A Multilevel Gradient Method for Optimization Problems

I problemi di ottimizzazione di dimensione finita di larga scala spesso derivano dalla discretizzazione di problemi di dimensione infinita. È perciò possibile descrivere il problema di ottimizzazione su più livelli discreti. Lavorando su un livello più basso di quello del problema considerato, si possono calcolare soluzioni approssimate che saranno poi punti di partenza per il problema di ottimizzazione al livello più fine. I metodi multilivello, già ampiamente presenti in letteratura a partire dagli anni Novanta, sfruttano tale caratteristica dei problemi di ottimizzazione per migliorare le prestazioni dei metodi di ottimizzazione standard. L’obiettivo di questa tesi è quello di implementare una variante multilivello del metodo del gradiente (MGM) e di testarlo su due diversi campi: la risoluzione delle Equazioni alle Derivate Parziali la ricostruzione di immagini. In questo elaborato viene illustrata la teoria dello schema multilivello e presentato l’algoritmo di MGM utilizzato nei nostri esperimenti. Sono poi discusse le modalità di utilizzo di MGM per i due problemi sopra presentati. Per il problema PDE, i risultati ottenuti mostrano un ottimo comportamento di MGM rispetto alla implementazione classica ad un livello. I risultati ottenuti per il problema di ricostruzione di immagini, al contrario delle PDEs, evidenziano come MGM sia efficace solo in determinate condizioni.

Compressed sensing in digital tomosynthesis reconstruction

In this work we study a model for the breast image reconstruction in Digital Tomosynthesis, that is a non-invasive and non-destructive method for the three-dimensional visualization of the inner structures of an object, in which the data acquisition includes measuring a limited number of low-dose two-dimensional projections of an object by moving a detector and an X-ray tube around the object within a limited angular range. The problem of reconstructing 3D images from the projections provided in the Digital Tomosynthesis is an ill-posed inverse problem, that leads to a minimization problem with an object function that contains a data fitting term and a regularization term. The contribution of this thesis is to use the techniques of the compressed sensing, in particular replacing the standard least squares problem of data fitting with the problem of minimizing the 1-norm of the residuals, and using as regularization term the Total Variation (TV). We tested two different algorithms: a new alternating minimization algorithm (ADM), and a version of the more standard scaled projected gradient algorithm (SGP) that involves the 1-norm. We perform some experiments and analyse the performance of the two methods comparing relative errors, iterations number, times and the qualities of the reconstructed images. In conclusion we noticed that the use of the 1-norm and the Total Variation are valid tools in the formulation of the minimization problem for the image reconstruction resulting from Digital Tomosynthesis and the new algorithm ADM has reached a relative error comparable to a version of the classic algorithm SGP and proved best in speed and in the early appearance of the structures representing the masses.

Regularization methods for the solution of a nonlinear least-squares problem in tomography

In this work we study a polyenergetic and multimaterial model for the breast image reconstruction in Digital Tomosynthesis, taking into consideration the variety of the materials forming the object and the polyenergetic nature of the X-rays beam. The modelling of the problem leads to the resolution of a high-dimensional nonlinear least-squares problem that, due to its nature of inverse ill-posed problem, needs some kind of regularization. We test two main classes of methods: the Levenberg-Marquardt method (together with the Conjugate Gradient method for the computation of the descent direction) and two limited-memory BFGS-like methods (L-BFGS). We perform some experiments for different values of the regularization parameter (constant or varying at each iteration), tolerances and stop conditions. Finally, we analyse the performance of the several methods comparing relative errors, iterations number, times and the qualities of the reconstructed images.

Ab-initio determination of x-ray absorption near edge structure (xanes) spectra in an ultrasoft and norm conserving pseudopotentials scheme

X-ray absorption spectroscopy (XAS) is a powerful means of investigation of structural and electronic properties in condensed -matter physics. Analysis of the near edge part of the XAS spectrum, the so – called X-ray Absorption Near Edge Structure (XANES), can typically provide the following information on the photoexcited atom: - Oxidation state and coordination environment. - Speciation of transition metal compounds. - Conduction band DOS projected on the excited atomic species (PDOS). Analysis of XANES spectra is greatly aided by simulations; in the most common scheme the multiple scattering framework is used with the muffin tin approximation for the scattering potential and the spectral simulation is based on a hypothetical, reference structure. This approach has the advantage of requiring relatively little computing power but in many cases the assumed structure is quite different from the actual system measured and the muffin tin approximation is not adequate for low symmetry structures or highly directional bonds. It is therefore very interesting and justified to develop alternative methods. In one approach, the spectral simulation is based on atomic coordinates obtained from a DFT (Density Functional Theory) optimized structure. In another approach, which is the object of this thesis, the XANES spectrum is calculated directly based on an ab – initio DFT calculation of the atomic and electronic structure. This method takes full advantage of the real many-electron final wavefunction that can be computed with DFT algorithms that include a core-hole in the absorbing atom to compute the final cross section. To calculate the many-electron final wavefunction the Projector Augmented Wave method (PAW) is used. In this scheme, the absorption cross section is written in function of several contributions as the many-electrons function of the finale state; it is calculated starting from pseudo-wavefunction and performing a reconstruction of the real-wavefunction by using a transform operator which contains some parameters, called partial waves and projector waves. The aim of my thesis is to apply and test the PAW methodology to the calculation of the XANES cross section. I have focused on iron and silicon structures and on some biological molecules target (myoglobin and cytochrome c). Finally other inorganic and biological systems could be taken into account for future applications of this methodology, which could become an important improvement with respect to the multiscattering approach.

Extracting evolutionary information from the spectral decomposition of early-type galaxies.

Holding the major share of stellar mass in galaxies and being also old and passively evolving, early-type galaxies (ETGs) are the primary probes in investigating these various evolution scenarios, as well as being useful means to provide insights on cosmological parameters. In this thesis work I focused specifically on ETGs and on their capability in constraining galaxy formation and evolution; in particular, the principal aims were to derive some of the ETGs evolutionary parameters, such as age, metallicity and star formation history (SFH) and to study their age-redshift and mass-age relations. In order to infer galaxy physical parameters, I used the public code STARLIGHT: this program provides a best fit to the observed spectrum from a combination of many theoretical models defined in user-made libraries. the comparison between the output and input light-weighted ages shows a good agreement starting from SNRs of ∼ 10, with a bias of ∼ 2.2% and a dispersion 3%. Furthermore, also metallicities and SFHs are well reproduced. In the second part of the thesis I performed an analysis on real data, starting from Sloan Digital Sky Survey (SDSS) spectra. I found that galaxies get older with cosmic time and with increasing mass (for a fixed redshift bin); absolute light-weighted ages, instead, result independent from the fitting parameters or the synthetic models used. Metallicities, instead, are very similar from each other and clearly consistent with the ones derived from the Lick indices. The predicted SFH indicates the presence of a double burst of star formation. Velocity dispersions and extinctiona are also well constrained, following the expected behaviours. As a further step, I also fitted single SDSS spectra (with SNR∼ 20), to verify that stacked spectra gave the same results without introducing any bias: this is an important check, if one wants to apply the method at higher z, where stacked spectra are necessary to increase the SNR. Our upcoming aim is to adopt this approach also on galaxy spectra obtained from higher redshift Surveys, such as BOSS (z ∼ 0.5), zCOSMOS (z 1), K20 (z ∼ 1), GMASS (z ∼ 1.5) and, eventually, Euclid (z 2). Indeed, I am currently carrying on a preliminary study to estabilish the applicability of the method to lower resolution, as well as higher redshift (z 2) spectra, just like the Euclid ones.