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

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

Local Improvements to Reduced-Order Approximations of Optimal Control Problems Governed by Diffusion-Convection-Reaction Equation

We consider the optimal control problem governed by diffusion convection reaction equation without control constraints. The proper orthogonal decomposition(POD) method is used to reduce the dimension of the problem. The POD method may be lack of accuracy if the POD basis depending on a set of parameters is used to approximate the problem depending on a different set of parameters. We are interested in the perturbation of diffusion term. To increase the accuracy and robustness of the basis, we compute three bases additional to the baseline POD. The first two of them use the sensitivity information to extrapolate and expand the POD basis. The other one is based on the subspace angle interpolation method. We compare these different bases in terms of accuracy and complexity and investigate the advantages and main drawbacks of them.Comment: 19 pages, 5 figures, 2 table

An Optimal EDG Method for Distributed Control of Convection Diffusion PDEs

We propose an embedded discontinuous Galerkin (EDG) method to approximate the solution of a distributed control problem governed by convection diffusion PDEs, and obtain optimal a priori error estimates for the state, dual state, their uxes, and the control. Moreover, we prove the optimize-then-discretize (OD) and discrtize-then-optimize (DO) approaches coincide. Numerical results confirm our theoretical results

An HDG Method for Distributed Control of Convection Diffusion PDEs

We propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a distributed optimal control problem governed by an elliptic convection diffusion PDE. We derive optimal a priori error estimates for the state, adjoint state, their fluxes, and the optimal control. We present 2D and 3D numerical experiments to illustrate our theoretical results.Comment: arXiv admin note: substantial text overlap with arXiv:1712.10106, arXiv:1712.01403, arXiv:1712.0293

A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity

In the first part of this work, we analyzed a Dirichlet boundary control problem for an elliptic convection diffusion PDE and proposed a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For the case of a 2D polygonal domain, we also proved an optimal superlinear convergence rate for the control under certain assumptions on the domain and on the target state. In this work, we revisit the convergence analysis without these assumptions; in this case, the solution can have low regularity and we use a different analysis approach. We again prove an optimal convergence rate for the control, and present numerical results to illustrate the convergence theory

The SIPG method of Dirichlet boundary optimal control problems with weakly imposed boundary conditions

In this paper, we consider the symmetric interior penalty Galerkin (SIPG) method which is one of Discontinuous Galerkin Methods for the Dirichlet optimal control problems governed by linear advection-diffusion-reaction equation on a convex polygonal domain and the difficulties which we faced while solving this problem numerically. Since standard Galerkin methods have failed when the boundary layers have occurred and advection diffusion has dominated, these difficulties can occur in the cases of higher order elements and non smooth Dirichlet data in using standard finite elements. We find the most convenient natural setting of Dirichlet boundary control problem for the Laplacian and the advection diffusion reaction equations.After converting the continuous problem to an optimization problem, we solve it by "discretize-then-optimize" approach. In final, we estimate the optimal priori error estimates in suitable norms of the solutions and then support the result and the features of the method with numerical examples on the different kinds of domain