Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation

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 consists 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 most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest

Optimal Control of Convective FitzHugh-Nagumo Equation

We investigate smooth and sparse optimal control problems for convective FitzHugh-Nagumo equation with travelling wave solutions in moving excitable media. The cost function includes distributed space-time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the travelling waves. 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

Fitzhugh-Nagumo denkleminin eniyilemeli kontrolü ve indirgenmiş dereceli modellemesi.

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.Ph.D. - Doctoral Progra

Structure Preserving Integration and Model Order Reduction of Skew Gradient Reaction Diffusion Systems

Activator-inhibitor FitzHugh-Nagumo (FHN) equation is an example for reaction-diffusion equations with skew-gradient structure. We discretize the FHN equation using symmetric interior penalty discontinuous Galerkin (SIPG) method in space and average vector field (AVF) method in time. The AVF method is a geometric integrator, i.e. it preserves the energy of the Hamiltonian systems and energy dissipation of the gradient systems. In this work, we show that the fully discrete energy of the FHN equation satisfies the mini-maximizer property of the continuous energy for the skew-gradient systems. We present numerical results with traveling fronts and pulses for one dimensional, two coupled FHN equations and three coupled FHN equations with one activator and two inhibitors in skew-gradient form. Turing patterns are computed for fully discretized two dimensional FHN equation in the form of spots and labyrinths. Because the computation of the Turing patterns is time consuming for different parameters, we applied model order reduction with the proper orthogonal decomposition (POD). The nonlinear term in the reduced equations is computed using the discrete empirical interpolation (DEIM) with SIPG discretization. Due to the local nature of the discontinuous Galerkin (DG) method, the nonlinear terms can be computed more efficiently than for the continuous finite elements. The reduced solutions are very close to the fully discretized ones. The efficiency and accuracy of the POD and POD-DEIM reduced solutions are shown for the labyrinth-like patterns

Reduced Order Modelling for Reaction-Diffusion Equations with Cross Diffusion

In this work, we present reduced-order models (ROMs) for a nonlinear cross-diffusion problem from population dynamics, the Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics. The formation of the patterns of the SKT equation consists of a fast transient phase and a long stationary phase. Reduced order solutions are computed by separating the time into two time-intervals. In numerical experiments, we show for one- and two-dimensional SKT equations with pattern formation, the reduced-order solutions obtained in the time-windowed form, i.e., principal decomposition framework, are more accurate than the global proper orthogonal decomposition solutions obtained in the whole time interval. The finite-difference discretization of the SKT equation in space results in a system of linear-quadratic ordinary differential equations. The ROMs have the same linear-quadratic structure as the full order model. Using the linear-quadratic structure of the ROMs, the computation of the reduced-order solutions is further accelerated by the use of proper orthogonal decomposition in a tensorial framework so that the computations in the reduced system are independent of the full-order solutions. Furthermore, the prediction capabilities of the ROMs are illustrated for one- and two-dimensional patterns. Finally, we show that the entropy is decreasing by the reduced solutions, which is important for the global existence of solutions to the nonlinear cross-diffusion equations such as the SKT equation

Adveksiyon- ve Reaksiyon-Difuzyon Sistemleri için Optimal Kontrol ve Parametre Tahmin Yöntemleri

Adveksiyon- ve reaksiyon-difuzyon sistemlerinin kontrol edildiği optimizasyon problemleri teknoloji ve çeşitli bilim dallarında önemli problemleri temsil ettiğinden, onların çözümleri için geliştirilen metod ve algoritmalar önemini korumakla birlikte geliştirilmesi için çaba harcanmaktadır. Yapacağımız çalışma ile bu problemlerin çözümlerinin daha hızlı ve daha doğru elde edilmesi sağlanmaya çalışılacaktır. Bu da bu problemleri içeren teknoloji ve bilim dallarının gelişimine ve uygulanan algoritmaların iyileştirilmesine önemli katkı sağlayacaktır