Problem adapted Hierarchical Model Reduction for the Fokker-Planck equation

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Julia Brunken Tobias Leibner Mario Ohlberger Kathrin Smetana

Abstract

In this paper we introduce a new hierarchical model reduction framework for the Fokker-Planck equation. We reduce the dimension of the equation by a truncated basis expansion in the velocity variable, obtaining a hyperbolic system of equations in space and time. Unlike former methods like the Legendre moment models, the new framework generates a suitable problem-dependent basis of the reduced velocity space that mimics the shape of the solution in the velocity variable. To that end, we adapt the framework of [M. Ohlberger and K. Smetana. \emph{A dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction}. SIAM J. Sci. Comput., 36(2):A714–A736, 2014] and derive initially a parametrized elliptic partial differential equation (PDE) in the velocity variable. Then, we apply ideas of the Reduced Basis method to develop a greedy algorithm that selects the basis from solutions of the parametrized PDE. Numerical experiments demonstrate the potential of this new method.

Article Details

How to Cite
BRUNKEN, Julia et al. Problem adapted Hierarchical Model Reduction for the Fokker-Planck equation. Proceedings of the Conference Algoritmy, [S.l.], p. 13-22, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/390>. Date accessed: 22 sep. 2017.
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References

[1] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non-Newton. Fluid Mech., 139(3):153–176, 2006.

[2] A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. II: Transient simulation using space-time separated representations. J. Non-Newton. Fluid Mech., 144(2-3):98–121, 2007.

[3] W. Dahmen, C. Huang, C. Schwab, and G. Welper. Adaptive Petrov–Galerkin methods for first order transport equations. SIAM J. Numer. Anal., 50(5):2420–2445, 2012.

[4] W. Dahmen, C. Plesken, and G. Welper. Double greedy algorithms: Reduced basis methods for transport dominated problems. ESAIM Math. Model. Numer. Anal., 48:623–663, 5 2014.

[5] R. DeVore, G. Petrova, and P. Wojtaszczyk. Greedy algorithms for reduced bases in Banach spaces. Constr. Approx., 37(3):455–466, 2013.

[6] DUNE. Distributed and unified numerics environment. www.dune-project.org/.

[7] M. Frank, H. Hensel, and A. Klar. A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy. SIAM J. Appl. Math., 67(2):582–603, 2007.

[8] E. Hairer, S. Nørsett, and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Lecture Notes in Economic and Mathematical Systems. Springer, 1993.

[9] H. Hensel, R. Iza-Teran, and N. Siedow. Deterministic model for dose calculation in photon radiotherapy. Phys. Med. Biol., 51(3):675, 2006.

[10] D. J. Knezevic and A. T. Patera. A certified reduced basis method for the Fokker-Planck equation of dilute polymeric fluids: FENE dumbbells in extensional flow. SIAM J. Sci. Comput., 32(2):793–817, 2010.

[11] C. Levermore. Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83(5-6):1021– 1065, 1996.

[12] R. Milk, S. Rave, and F. Schindler. pyMOR – generic algorithms and interfaces for model order reduction. arxiv:1506.07094, preprint, 2015.

[13] M. Ohlberger and S. Rave. Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing. C. R. Math. Acad. Sci. Paris, 351(23-24):901–906, 2013.

[14] M. Ohlberger and K. Smetana. A dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction. SIAM J. Sci. Comput., 36(2):A714–A736, 2014.

[15] S. Perotto, A. Ern, and A. Veneziani. Hierarchical local model reduction for elliptic problems: a domain decomposition approach. Multiscale Model. Simul., 8(4):1102–1127, 2010.

[16] G. Pomraning. The Fokker-Planck operator as an asymptotic limit. Math. Models Methods Appl. Sci., 02(01):21–36, 1992.

[17] G. Rozza, D. Huynh, and A. Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Arch. Comput. Meth. Eng., 15(3):229–275, 2008.

[18] F. Schneider, G. Alldredge, M. Frank, and A. Klar. Higher order mixed moment approximations for the Fokker-Planck equation in one space dimension. SIAM J. Appl. Math., 74(4):1087– 1114, 2014.

[19] K. Smetana and M. Ohlberger. Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques. arxiv:1401.0851, preprint, 2014.

[20] T. Taddei, S. Perotto, and A. Quarteroni. Reduced basis techniques for nonlinear conservation laws. ESAIM Math. Model. Numer. Anal., 49(3):787–814, 2015.

[21] M. Vogelius and I. Babuˇska. On a dimensional reduction method. I. The optimal selection of basis functions. Math. Comp., 37(155):31–46, 1981.

[22] G. Welper. Transformed snapshot interpolation. arxiv:1505.01227, preprint, 2015.