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In previous works, Bohemian matrices have attracted the attention of several researchers for theirrich combinatorial structure, and they have been studied intensively from several points of view, including height,determinants, characteristic polynomials, normality, and stability. Here we consider a selected number of examples ofupper Hessenberg and Toeplitz Bohemian matrix sequences whose entries belong to the population P = {0, ±1},and we propose a connection with the spectral theory of Toeplitz matrix sequences and Generalized Locally Toeplitz(GLT) matrix sequences in order to give results on the localization and asymptotical distribution of their spectra andsingular values. Numerical experiments that support the mathematical study are reported. A conclusion section endsthe note in order to illustrate the applicability of the proposed tools to more general cases.

Recent years have seen the rapid growth in interest towards fractional calculus. Fractional calculus plays an important role in modelling anomalous diffusion phenomena, however closed-form analytical solutions of such equations are rarely available, hence numerical estimates are needed. In this thesis we consider various fractional diffusion equations (FDEs), where different fractional derivative definitions and related discretizations are involved and we focus on multigrid-based approaches for solving the associated linear systems. Precisely, we will leverage the spectral properties of the coefficient matrix, retrieved by exploiting its structure, to design ad-hoc (tailored) multigrid solvers or preconditioners for Krylov methods. We develop a new approach to compute the Jacobi weight, which is versatile enough to work with various FDEs and allows to build a parameter free multigrid method. Moreover, in the case of uniform meshes, we exploit the knowledge about anisotropic integer-order partial diffusion equations to deal with anisotropic FDEs, by building a robust multigrid-based solver. Furthermore, we study the behavior of multigrid methods as parallel-in-time solvers and, then, we provide a new second-order accurate finite volume approximation and related ad-hoc multigrid solver. Finally, we extend our multigrid strategies to deal with a singular one-dimensional space-FDE discretized over non-uniform meshes.

Recent years have seen the rapid growth in interest towards fractional calculus. Fractional calculus plays an important role in modelling anomalous diffusion phenomena, however closed-form analytical solutions of such equations are rarely available, hence numerical estimates are needed. In this thesis we consider various fractional diffusion equations (FDEs), where different fractional derivative definitions and related discretizations are involved and we focus on multigrid-based approaches for solving the associated linear systems. Precisely, we will leverage the spectral properties of the coefficient matrix, retrieved by exploiting its structure, to design ad-hoc (tailored) multigrid solvers or preconditioners for Krylov methods. We develop a new approach to compute the Jacobi weight, which is versatile enough to work with various FDEs and allows to build a parameter free multigrid method. Moreover, in the case of uniform meshes, we exploit the knowledge about anisotropic integer-order partial diffusion equations to deal with anisotropic FDEs, by building a robust multigrid-based solver. Furthermore, we study the behavior of multigrid methods as parallel-in-time solvers and, then, we provide a new second-order accurate finite volume approximation and related ad-hoc multigrid solver. Finally, we extend our multigrid strategies to deal with a singular one-dimensional space-FDE discretized over non-uniform meshes.