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We are interested in the numerical solution of large-scale linear matrix equations. In particular, due to their occurrence in many applications, we study the so-called Sylvester and Lyapunov equations. A characteristic aspect of the large-scale setting is that although data are sparse, the solution is in general dense so that storing it may be unfeasible. Therefore, it is necessary that the solution allows for a memory-saving approximation that can be cheaply stored. An extensive literature treats the case of the aforementioned equations with low-rank right-hand side. This assumption, together with certain hypotheses on the spectral distribution of the matrix coefficients, is a sufficient condition for proving a fast decay in the singular values of the solution. This decay motivates the search for a low-rank approximation so that only low-rank matrices are actually computed and stored remarkably reducing the storage demand. This is the task of the so-called low-rank methods and a large amount of work in this direction has been carried out in the last years. Projection methods have been shown to be among the most effective low-rank methods and in the first part of this thesis we propose some computational enhanchements of the classical algorithms. The case of equations with not necessarily low rank right-hand side has not been deeply analyzed so far and efficient methods are still lacking in the literature. In this thesis we aim to significantly contribute to this open problem by introducing solution methods for this kind of equations. In particular, we address the case when the coefficient matrices and the right-hand side are banded and we further generalize this structure considering quasiseparable data. In the last part of the thesis we study large-scale generalized Sylvester equations and, under some assumptions on the coefficient matrices, novel approximation spaces for their solution by projection are proposed.

We are interested in the numerical solution of large-scale linear matrix equations. In particular, due to their occurrence in many applications, we study the so-called Sylvester and Lyapunov equations. A characteristic aspect of the large-scale setting is that although data are sparse, the solution is in general dense so that storing it may be unfeasible. Therefore, it is necessary that the solution allows for a memory-saving approximation that can be cheaply stored. An extensive literature treats the case of the aforementioned equations with low-rank right-hand side. This assumption, together with certain hypotheses on the spectral distribution of the matrix coefficients, is a sufficient condition for proving a fast decay in the singular values of the solution. This decay motivates the search for a low-rank approximation so that only low-rank matrices are actually computed and stored remarkably reducing the storage demand. This is the task of the so-called low-rank methods and a large amount of work in this direction has been carried out in the last years. Projection methods have been shown to be among the most effective low-rank methods and in the first part of this thesis we propose some computational enhanchements of the classical algorithms. The case of equations with not necessarily low rank right-hand side has not been deeply analyzed so far and efficient methods are still lacking in the literature. In this thesis we aim to significantly contribute to this open problem by introducing solution methods for this kind of equations. In particular, we address the case when the coefficient matrices and the right-hand side are banded and we further generalize this structure considering quasiseparable data. In the last part of the thesis we study large-scale generalized Sylvester equations and, under some assumptions on the coefficient matrices, novel approximation spaces for their solution by projection are proposed.

This thesis initiates a comprehensive exploration of environmental data provided by ARPAE, employing a structured approach to data processing, analytics, and predictive modeling. The primary objective is to clarify the complexities of environmental quality, spanning from data collection and cleansing to in-depth analysis and future forecasting. The initial chapter provides a detailed overview of the Extract, Transform, Load (ETL) processes, explaining the theoretical framework behind these processes, the issues encountered and the solutions applied. Talend Open Studio for Data Integration is introduced along with its components, showcasing their role in transforming raw ARPAE data into a structured and usable format. Additionally, DBeaver is presented as a database management tool facilitating data organization. Then, all the tables created on the database and the jobs are shown in detail. In Chapter 2 the focus shifts to data analytics, where Power BI takes center stage. The aim of this chapter is to visualize and analyze the data collected in the previous one. The creation of informative dashboards becomes pivotal, visually representing trends in key environmental parameters. We carefully examine the data, paying close attention to whether the environmental data complies with legal standards. The final chapter elevates the exploration to predictive analysis, introducing linear regression, ETS, and ARIMA models as tools for forecasting future environmental data based on historical information. These models are applied specifically to critical parameters such as PM10, PM2.5 and O3, aiming to predict their values for the year 2023. Subsequently, we compare these predictions with the available partial data from 2023. By integrating technical methodologies, analytical insights, and predictive capabilities, this thesis aims to contribute to a rich and detailed understanding of both historical trends and potential future trajectories in environmental quality.

Il trattamento numerico dell'equazione di convezione-diffusione con le relative condizioni al bordo, comporta la risoluzione di sistemi lineari algebrici di grandi dimensioni in cui la matrice dei coefficienti è non simmetrica. Risolutori iterativi basati sul sottospazio di Krylov sono ampiamente utilizzati per questi sistemi lineari la cui risoluzione risulta particolarmente impegnativa nel caso di convezione dominante. In questa tesi vengono analizzate alcune strategie di precondizionamento, atte ad accelerare la convergenza di questi metodi iterativi. Vengono confrontati sperimentalmente precondizionatori molto noti come ILU e iterazioni di tipo inner-outer flessibile. Nel caso in cui i coefficienti del termine di convezione siano a variabili separabili, proponiamo una nuova strategia di precondizionamento basata sull'approssimazione, mediante equazione matriciale, dell'operatore differenziale di convezione-diffusione. L'azione di questo nuovo precondizionatore sfrutta in modo opportuno recenti risolutori efficienti per equazioni matriciali lineari. Vengono riportati numerosi esperimenti numerici per studiare la dipendenza della performance dei diversi risolutori dalla scelta del termine di convezione, e dai parametri di discretizzazione.

Il trattamento numerico dell'equazione di convezione-diffusione con le relative condizioni al bordo, comporta la risoluzione di sistemi lineari algebrici di grandi dimensioni in cui la matrice dei coefficienti è non simmetrica. Risolutori iterativi basati sul sottospazio di Krylov sono ampiamente utilizzati per questi sistemi lineari la cui risoluzione risulta particolarmente impegnativa nel caso di convezione dominante. In questa tesi vengono analizzate alcune strategie di precondizionamento, atte ad accelerare la convergenza di questi metodi iterativi. Vengono confrontati sperimentalmente precondizionatori molto noti come ILU e iterazioni di tipo inner-outer flessibile. Nel caso in cui i coefficienti del termine di convezione siano a variabili separabili, proponiamo una nuova strategia di precondizionamento basata sull'approssimazione, mediante equazione matriciale, dell'operatore differenziale di convezione-diffusione. L'azione di questo nuovo precondizionatore sfrutta in modo opportuno recenti risolutori efficienti per equazioni matriciali lineari. Vengono riportati numerosi esperimenti numerici per studiare la dipendenza della performance dei diversi risolutori dalla scelta del termine di convezione, e dai parametri di discretizzazione.