Johann Radon Institute for Computational and Applied Mathematics

The problem of foot and mouth disease (FMD) is of serious concern to the livestock sector in most nations, especially in developing countries. This paper presents the formulation and analysis of a deterministic model for the transmission dynamics of FMD through a contaminated environment. It is shown that the key parameters that drive the transmission of FMD in a contaminated environment are the shedding, transmission, and decay rates of the virus. Using numerical results, it is depicted that the host-to-host route is more severe than the environmental-to-host route. The model is then transformed into an optimal control problem. Using the Pontryagin’s Maximum Principle, the optimality system is determined. Utilizing a gradient type algorithm with projection, the optimality system is solved for three control strategies: optimal use of vaccination, environmental decontamination, and a combination of vaccination and environmental decontamination. Results show that a combination of vacc...

We present a boundary optimal control approach for a Bernoulli free boundary problem. Our control objective consists in tracking a free boundary to an a priori given desired one. An appropriate cost functional is chosen for this purpose and its gradient is analytically derived. The resulting optimality conditions are discretized and the minimization problem is solved numerically by a steepest descent method. Numerical simulations show that the proposed method is in agreement with known results.

Eigenvector centrality is one of the outstanding measures of central tendency in graph theory. In this paper we consider the problem of calculating eigenvector centrality of graph partitioned into components and how this partitioning can be used. Two cases are considered; first where the a single component in the graph has the dominant eigenvalue, secondly when there are at least two components that share the dominant eigenvalue for the graph. In the first case we implement and compare the method to the usual approach (power method) for calculating eigenvector centrality while in the second case with shared dominant eigenvalues we show some theoretical and numerical results.

A general framework for calculating shape derivatives for domain optimization problems with partial differential equations as constraints is presented. The first order approximation of the cost with respect to the geometry perturbation is arranged in an efficient manner that allows the computation of the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In doing so, the state variable is only required to be Lipschitz continuous with respect to geometry perturbations. Application to shape optimization with the Navier-Stokes equations as PDE constraint is given.

In this paper, an optimal actuator placement problem, with a linear wave equation as the constraint, is considered. In particular, this work presents the framework for finding the best location of actuators depending upon the given initial conditions, and where the dependence on the initial conditions is averaged out. The problem is motivated by the need to control vibrations induced by pedestrian-bridge interactions. An approach based on the shape optimization techniques is used to solve the problem. Specifically, the shape sensitivities involving a cost functional are determined using the averaged adjoint approach. A numerical algorithm based on these sensitivities is used as a solution strategy. Numerical tests illustrate the theoretical results.