Symbol Based Convergence Analysis in Block Multigrid Methods with applications for Stokes problems

The main focus of this paper is the study of efficient multigrid methods for large linear system with a particular saddle-point structure. In particular, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function $\mathbf{f}$, which consequently is a matrix-valued function with dimension depending on the block of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes equation. We show the efficiency of the methods studying the hidden $9\times 9$ block structure of the obtained matrix sequence proposing an efficient algebraic multigrid method with convergence rate independent of the matrix size. Moreover, we present several numerical tests comparing the results with different known strategies

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

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A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization with space-time grid for a parabolic diffusion problem and from the approximation of distributed order fractional equations. For this purpose we will use the classical GLT theory and the new concept of GLT momentary symbols. The first permits to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices, the latter permits to derive a function, which describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix-sizes but under given assumptions. The note is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

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Spectral analysis and fast methods for structured matrix sequences and PDE discretizations.

When simulating phenomena in physics, engineering, or applied sciences, often one has to deal with functional equations that do not admit an analytical solution. Describing these real situations is, however, possible, resorting to one of its numerical approximations and treating the resulting mathematical representation. This thesis is placed in this context: Indeed the purpose is that of furnishing several useful tools to deal with some computational problems, stemming from discretization techniques. In most of the cases the numerical methods we analyse are the classical Qp Lagrangian FEM and the more recent Galerkin B-spline Isogeometric Analysis (IgA) approximation and Staggered Discontinuous Galerkin (DG) methods. As our model PDE, we consider classical second-order elliptic differential equations and the Incompressible Navier-Stokes equations. In all these situations the resulting matrix sequences {An}n possess a structure, namely they belong to the class of Toeplitz matrix sequences or to the more general class of Generalized Locally Toeplitz (GLT) matrix sequences, in the most general block k-level case. Consequently, the spectral analysis of the coefficient matrices plays a crucial role for an efficient and fast resolution. Indeed the convergence properties of iterative methods proposed, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol of the coefficient matrix sequence. In our setting the symbol is a function which asymptotically provides a reasonable approximation of the eigenvalues [singular values] of An by its evaluations of an uniform grid on its domain. These reasons, and many others, make the research of more and more efficient eigensolvers relevant and topical. In this direction, the second goal of this thesis is to provide new tools for computing the spectrum of preconditioned banded symmetric Toeplitz matrices, Toeplitz-like matrices, n-1K[p]n , nM[p]n , n-2L[p]n , coming from the B-spline IgA approximation of \u2013u\u201d = \uf06cu, plus its multivariate counterpart for -\uf044u = \uf06cu, and block and preconditioned block banded symmetric Toeplitz matrices. For all the above cases we propose new algorithms based on the classical concept of symbol, but with an innovative view on the errors of the approximation of eigenvalues by the uniform sampling of the symbol. The algorithms devised are special interpolation-extrapolation procedures performed with a high level of accuracy and only at the cost of computing of the eigenvalues of a moderate number of small sized matrices

Spectral analysis and fast methods for structured matrix sequences and PDE discretizations.

When simulating phenomena in physics, engineering, or applied sciences, often one has to deal with functional equations that do not admit an analytical solution. Describing these real situations is, however, possible, resorting to one of its numerical approximations and treating the resulting mathematical representation. This thesis is placed in this context: Indeed the purpose is that of furnishing several useful tools to deal with some computational problems, stemming from discretization techniques. In most of the cases the numerical methods we analyse are the classical Qp Lagrangian FEM and the more recent Galerkin B-spline Isogeometric Analysis (IgA) approximation and Staggered Discontinuous Galerkin (DG) methods. As our model PDE, we consider classical second-order elliptic differential equations and the Incompressible Navier-Stokes equations. In all these situations the resulting matrix sequences {An}n possess a structure, namely they belong to the class of Toeplitz matrix sequences or to the more general class of Generalized Locally Toeplitz (GLT) matrix sequences, in the most general block k-level case. Consequently, the spectral analysis of the coefficient matrices plays a crucial role for an efficient and fast resolution. Indeed the convergence properties of iterative methods proposed, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol of the coefficient matrix sequence. In our setting the symbol is a function which asymptotically provides a reasonable approximation of the eigenvalues [singular values] of An by its evaluations of an uniform grid on its domain. These reasons, and many others, make the research of more and more efficient eigensolvers relevant and topical. In this direction, the second goal of this thesis is to provide new tools for computing the spectrum of preconditioned banded symmetric Toeplitz matrices, Toeplitz-like matrices, n-1K[p]n , nM[p]n , n-2L[p]n , coming from the B-spline IgA approximation of –u” = u, plus its multivariate counterpart for -u = u, and block and preconditioned block banded symmetric Toeplitz matrices. For all the above cases we propose new algorithms based on the classical concept of symbol, but with an innovative view on the errors of the approximation of eigenvalues by the uniform sampling of the symbol. The algorithms devised are special interpolation-extrapolation procedures performed with a high level of accuracy and only at the cost of computing of the eigenvalues of a moderate number of small sized matrices

Spectral analysis and fast methods for structured matrix sequences and PDE discretizations.

When simulating phenomena in physics, engineering, or applied sciences, often one has to deal with functional equations that do not admit an analytical solution. Describing these real situations is, however, possible, resorting to one of its numerical approximations and treating the resulting mathematical representation. This thesis is placed in this context: Indeed the purpose is that of furnishing several useful tools to deal with some computational problems, stemming from discretization techniques. In most of the cases the numerical methods we analyse are the classical Qp Lagrangian FEM and the more recent Galerkin B-spline Isogeometric Analysis (IgA) approximation and Staggered Discontinuous Galerkin (DG) methods. As our model PDE, we consider classical second-order elliptic differential equations and the Incompressible Navier-Stokes equations. In all these situations the resulting matrix sequences {An}n possess a structure, namely they belong to the class of Toeplitz matrix sequences or to the more general class of Generalized Locally Toeplitz (GLT) matrix sequences, in the most general block k-level case. Consequently, the spectral analysis of the coefficient matrices plays a crucial role for an efficient and fast resolution. Indeed the convergence properties of iterative methods proposed, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol of the coefficient matrix sequence. In our setting the symbol is a function which asymptotically provides a reasonable approximation of the eigenvalues [singular values] of An by its evaluations of an uniform grid on its domain. These reasons, and many others, make the research of more and more efficient eigensolvers relevant and topical. In this direction, the second goal of this thesis is to provide new tools for computing the spectrum of preconditioned banded symmetric Toeplitz matrices, Toeplitz-like matrices, n-1K[p]n , nM[p]n , n-2L[p]n , coming from the B-spline IgA approximation of –u” = u, plus its multivariate counterpart for -u = u, and block and preconditioned block banded symmetric Toeplitz matrices. For all the above cases we propose new algorithms based on the classical concept of symbol, but with an innovative view on the errors of the approximation of eigenvalues by the uniform sampling of the symbol. The algorithms devised are special interpolation-extrapolation procedures performed with a high level of accuracy and only at the cost of computing of the eigenvalues of a moderate number of small sized matrices