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

Fractional Order Version of the HJB Equation

We consider an extension of the well-known Hamilton-Jacobi-Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Nonlinear Dynamics', ISSN 1555-1415, eISSN 1555-1423, CODEN: JCNDDM. Submitted 28-June-2018; Revised 15-Sept-2018; Accepted 28-Oct-201

Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method

In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework of multi-level Monte Carlo for parametric expectations and propose modifications of the MLMC estimator, error estimation procedure, and adaptive MLMC parameter selection to ensure the estimation of the CVaR and sensitivities for a given design with a prescribed accuracy. We then propose combining the MLMC framework with an alternating inexact minimisation-gradient descent algorithm, for which we prove exponential convergence in the optimisation iterations under the assumptions of strong convexity and Lipschitz continuity of the gradient of the objective function. We demonstrate the performance of our approach on two numerical examples of practical relevance, which evidence the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods for fixed design computations of output expectations.Comment: 26 pages, 18 figures, 1 table, Related to arXiv:2208.07252, Data available at https://zenodo.org/record/719344

Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems

Orthogonal subgrid-scale stabilization for nonlinear reaction-convection-diffusion equations

Nonlinear reaction-convection-diffusion equations are encountered in modeling of a variety of natural phenomena such as in chemical reactions, population dynamics and contaminant dispersal. When the scale of convective and reactive phenomena are large, Galerkin finite element solution fails. As a remedy, Orthogonal Subgrid Scale stabilization is applied to the finite element formulation. It has its origins in the Variational Multi Scale approach. It is based on a fine grid - coarse grid component sum decomposition of solution and utilizes the fine grid solution orthogonal to the residual of the finite element coarse grid solution as a correction term. With selective mesh refinement, a stabilized oscillation-free solution that can capture sharp layers is obtained. Newton Raphson method is utilized for the linearization of nonlinear reaction terms. Backward difference scheme is used for time integration. The formulation is tested for cases with standalone and coupled systems of transient nonlinear reaction-convection-diffusion equations. Method of manufactured solution is used to test for correctness and bug-free implementation of the formulation. In the error analysis, optimal convergence is achieved. Applications in channel flow, cavity flow and predator-prey model is used to highlight the need and effectiveness of the stabilization technique

Control of Spiral Waves in Reaction-Diffusion Systems Using Response Function

This thesis is motivated by the desire to understand spiral wave dynamics in reactiondiffusion systems with particular focus on the FitzHugh-Nagumo model. We attempt to control the behaviour of spiral waves using controller dynamics. Response functions characterise the behaviour of spiral waves under perturbations, and so it is natural to use these for control purposes. In this project, we consider perturbations of the FitzHugh-Nagumo equation using control functions with different support. We calculate the response functions using the adjoint linear system of the FitzHugh-Nagumo equation with 1D controller dynamics and also characterise the control functions with the smallest support function which can be used to control the system in periodic and meander regimes. We find the minimum size of the support function that the radius is comparable to the region of the non zero response function

Bivariate pseudospectral collocation algorithms for nonlinear partial differential equations.

Doctor of Philosophy in Applied Matheatics. University of KwaZulu-Natal, Pietermaritzburg 2016.Abstract available in PDF file