Modeling Ultrafast Spectroscopy of the NADH Dimer: From UV-VIS to X-rays

Ultrafast pump-probe spectroscopy is a conceptually simple and versatile tool for resolving photoinduced dynamics in molecular systems. Due to the fast development of new experimental setups, such as synchrotron light sources and X-ray free electron lasers (XFEL), new spectral windows are becoming accessible. On the one hand, these sources have enabled scientist to access faster and faster time scales and to reach unprecedent insights into dynamical properties of matter. On the other hand, the complementarity of well-developed and novel techniques allows to study the same physical process from different points of views, integrating the advantages and overcoming the limitations of each approach. In this context, it is highly desirable to reach a clear understanding of which type of spectroscopy is more suited to capture a certain facade of a given photo-induced process, that is, to establish a correlation between the process to be unraveled and the technique to be used. In this thesis, I will show how computational spectroscopy can be a tool to establish such a correlation. I will study a specific process, which is the ultrafast energy transfer in the nicotinamide adenine dinucleotide dimer (NADH). This process will be observed in different spectral windows (from UV-VIS to X-rays), accessing the ability of different spectroscopic techniques to unravel the system evolution by means of state-of-the-art theoretical models and methodologies. The comparison of different spectroscopic simulations will demonstrate their complementarity, eventually allowing to identify the type of spectroscopy that is best suited to resolve the ultrafast energy transfer.

Graph theory and geometric deep learning

La seguente tesi propone un’introduzione al geometric deep learning. Nella prima parte vengono presentati i concetti principali di teoria dei grafi ed introdotta una dinamica di diffusione su grafo, in analogia con l’equazione del calore. A seguire, iniziando dal linear classifier verranno introdotte le architetture che hanno portato all’ideazione delle graph convolutional networks. In conclusione, si analizzano esempi di alcuni algoritmi utilizzati nel geometric deep learning e si mostra una loro implementazione sul Cora dataset, un insieme di dati con struttura a grafo.

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.