Analisi di strutture nella ricostruzione di immagini e monumenti

1. Definizione ed analisi teorica di nuovi operatori di proiezione per metodi multigrid basati solo sulle informazioni algebriche del problema, applicabili sia a discretizzazioni agli elementi finiti, sia a problemi di ricostruzione di immagini ed a matrici di grafo. 2. Definizione e studio di metodi multilivello regolarizzanti per la ricostruzione di immagini sfocate ed affette da rumore, combinando tecniche nonlineari di edge-preserving con operatori di trasferimento di griglia regolarizzanti che preservano la struttura. 3. Applicazione di condizioni al contorno in grado di preservare segnali smooth a tecniche di regolarizzazione accurate e solitamente computazionalmente costose, (e.g., Total Variation (TV), Regularized Total Least Square (RTLS), preconditioned GMRES, etc.), ricorrendo a trasformate discrete veloci di recente sviluppo (generalizzazione di FFT). 4. Studio di metodi impliciti per EDP paraboliche degeneri con applicazioni sia ai modelli di degrado monumentale sia a problemi di ricostruzione di immagini sfocate con termine regolarizzante non lineare. 5. Analisi spettrale di matrici, con struttura nascosta, non Hermitiane associate a simboli a blocchi con applicazioni al precondizionamento di EDP, alla regolarizzazione non lineare, ed a problemi di ricostruzione di segnali o immagini in cui alcuni campionamenti non sono disponibili o in cui le dimensioni del dominio introducono evidenti distorsioni di tipo prospettico

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8

Simplification of a result on banded Toeplitz matrices and BVM methods

In a recent paper, motivated by the analysis of some BVM methods, Aceto and Trigiante consider tests for proving the positive definiteness of real banded Toeplitz matrices. Here we furnish a new test and we show that the analysis provided by Aceto and Trigiante can be simplified by using known facts on the symbol