Optimal control and reduced order modelling of Fitzhugh–Nagumo equation

Download
2017
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.

Suggestions

Sturm comparison theory for impulsive differential equations
Özbekler, Abdullah; Ağacık, Zafer; Department of Mathematics (2005)
In this thesis, we investigate Sturmian comparison theory and oscillation for second order impulsive differential equations with fixed moments of impulse actions. It is shown that impulse actions may greatly alter the oscillation behavior of solutions. In chapter two, besides Sturmian type comparison results, we give Leightonian type comparison theorems and obtain Wirtinger type inequalities for linear, half-linear and non-selfadjoint equations. We present analogous results for forced super linear and super...
Periodic solutions and stability of differential equations with piecewise constant argument of generalized type
Büyükadalı, Cemil; Akhmet, Marat; Department of Mathematics (2009)
In this thesis, we study periodic solutions and stability of differential equations with piecewise constant argument of generalized type. These equations can be divided into three main classes: differential equations with retarded, alternately advanced-retarded, and state-dependent piecewise constant argument of generalized type. First, using the method of small parameter due to Poincaré, the existence and stability of periodic solutions of quasilinear differential equations with retarded piecewise constant...
Inverse problems for parabolic equations
Baysal, Arzu; Çelebi, Okay; Department of Mathematics (2004)
In this thesis, we study inverse problems of restoration of the unknown function in a boundary condition, where on the boundary of the domain there is a convective heat exchange with the environment. Besides the temperature of the domain, we seek either the temperature of the environment in Problem I and II, or the coefficient of external boundary heat emission in Problem III and IV. An additional information is given, which is the overdetermination condition, either on the boundary of the domain (in Proble...
Inverse problems for a semilinear heat equation with memory
Kaya, Müjdat; Çelebi, Okay; Department of Mathematics (2005)
In this thesis, we study the existence and uniqueness of the solutions of the inverse problems to identify the memory kernel k and the source term h, derived from First, we obtain the structural stability for k, when p=1 and the coefficient p, when g( )= . To identify the memory kernel, we find an operator equation after employing the half Fourier transformation. For the source term identification, we make use of the direct application of the final overdetermination conditions.
Asymptotic integration of impulsive differential equations
Doğru Akgöl, Sibel; Ağacık, Zafer; Özbekler, Abdullah; Department of Mathematics (2017)
The main objective of this thesis is to investigate asymptotic properties of the solutions of differential equations under impulse effect, and in this way to fulfill the gap in the literature about asymptotic integration of impulsive differential equations. In this process our main instruments are fixed point theorems; lemmas on compactness; principal and nonprincipal solutions of impulsive differential equations and Cauchy function for impulsive differential equations. The thesis consists of five chapters....
Citation Formats
T. Küçükseyhan, “Optimal control and reduced order modelling of Fitzhugh–Nagumo equation,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.