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 of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper 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

Protective versus pathogenic anti-CD4 immunity: insights from the study of natural resistance to HIV infection

HIV-1 exposure causes several dramatic unbalances in the immune system homeostasis. Here, we will focus on the paradox whereby CD4 specific autoimmune responses, which are expected to contribute to the catastrophic loss of most part of the T helper lymphocyte subset in infected patients, may display the characteristics of an unconventional protective immunity in individuals naturally resistant to HIV-1 infection. Reference to differences in fine epitope mapping of these two oppositely polarized outcomes will be presented, with particular reference to partially or totally CD4-gp120 complex-specific antibodies. The fine tuning of the anti-self immune response to the HIV-1 receptor may determine whether viral exposure will result in infection or, alternatively, protective immunity

Cationic Host Defence Peptides:Potential as Antiviral Therapeutics

There is a pressing need to develop new antiviral treatments; of the 60 drugs currently available, half are aimed at HIV-1 and the remainder target only a further six viruses. This demand has led to the emergence of possible peptide therapies, with 15 currently in clinical trials. Advancements in understanding the antiviral potential of naturally occurring host defence peptides highlights the potential of a whole new class of molecules to be considered as antiviral therapeutics. Cationic host defence peptides, such as defensins and cathelicidins, are important components of innate immunity with antimicrobial and immunomodulatory capabilities. In recent years they have also been shown to be natural, broad-spectrum antivirals against both enveloped and non-enveloped viruses, including HIV-1, influenza virus, respiratory syncytial virus and herpes simplex virus. Here we review the antiviral properties of several families of these host peptides and their potential to inform the design of novel therapeutics

Spectral analysis of saddle\u2013point matrices from optimization problems with elliptic PDE constraints

The main focus of this paper is the characterization and exploitation of the asymptotic spectrum of the saddle--point matrix sequences arising from the discretization of optimization problems constrained by elliptic partial differential equations. We uncover the existence of a hidden structure in these matrix sequences, namely, we show that these are indeed an example of Generalized Locally Toeplitz (GLT) sequences. We show that this enables a sharper characterization of the spectral properties of such sequences than the one that is available by using only the fact that we deal with saddle--point matrices. Finally, we exploit it to propose an optimal preconditioner strategy for the GMRES, and Flexible-GMRES methods.Comment: 26 pages, 5 figure

Symbol based convergence analysis in multigrid methods for saddle point problems

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the two-grid method. The present paper analyzes saddle point problems where the blocks are circulant within this framework. It contains sufficient conditions for convergence and optimal parameters for the preconditioning of the unilevel and multilevel saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, it is shown that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and studying the “level independency” property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case

A SYMBOL-BASED ANALYSIS for MULTIGRID METHODS for BLOCK-CIRCULANT and BLOCK-TOEPLITZ SYSTEMS

In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an elegant form in the case of Toeplitz matrices generated by a scalar-valued function. In the block-Toeplitz setting, that is, in the case where the matrix entries are small generic matrices instead of scalars, some algorithms have already been proposed regarding specific applications, and a first rigorous convergence analysis has been performed in [M. Donatelli et al., Numer. Linear Algebra Appl., 28 (2021), e2356]. However, with the existent symbol-based theoretical tools, it is still not possible to prove the convergence of many multigrid methods known in the literature. This paper aims to generalize the previous results, giving more general sufficient conditions on the symbol of the grid transfer operators. In particular, we treat matrix-valued trigonometric polynomials which can be nondiagonalizable and singular at all points, and we express the new conditions in terms of the eigenvectors associated with the ill-conditioned subspace. Moreover, we extend the analysis to the V-cycle method, proving a linear convergence rate under stronger conditions, which resemble those given in the scalar case. In order to validate our theoretical findings, we present a classical block structured problem stemming from an FEM approximation of a second order differential problem. We focus on two multigrid strategies that use the geometric and the standard bisection grid transfer operators and prove that both fall into the category of projectors satisfying the proposed conditions. In addition, using a tensor product argument, we provide a strategy to construct efficient V-cycle procedures in the block multilevel setting

Multigrid methods for block-Toeplitz linear systems: convergence analysis and applications

5nononeIn the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been widely studied. On the other hand, only few papers have investigated the case of block entries, where the entries are small generic matrices of fixed size instead of scalars. In that case the efforts of the researchers have been mainly devoted to specific applications, focusing on algorithmic proposals, but with marginal theoretical results. In this paper, we propose a general two-grid convergence analysis, proving an optimal convergence rate independent of the matrix size, in the case of positive definite block-Toeplitz matrices with generic blocks. In particular, the proof of the approximation property is not a straightforward generalization of the scalar case and, in fact, we have to require a specific commutativity condition on the block symbol of the grid transfer operator. According to the analysis, we define a class of grid transfer operators satisfying the previous theoretical conditions and we propose a strategy to ensure fast multigrid convergence even for more than two grids. Among the numerous applications that lead to the block-Toeplitz structure, high-order Lagrangian finite element methods and staggered discontinuous Galerkin methods are considered in the numerical results, confirming the effectiveness of our proposal and the correctness of the proposed theoretical analysis.noneDonatelli M.; Ferrari P.; Furci I.; Serra-Capizzano S.; Sesana D.Donatelli, M.; Ferrari, P.; Furci, I.; Serra-Capizzano, S.; Sesana, D

Toeplitz momentary symbols: definition, results, and limitations in the spectral analysis of structured matrices

A powerful tool for analyzing and approximating the singular values and eigenvalues of structured matrices is the theory of Generalized Locally Toeplitz (GLT) sequences. By the GLT theory one can derive a function, called the symbol, which describes the singular value or the eigenvalue distribution of the sequence, the latter under precise assumptions. However, for small values of the matrix-size of the considered sequence, the approximations may not be as good as it is desirable, since in the construction of the GLT symbol one disregards small norm and low-rank perturbations. On the other hand, Local Fourier Analysis (LFA) can be used to construct polynomial symbols in a similar manner for discretizations, where the geometric information is present, but the small norm perturbations are retained. The main focus of this paper is the introduction of the concept of sequence of “Toeplitz momentary symbols”, associated with a given sequence of truncated Toeplitz-like matrices. We construct the symbol in the same way as in the GLT theory, but we keep the information of the small norm contributions. The low-rank contributions are still disregarded, and we give an idea on the reason why this is negligible in certain cases and why it is not in other cases, being aware that in presence of high nonnormality the same low-rank perturbation can produce a dramatic change in the eigenvalue distribution. Moreover, a difference with respect to the LFA symbols is that GLT symbols and Toeplitz momentary symbols are more general - just Lebesgue measurable - and are applicable to a larger class of matrices, while in the LFA setting only trigonometric polynomials are considered and more specifically those related to the approximation stencils. We show the applicability of the approach which leads to higher accuracy in some cases, when approximating the singular values and eigenvalues of Toeplitz-like matrices using Toeplitz momentary symbols, compared with the GLT symbol. Finally, since for many applications and their analysis it is often necessary to consider non-square Toeplitz matrices, we formalize and provide some useful definitions, applicable for non-square Toeplitz momentary symbols

The eigenvalue distribution of special 2-by-2 block matrix-sequences with applications to the case of symmetrized toeplitz structures 17

Given a Lebesgue integrable function f over [ 12\u3c0, \u3c0], we consider the sequence of matrices {YnTn[f]}n, where Tn[f] is the n-by-n Toeplitz matrix generated by f and Yn is the anti-identity matrix. Because of the unitary nature of Yn, the singular values of Tn[f] and YnTn[f] coincide. However, the eigenvalues are affected substantially by the action of Yn. Under the assumption that the Fourier coefficients of f are real, we prove that {YnTn[f]}n is distributed in the eigenvalue sense as \ub1|f|. A generalization of this result to the block Toeplitz case is also shown. We also consider the preconditioning introduced by [J. Pestana and A. Wathen, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 273\u2013288] and prove that the preconditioned matrix-sequence is distributed in the eigenvalue sense as \u3c61 under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed