Isabella Furci

We present novel improvements in the context of symbol-based multigrid procedures for solving large block structured linear systems. We study the application of an aggregation-based grid transfer operator that transforms the symbol of a block Toeplitz matrix from matrix-valued to scalar-valued at the coarser level. Our convergence analysis of the two-grid method reveals the connection between the features of the scalar-valued symbol at the coarser level and the properties of the original matrix-valued one. This allows us to prove the convergence of a V-cycle multigrid with standard grid transfer operators for scalar Toeplitz systems at the coarser levels. Consequently, we extend the class of suitable smoothers for block Toeplitz matrices, focusing on the efficiency of block strategies, particularly the relaxed block Jacobi method. General conditions on smoothing parameters are derived, with emphasis on practical applications where these parameters can be calculated with negligible computational cost. We test the proposed strategies on linear systems stemming from the discretization of differential problems with the [Formula: see text] Lagrangian finite element method or B-spline with nonmaximal regularity. The numerical results show in both cases computational advantages compared to existing methods for block structured linear systems.

Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show 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 proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

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 f...

A powerful tool for analyzing and approximating the singular values and eigenvalues of structured matrices is the theory of GLT sequences. By the GLT theory one can derive a function, 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, 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 and are applicable to a larger class of matrices. We show the applicability of the approach which leads to higher accuracy in some cases when 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.

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 [7]. 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 non-diagonalizable 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 a 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 we 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.