999) 493–516.  
  10. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis, volume VII, North-Holland, Amsterdam (2000) 713–1020.  
  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.  
  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.  
  13. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996).  
  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: M2AN39 (2005) 157–181.  
  16. P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées 22 [Research in Applied Mathematics]. Masson, Paris (1992).  
  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.  
  18. R.L. Higdon, Initial-boundary value problems for linear hyperbolic systems. SIAM Rev.28 (1986) 177–217.  
  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 Pascal10 (2003) 269–296.  
  20. D. Kröner, Numerical Schemes for Conservation Laws. John Wiley & Sons and Teubner (1997).  
  21. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).  
  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.  
  23. M. Mangold and M. Sheng, Nonlinear model reduction of a 2D MCFC model with internal reforming. Fuel Cells4 (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.  
  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: M2AN35 (2001) 355–387.  
  28. M. Ohlberger, A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Numer. Math.87 (2001) 737–761.  
  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.  
  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.  
  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: M2AN36 (2002) 747–771.  
  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 Engineering124 (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).  
  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.  
  36. C.W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcat. Chaos15 (2005) 997–1013.  
  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. Fluids47 (2005) 773–788.  
  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.  

NotesEmbed ?

top

You must be logged in to post comments.