# Problem adapted Hierarchical Model Reduction for the Fokker-Planck equation

## Main Article Content

## 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, J., Leibner, T., Ohlberger, M., & Smetana, K.
(2016).
Problem adapted Hierarchical Model Reduction for the Fokker-Planck equation.

*Proceedings Of The Conference Algoritmy,*, 13-22. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/390/307
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Articles

## References

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[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/.

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[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.

[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.