Model order reduction for pattern formation in reaction-diffusion systems

Date

2017-09-22

Author

Karasözen, Bülent
Küçükseyhan, Tuğba
Mülayim, Gülden

Metadata

Show full item record

Item Usage Stats

275
views

0
downloads

We compare three reduced order modelling (ROM) techniques: the proper orthogonal decomposition (POD), discrete empirical interpolation (DEIM) [2], and dynamical mode decomposition (DMD) [1] to reaction diusion equations in biology. The formation of patterns in reaction-diusion equations require highly accurate solutions in space and time and therefore require large computational time to reach the steady states. The three reduced order methods are applied to the diusive FitzHugh-Nagumo equation [3] and the Brusselator model with cross diusion [4]. DMD is an equation-free, data driven method which extracts dynamically relevant information content without explicitly knowing the dynamical operator. We use DMD as an alternative method to DEIM in order to approximate the nonlinear reaction terms. Application of the POD-DMD Galerkin projection gives rise to a linear system of equations. The high delity full order solutions (FOMs) are obtained by the discontinuous Galerkin discretization in space and semi-implicit Euler method in time. We compare the accuracy and CPU times of three reduced order model (ROM) solutions with the ones for FOM solutions. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest.

Subject Keywords

Cross diffusion, FitzHugh-Nagumo model, Pattern formation, Reduced order modelling, Turing-Hopf bifurcation

URI

https://hdl.handle.net/11511/88164
https://acomen.ugent.be/BoA_ACOMEN2017.pdf

Conference Name

7th International Conference on Advanced Computational Methods in Engineering, ACOMEN 2017 18–22 September 2017

Collections

Department of Mathematics, Conference / Seminar

Suggestions

OpenMETU
Core

Reduced order optimal control of the convective FitzHugh-Nagumo equations Karasözen, Bülent; KÜÇÜKSEYHAN, TUĞBA (2020-02-15) In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh-Nagumo (FHN) equations. The convective FHN equations consist of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The ...
Model order reduction for nonlinear Schrodinger equation Karasözen, Bülent; Uzunca, Murat (2015-05-01) We apply the proper orthogonal decomposition (POD) to the nonlinear Schrodinger (NLS) equation to derive a reduced order model. The NLS equation is discretized in space by finite differences and is solved in time by structure preserving symplectic mid-point rule. A priori error estimates are derived for the POD reduced dynamical system. Numerical results for one and two dimensional NLS equations, coupled NLS equation with soliton solutions show that the low-dimensional approximations obtained by POD reprodu...
Analysis of composite nanoparticles with surface integral equations and the multilevel fast multipole algorithm Ergül, Özgür Salih (IOP Publishing, 2012-06-01) Composite nanoparticles involving multiple parts with different material properties are analyzed rigorously with surface integral equations and the multilevel fast multipole algorithm. Accuracy and efficiency of the developed parallel implementation are demonstrated on spherical objects with dielectric, perfectly conducting, plasmonic, and double-negative regions. Significant effects of the formulation on numerical solutions are also considered to show the tradeoff between the efficiency and accuracy.
Model Order Reduction for Pattern Formation in FitzHugh-Nagumo Equations Karasözen, Bülent; Kucukseyhan, Tugba (2015-09-18) 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...
Annulus criteria for mixed nonlinear elliptic differential equations ŞAHİNER, YETER; Zafer, Ağacık (Elsevier BV, 2011-05-01) New oscillation criteria are obtained for forced second order elliptic partial differential equations with damping and mixed nonlinearities of the form

Citation Formats

B. Karasözen, T. Küçükseyhan, and G. Mülayim, “ Model order reduction for pattern formation in reaction-diffusion systems,” Ghent, Belgium, 2017, p. 93, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/88164.