Abstract
We developed a reduced order model (ROM) using the proper orthogonal decomposition (POD) to compute efficiently the labyrinth and spot like patterns of the FitzHugh-Nagumo (FNH) equation. The FHN equation is discretized in space by the discontinuous Galerkin (dG) method and in time by the backward Euler method. Applying POD-DEIM (discrete empirical interpolation method) to the full order model (FOM) for different values of the parameter in the bistable nonlinearity, we show that using few POD and DEIM modes, the patterns can be computed accurately. Due to the local nature of the dG discretization, the POD-DEIM requires less number of connected nodes than continuous finite element for the nonlinear terms, which leads to a significant reduction of the computational cost for dG POD-DEIM.
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References
H. Antil, M. Heinkenschloss, D.C. Sorensen, Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems, in Reduced Order Methods for Modeling and Computational Reduction, ed. by A. Quarteroni, G. Rozza. Modeling, Simulation and Applications, vol. 9 (Springer, Cham, 2014), pp. 101–136
M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Math. 339 (9), 667–672 (2004)
S. Chaturantabut, D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010)
J.S. Hesthaven, G. Rozza, B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics (Springer/Basque Center for Applied Mathematics, Bilbao, 2016)
M. Hinze, S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control, in Dimension Reduction of Largescale Systems, ed. by P. Benner, D.C. Sorensen, V. Mehrmann. Lecture Notes in Computational Science and Engineering, vol. 45 (Springer, Berlin/New York, 2005), pp. 261–306
T.T. Marquez-Lago, P. Padilla, A selection criterion for patterns in reaction diffusion systems. Theor. Biol. Med. Model. 11, 1–17 (2014)
B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation (SIAM, Philadelphia, 2008)
Acknowledgements
This work has been supported by METU BAP-07-05-2015 009.
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Karasözen, B., Uzunca, M., Küçükseyhan, T. (2016). Model Order Reduction for Pattern Formation in FitzHugh-Nagumo Equations. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_35
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DOI: https://doi.org/10.1007/978-3-319-39929-4_35
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