In this paper we study the structure of solutions of the one dimensional weighted total
variation regularisation problem, motivated by its application in signal recovery tasks. We study
in depth the relationship between the weight function and the creation of new discontinuities in
the solution. A partial semigroup property relating the weight function and the solution is shown
and analytic solutions for simply data functions are computed. We prove that the weighted total
variation minimisation problem is well-posed even in the case of vanishing weight function, despite
the lack of coercivity. This is based on the fact that the total variation of the solution is bounded
by the total variation of the data, a result that it also shown here. Finally the relationship to the
corresponding weighted fidelity problem is explored, showing that the two problems can produce
completely different solutions even for very simple data functions.
We address the problem of optimally placing sensor networks for convection-diffusion
processes where the convective part is perturbed. The problem is formulated as an optimal control
problem where the integral Riccati equation is a constraint and the design variables are sensor
locations. The objective functional involves a term associated to the trace of the solution to the
Riccati equation and a term given by a constrained optimization problem for the directional derivative
of the previous quantity over a set of admissible perturbations. The paper addresses the existence
of the derivative with respect to the convective part of the solution to the Riccati equation, the
well-posedness of the optimization problem and finalizes with a range of numerical tests.
Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme, respectively. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.