Weighted geometric mean of large-scale matrices: numerical analysis and algorithms

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators arising in domain decomposition of elliptic problems. In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v. Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

Axis control optimization: autotuning techniques and frequency response analysis of industrial mechatronic systems

In the field of industrial automation, there is an increasing need to use optimal control systems that have low tracking errors and low power and energy consumption. The motors we are dealing with are mainly Permanent Magnet Synchronous Motors (PMSMs), controlled by 3 different types of controllers: a position controller, a speed controller, and a current controller. In this thesis, therefore, we are going to act on the gains of the first two controllers by going to find, through the TwinCAT 3 software, what might be the best set of parameters. To do this, starting with the default parameters recommended by TwinCAT, two main methods were used and then compared: the method of Ziegler and Nichols, which is a tabular method, and advanced tuning, an auto-tuning software method of TwinCAT. Therefore, in order to analyse which set of parameters was the best,several experiments were performed for each case, using the Motion Control Function Blocks. Moreover, some machines, such as large robotic arms, have vibration problems. To analyse them in detail, it was necessary to use the Bode Plot tool, which, through Bode plots, highlights in which frequencies there are resonance and anti-resonance peaks. This tool also makes it easier to figure out which and where to apply filters to improve control.

Data sanitization in a clearing system for public transport operators

In this work we will discuss about a project started by the Emilia-Romagna Regional Government regarding the manage of the public transport. In particular we will perform a data mining analysis on the data-set of this project. After introducing the Weka software used to make our analysis, we will discover the most useful data mining techniques and algorithms; and we will show how these results can be used to violate the privacy of the same public transport operators. At the end, despite is off topic of this work, we will spend also a few words about how it's possible to prevent this kind of attack.