Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol

This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let $I_k:=(-\pi,\pi)^k$, let $\mathcal M_s$ be the linear space of complex $s\times s$ matrices, and let $f,g:I_k\to\mathcal M_s$ be functions whose components $f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s,$ belong to $L^\infty$. Consider the matrices $T_n^{-1}(g)T_n(f)$, where $n:=(n_1,\ldots,n_k)$ varies in $\mathbb N^k$ and $T_n(f),T_n(g)$ are the multilevel block Toeplitz matrices of size $n_1\cdots n_ks$ generated by $f,g$. Then $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_\lambda g^{-1}f$, i.e. the family of matrices $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}$ has a global (asymptotic) spectral distribution described by the function $g^{-1}f$, provided $g$ possesses certain properties (which ensure in particular the invertibility of $T_n^{-1}(g)$ for all $n$) and the following topological conditions are met: the essential range of $g^{-1}f$, defined as the union of the essential ranges of the eigenvalue functions $\lambda_j(g^{-1}f),\ j=1,\ldots,s$, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case $g=1$. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems.Comment: 18 pages, 26 figure

Approximation and spectral analysis for large structured linear systems.

In this work we are interested in standard and less standard structured linear systems coming from applications in various _elds of computational mathematics and often modeled by integral and/or di_erential equations. Starting from classical Toeplitz and Circulant structures, we consider some extensions as g-Toeplitz and g-Circulants matrices appearing in several contexts in numerical analysis and applications. Then we consider special matrices arising from collocation methods for di_erential equations: also in this case, under suitable assumptions we observe a Toeplitz structure. More in detail we _rst propose a detailed study of singular values and eigenvalues of g-circulant matrices and then we provide an analysis of distribution of g-Toeplitz sequences. Furthermore, when possible, we consider Krylov space methods with special attention to the minimization of the computational work. When the involved dimensions are large, the Preconditioned Conjugate Gradient (PCG) method is recommended because of the much stronger robustness with respect to the propagation of errors. In that case, crucial issues are the convergence speed of this iterative solver, the use of special techniques (preconditioning, multilevel techniques) for accelerating the convergence, and a careful study of the spectral properties of such matrices. Finally, the use of radial basis functions allow of determining and studying the asymptotic behavior of the spectral radii of collocation matrices approximating elliptic boundary value problems

Multigrid for Qk finite element matrices using a (block) Toeplitz symbol approach

In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the Qk Finite Elements approximation of one-and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div (-a(x) 07\u2022), with a continuous and positive over \u3a9, \u3a9 being an open and bounded subset of R2. While the analysis is performed in one dimension, the numerics are carried out also in higher dimension d 65 2, showing an optimal behavior in terms of the dependency on the matrix size and a substantial robustness with respect to the dimensionality d and to the polynomial degree k

Symmetrization Techniques in Image Deblurring

This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, we first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. The results of many numerical experiments are reported to show the effectiveness of the new preconditioning techniques, including when considering the deblurring of sparse images

Structured matrices coming from PDE approximation theory: spectral analysis, spectral symbol and design of fast iterative solvers.

Partial Differential Equations (PDE) are extensively used in Applied Sciences to model real-world problems. The solution u of a PDE is normally not available in closed form, and so it is necessary to approximate it by means of some numerical method. Despite the differences among the various methods, the principle on which all of them are based is essentially the same: they first discretize the PDE by introducing a mesh, related to some discretization parameter n, and then they compute the corresponding numerical solution u_n, which will converge to u when n tends to infinity, i.e., when the mesh is progressively refined. Now, if both the PDE and the numerical method are linear, the computation of u_n reduces to solving a certain linear system A_n * u_n = f_n whose size d_n tends to infinity with n. In addition, the sequence of discretization matrices A_n often enjoys an asymptotic spectral distribution described by a certain matrix-valued function f, which takes values in the space of Hermitian matrices of a certain size s. This means that, for large n, the eigenvalues of A_n are approximately given by a uniform sampling of the eigenvalue functions lambda_i(f), i=1,...,s, over the domain of f. In this situation, f is called the (spectral) symbol of the sequence of matrices A_n. The identification and the study of the symbol are two interesting issues in themselves, because they provide an accurate information about the asymptotic global behavior of the eigenvalues of A_n. In particular, the number s coincides with the number of "branches" that compose the asymptotic spectrum of A_n. However, the knowledge of the symbol f and of its properties is not only interesting in itself, but can also be used for practical purposes. In particular, it can be used to design effective preconditioned Krylov and multigrid solvers for the linear systems associated with A_n. The reason is clear: the convergence properties of preconditioned Krylov and multigrid methods strongly depend on the spectral features of the matrix to which they are applied. Hence, the spectral information provided by the symbol can be conveniently used for designing fast solvers of this kind. The purpose of this thesis is to present some specific examples, of interest in practical applications, in which the above philosophical discussion comes to life. As our model PDE, we consider classical second-order elliptic differential equations. Concerning the numerical methods that we employ for their solution, we make three choices: the classical Qp Lagrangian Finite Element Method (FEM), the Galerkin B-spline Isogeometric Analysis (IgA) and the B-spline IgA Collocation Method. We first identify and study the symbol f that characterizes the asymptotic spectrum of the discretization matrices A_n arising from these approximation techniques. Then, by exploiting the properties of the symbol, we design fast iterative solvers for the matrices A_n associated with the two numerical methods based on IgA (the Galerkin B-spline IgA and the B-spline IgA Collocation Method)