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Problem adapted Hierarchical Model Reduction for the Fokker-Planck equation

Proceedings of ALGORITMY 2016, 20th Conference on Scientific Computing, Vysoke Tatry, Podbanske, Slovakia, March 13-18, 2016, page 13--22 - 2016
Télécharger la publication : 390-2522-1-PB.pdf [391Ko]  
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):A714A736, 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.

Références BibTex

@InProceedings{BLOS16,
  author       = {Brunken, J. and Leibner, T. and Ohlberger, M. and Smetana, K.},
  title        = {Problem adapted Hierarchical Model Reduction for the Fokker-Planck equation},
  booktitle    = {Proceedings of ALGORITMY 2016, 20th Conference on Scientific Computing, Vysoke Tatry, Podbanske, Slovakia, March 13-18, 2016},
  pages        = {13--22},
  year         = {2016},
  editor       = {A. Handlovi\čova and D. Sev\čovi\č},
  publisher    = {Publishing House of Slovak University of Technology in Bratislava},
  note         = {http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/390/307},
  issn         = {978-80-227-4454-4},
  url          = \{/2016/BLOS16},
}

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