Computing primal solutions with exact arithmetics in SCIP.

The research for exact solutions of mixed integer problems is an active topic in the scientific community. State-of-the-art MIP solvers exploit a floating- point numerical representation, therefore introducing small approximations. Although such MIP solvers yield reliable results for the majority of problems, there are cases in which a higher accuracy is required. Indeed, it is known that for some applications floating-point solvers provide falsely feasible solutions, i.e. solutions marked as feasible because of approximations that would not pass a check with exact arithmetic and cannot be practically implemented. The framework of the current dissertation is SCIP, a mixed integer programs solver mainly developed at Zuse Institute Berlin. In the same site we considered a new approach for exactly solving MIPs. Specifically, we developed a constraint handler to plug into SCIP, with the aim to analyze the accuracy of provided floating-point solutions and compute exact primal solutions starting from floating-point ones. We conducted a few computational experiments to test the exact primal constraint handler through the adoption of two main settings. Analysis mode allowed to collect statistics about current SCIP solutions' reliability. Our results confirm that floating-point solutions are accurate enough with respect to many instances. However, our analysis highlighted the presence of numerical errors of variable entity. By using the enforce mode, our constraint handler is able to suggest exact solutions starting from the integer part of a floating-point solution. With the latter setting, results show a general improvement of the quality of provided final solutions, without a significant loss of performances.

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.

Solvent regeneration of activated carbon mixed-matrix membranes for the removal of micropollutants from wastewater

The constantly increasing demand of clean water has become challenging to deal with over the past years, water being an ever more precious resource. In recent times, the existing wastewater treatments had to be integrated with new steps, due to the detection of so-called organic micropollutants (OMPs). These compounds have been shown to adversely affect the environment and possibly human health, even when found in very low concentrations. In order to remove OMPs from wastewater, one possible technique is a hybrid process combining filtration and adsorption. In this work, polyethersulfone multi-channel mixed-matrix membranes with embedded powdered activated carbon (PAC) were tested to investigate the membrane’s adsorption and desorption performance. Micropollutants retention was analyzed using the pharmaceutical compounds diclofenac (DCF), paracetamol (PARA) and carbamazepine (CBZ) in filtration mode, combining the PAC adsorption process with the membrane’s ultrafiltration. Desorption performance was studied through solvent regeneration, using seven different solvents: pure water, pure ethanol, mixture of ethanol and water in different concentration, sodium hydroxide and a mixture of ethanol and sodium hydroxide. Regeneration experiments were carried out in forward-flushing. At first regeneration efficiency was investigated using a single-solute solution (diclofenac in water). The mixture Ethanol/Water (50:50) was found to be the most efficient with long-term retention of 59% after one desorption cycle. It was, therefore, later tested on a membrane previously loaded with a multi-solute solution. Three desorption cycles were performed after which, retention (after 30 min) reached values of 87% for PARA and 72% for CBZ and 55% for DCF, which indicates decent regenerability. A morphological analysis on the membranes confirmed that, in any case, the regeneration cycles did not affect either the membranes’ structure, or the content and distribution of PAC in the matrix.