Generalized Locally Toeplitz Research Articles

From convergence in measure to convergence of matrix-sequences through concave functions and singular values

Sequences of matrices with increasing size naturally arise in several areas of science, such as, for example, the numerical discretization of differential and integral equations. An approximation theory for sequences of this kind has recently been developed, with the aim of providing tools for computing their asymptotic singular value and eigenvalue distributions. The cornerstone of this theory is the notion of approximating classes of sequences (a.c.s.), which is also fundamental to the theory of generalized locally Toeplitz (GLT) sequences, and hence to the spectral analysis of PDE discretization matrices. Drawing inspiration from measure theory, here it is introduced a class of functions which are proved to be complete pseudometrics inducing the a.c.s.\ convergence. It is also shown that each of these pseudometrics gives rise to a natural isometry between the spaces of GLT sequences and measurable functions. Furthermore, it is highlighted that the a.c.s.\ convergence is an asymptotic matrix version of the convergence in measure, thus suggesting a way to obtain matrix theory results from measure theory results.

Lusin theorem, GLT sequences and matrix computations: An application to the spectral analysis of PDE discretization matrices

In this paper we consider a general d-dimensional second-order elliptic Partial Differential Equation (PDE) with variable coefficients, and we extend previous results on the spectral distribution of discretization matrices arising from B-spline Isogeometric Analysis (IgA). First, we provide the spectral symbol of the Galerkin B-spline IgA stiffness matrices, under the assumption that the PDE coefficients only belong to L∞. This symbol describes the asymptotic spectral distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). Second, we prove the positive semi-definiteness of the d×d symmetric matrix in the Fourier variables (θ1,…,θd), which appears in the expression of the symbol. This matrix is related to the discretization of the (negative) Hessian operator, and its positive semi-definiteness implies the non-negativity of the symbol. The mathematical arguments used in our derivation are based on the Lusin theorem, on the theory of Generalized Locally Toeplitz (GLT) sequences, and on careful Linear Algebra manipulations of matrix determinants. These arguments are very general and can also be applied to other (local) PDE discretization methods, different from B-spline IgA.

Spectral analysis of coupled PDEs and of their Schur complements via Generalized Locally Toeplitz sequences in 2D

We consider large linear systems of algebraic equations arising from the Finite Element approximation of coupled partial differential equations. As case study we focus on the linear elasticity equations, formulated as a saddle point problem to allow for modelling of purely incompressible materials. To analyse the properties of the arising matrices we use the notion of the so-called spectral symbol in the Generalized Locally Toeplitz (GLT) setting. The fruitful idea behind GLT is that it allows to associate a function, a symbol, with instances of a broad class of structured matrices. The symbol describes the spectrum of the corresponding matrix, easing in this way its analysis and guiding the construction of efficient approximations of the matrix to be used as preconditioners or solvers in a multigrid context. We derive the GLT symbol of the sequence of matrices {An} approximating the elasticity equations. Further, exploiting the property that the GLT class defines an algebra of matrix sequences and the fact that the Schur complements are obtained via elementary algebraic operation on the blocks of An, we derive the symbols fS of the associated sequences of Schur complements {Sn}. As a consequence of the GLT theory, the eigenvalues of Sn for large n are described by a sampling of fS on a uniform grid of its domain of definition. We extend the existing GLT technique with novel elements, related to block-matrices and Schur complement matrices, and illustrate the theoretical findings with numerical tests.

Spectral analysis and spectral symbol of matrices in isogeometric collocation methods

A linear full elliptic second order Partial Dierential Equation (PDE), defined on a d-dimensional domain , is approximated by the isogeometric Galerkin method based on uniform tensor-product Bsplines of degrees (p1;:::; pd). The considered approximation process leads to a d-level stiness matrix, banded in a multilevel sense. This matrix is close to a d-level Toeplitz structure when the PDE coecients are constant and the physical domain is just the hypercube (0; 1) d without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiness matrices has been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coecients and an arbitrary domain , parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiness matrices. This symbol describes the asymptotic eigenvalue distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). The mathematical technique used for computing the symbol is based on the theory of Generalized Locally Toeplitz (GLT) sequences.

The GLT class as a generalized Fourier analysis and applications

Recently, the class of Generalized Locally Toeplitz (GLT) sequences has been introduced as a generalization both of classical Toeplitz sequences and of variable coefficient differential operators and, for every sequence of the class, it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function (the symbol) that can be easily identified. This generalizes the notion of a symbol for differential operators (discrete and continuous) or for Toeplitz sequences for which it is identified through the Fourier coefficients and is related to the classical Fourier Analysis. The GLT class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products: in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol (sparsely vanishing symbol = a symbol which vanishes at most in a set of zero Lebesgue measure). Furthermore, we show that the GLT class virtually includes any Finite Difference or Finite Element discretization of PDEs and, based on this, we demonstrate that our results on GLT sequences can be used in a PDE setting in various directions: (1) as a generalized Fourier Analysis for the study of iterative and semi-iterative methods when dealing with variable coefficients, non-rectangular domains, non-uniform gridding or triangulations, (2) in order to provide a tool for the stability analysis of PDE numerical schemes (e.g., a necessary von Neumann criterium for variable coefficient systems of PDEs is obtained, uniformly with respect to the boundary conditions), (3) for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case, etc. The final part of the paper deals indeed with problems (1)–(3) and other possible directions in which the GLT analysis can be conveniently employed.