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Optimal control and reduced order modelling of Fitzhugh–Nagumo equation

Küçükseyhan , Tuğba
In this thesis, we investigate model order reduction and optimal control of FitzHugh-Nagumo equation (FHNE). FHNE is coupled partial differential equations (PDEs) of activator-inhibitor types. Diffusive FHNE is a model for the transmission of electrical impulses in a nerve axon, whereas the convective FHNE is a model for blood coagulation in a moving excitable media. We discretize these state FHNEs using a symmetric interior penalty Galerkin (SIPG) method in space and an average vector field (AVF) method in time for diffusive FHNE. For time discretization of the convective FHNE, we use a backward Euler method. The diffusive FHNE has a skew-gradient structure. We show that the fully discrete energy of the diffusive FHNE satisfying the mini-maximizing property of the discrete energy of the skew-gradient system is preserved by SIPG-AVF discretization. Depending on the parameters and the non-linearity, specific patterns in one and two dimensional FHNEs occur like travelling waves and Turing patterns. Formation of fronts and pulses for the one dimensional (1D) diffusive FHNE, patterns and travelling waves for the two dimensional (2D) diffusive and convective FHNEs are studied numerically. Because the computation of the pattern formations is very time consuming, we apply three different model order reduction (MOR) techniques; proper orthogonal decomposition (POD), discrete empirical interpolation (DEIM), and dynamic mode decomposition (DMD). All these MOR techniques are compared with the high fidelity fully discrete SIPG-AVF solutions in terms of accuracy and computational time. Due to the local nature of the discontinuous Galerkin (DG) method, the nonlinear terms can be computed more efficiently by DEIM and DMD than for the continuous finite elements method (FEM). The numerical results reveal that the POD is the most accurate, the DMD the fastest, and the DEIM in between both. We also investigate sparse and non-sparse optimal control problems governed by the travelling wave solutions of the convective FHNE. We also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing Tikhonov regularization parameter. Further, we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced Hessian. We use the DMD as an alternative method to DEIM in order to approximate the nonlinear term in the convective FHNE. Applying the POD-DMD Galerkin projection gives rise to a linear discrete equation for the activator, and the discrete optimal control problem becomes convex. FOM and sub-optimal control solutions with the above mentioned MOR techniques are compared for a variety of numerical examples.