arXiv subjects

Sven-Erik Ekström

Publications and source records attributed to Sven-Erik Ekström.

13 recordsLinked to original sources

arXiv2023

Spectral properties of flipped Toeplitz matrices

We study the spectral properties of flipped Toeplitz matrices of the form $H_n(f)=Y_nT_n(f)$, where $T_n(f)$ is the $n\times n$ Toeplitz matrix generated by the function $f$ and $Y_n$ is the $n\times n$ exchange (or flip) matrix having $1$ on the main anti-diagonal and $0$ elsewhere. In particular, under suitable assumptions on $f$, we establish an alternating sign relationship between the eigenvalues of $H_n(f)$, the eigenvalues of $T_n(f)$, and the quasi-uniform samples of $f$. Moreover, after fine-tuning a few known theorems on Toeplitz matrices, we use them to provide localization results for the eigenvalues of $H_n(f)$. Our study is motivated by the convergence analysis of the minimal residual (MINRES) method for the solution of real non-symmetric Toeplitz linear systems of the form $T_n(f)\mathbf x=\mathbf b$ after pre-multiplication of both sides by $Y_n$, as suggested by Pestana and Wathen.

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arXiv2023

From asymptotic distribution and vague convergence to uniform convergence, with numerical applications

Let $\{Λ_n=\{λ_{1,n},\ldots,λ_{d_n,n}\}\}_n$ be a sequence of finite multisets of real numbers such that $d_n\to\infty$ as $n\to\infty$, and let $f:Ω\subset\mathbb R^d\to\mathbb R$ be a Lebesgue measurable function defined on a domain $Ω$ with $0<μ_d(Ω)<\infty$, where $μ_d$ is the Lebesgue measure in $\mathbb R^d$. We say that $\{Λ_n\}_n$ has an asymptotic distribution described by $f$, and we write $\{Λ_n\}_n\sim f$, if \[ \lim_{n\to\infty}\frac1{d_n}\sum_{i=1}^{d_n}F(λ_{i,n})=\frac1{μ_d(Ω)}\int_ΩF(f({\boldsymbol x})){\rm d}{\boldsymbol x}\qquad\qquad(*) \] for every continuous function $F$ with bounded support. If $Λ_n$ is the spectrum of a matrix $A_n$, we say that $\{A_n\}_n$ has an asymptotic spectral distribution described by $f$ and we write $\{A_n\}_n\sim_λf$. In the case where $d=1$, $Ω$~is a bounded interval, $Λ_n\subseteq f(Ω)$ for all $n$, and $f$ satisfies suitable conditions, Bogoya, Böttcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to $0$ of the difference between the properly sorted vector $[λ_{1,n},\ldots,λ_{d_n,n}]$ and the vector of samples $[f(x_{1,n}),\ldots,f(x_{d_n,n})]$, i.e., \[ \lim_{n\to\infty}\,\max_{i=1,\ldots,d_n}|f(x_{i,n})-λ_{τ_n(i),n}|=0, \qquad\qquad(**) \] where $x_{1,n},\ldots,x_{d_n,n}$ is a uniform grid in $Ω$ and $τ_n$ is the sorting permutation. We extend this result to the case where $d\ge1$ and $Ω$ is a Peano--Jordan measurable set (i.e., a bounded set with $μ_d(\partialΩ)=0$). See the rest of the abstract in the manuscript.

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arXiv2023

On the eigenvalues of Toeplitz matrices with two off-diagonals

Consider the Toeplitz matrix $T_n(f)$ generated by the symbol $f(θ)=\hat{f}_r e^{\mathbf{i}rθ}+\hat{f}_0+\hat{f}_{-s} e^{-\mathbf{i}sθ}$, where $\hat{f}_r, \hat{f}_0, \hat{f}_{-s} \in \mathbb{C}$ and $0<r<n,~0<s<n$. For $r=s=1$ we have the classical tridiagonal Toeplitz matrices, for which the eigenvalues and eigenvectors are known. Similarly, the eigendecompositions are known for $1<r=s$, when the generated matrices are ``symmetrically sparse tridiagonal''. In the current paper we study the eigenvalues of $T_n(f)$ for $1\leq r<s$, which are ``non-symmetrically sparse tridiagonal''. We propose an algorithm which constructs one or two ad hoc matrices smaller than $T_n(f)$, whose eigenvalues are sufficient for determining the full spectrum of $T_n(f)$. The algorithm is explained through use of a conjecture for which examples and numerical experiments are reported for supporting it and for clarifying the presentation. Open problems are briefly discussed.

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arXiv2022

Theoretical results for eigenvalues, singular values, and eigenvectors of (flipped) Toeplitz matrices and related computational proposals

In a series of recent papers the spectral behavior of the matrix sequence $\{Y_nT_n(f)\}$ is studied in the sense of the spectral distribution, where $Y_n$ is the main antidiagonal (or flip matrix) and $T_n(f)$ is the Toeplitz matrix generated by the function $f$, with $f$ being Lebesgue integrable and with real Fourier coefficients. This kind of study is also motivated by computational purposes for the solution of the related large linear systems using the (preconditioned) MINRES algorithm. Here we complement the spectral study with more results holding both asymptotically and for a fixed dimension $n$, and with regard to eigenvalues, singular values, and eigenvectors of $T_n(f), Y_nT_n(f)$ and to several relationships among them: beside fast linear solvers, a further target is the design of ad hoc procedures for the computation of the related spectra via matrix-less algorithms, with a cost being linear in the number of computed eigenvalues. We emphasize that the challenge of the case of non-monotone generating functions is considered in the current work, for which the previous matrix-less algorithms fail. Numerical experiments are reported and commented, with the aim of showing in a visual way the theoretical analysis.

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arXiv2022

Preconditioners for Fractional Diffusion Equations Based on the Spectral Symbol

It is well known that the discretization of fractional diffusion equations (FDEs) with fractional derivatives $α\in(1,2)$, using the so-called weighted and shifted Grünwald formula, leads to linear systems whose coefficient matrices show a Toeplitz-like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so-called Generalized Locally Toeplitz (GLT) class. Conversely, when the given FDE have constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function $\mathcal{F}_α$, called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one and two dimensional cases. In this note we propose a new preconditioner denoted by $\mathcal{P}_{\mathcal{F}_α}$ which belongs to the $τ$ algebra and it is based on the spectral symbol $\mathcal{F}_α$. Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one-dimensional case, the new preconditioner performs better in the more challenging multi-dimensional setting.

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arXiv2022

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.

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arXiv2022

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 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.

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arXiv2021

Spectral and norm estimates for matrix sequences arising from a finite difference approximation of elliptic operators

When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for matrix sequences arising from the approximation of the Laplacian via ad hoc finite differences. The analysis involves several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix sequences. Several numerical experiments are conducted, which confirm the correctness of the theoretical findings.

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arXiv2020

Eigenvalues and Eigenvectors of Tau Matrices with Applications to Markov Processes and Economics

In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called $τ_{\varepsilon,φ}$ algebra, a generalization of the more known $τ$ algebra originally proposed by Bini and Capovani. We study the properties of eigenvalues and eigenvectors of the generator $T_{n,\varepsilon,φ}$ of the $τ_{\varepsilon,φ}$ algebra. In particular, we derive the asymptotics for the outliers of $T_{n,\varepsilon,φ}$ and the associated eigenvectors; we obtain equations for the eigenvalues of $T_{n,\varepsilon,φ}$, which provide also the eigenvectors of $T_{n,\varepsilon,φ}$; and we compute the full eigendecomposition of $T_{n,\varepsilon,φ}$ in the specific case $\varepsilonφ=1$. We also present applications of our results in the context of queuing models, random walks, and diffusion processes, with a special attention to their implications in the study of wealth/income inequality and portfolio dynamics.

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arXiv2019

A Matrix-Less Method to Approximate the Spectrum and the Spectral Function of Toeplitz Matrices with Complex Eigenvalues

It is known that the generating function $f$ of a sequence of Toeplitz matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the eigenvalues of $T_n(f)$ if $f$ is not real. In a recent paper, we assume as a working hypothesis that, if the eigenvalues of $T_n(f)$ are real for all $n$, then they admit an asymptotic expansion where the first function $g$ appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of $T_n(f)$. In this paper we extend this idea to Toeplitz matrices with complex eigenvalues. The paper is predominantly a numerical exploration of different typical cases, and presents several avenues of possible future research.

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arXiv2019

A Matrix-Less Method to Approximate the Spectrum and the Spectral Function of Toeplitz Matrices with Real Eigenvalues

It is known that the generating function $f$ of a sequence of Toeplitz matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the eigenvalues of $T_n(f)$ if $f$ is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of $T_n(f)$ are real for all $n$, then they admit an asymptotic expansion of the same type as considered in previous works [1,10,12,13], where the first function $g$ appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of $T_n(f)$. After validating this working hypothesis through a number of numerical experiments, drawing inspiration from [12], we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $g$. The proposed algorithm is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $g$. Future research directions are outlined at the end of the paper.

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arXiv2019

Approximating the Perfect Sampling Grids for Computing the Eigenvalues of Toeplitz-like Matrices Using the Spectral Symbol

In a series of papers the author and others have studied an asymptotic expansion of the errors of the eigenvalue approximation, using the spectral symbol, in connection with Toeplitz (and Toeplitz-like) matrices, that is, $E_{j,n}$ in $λ_j(A_n)=f(θ_{j,n})+E_{j,n}$, $A_n=T_n(f)$, $f$ real-valued cosine polynomial. In this paper we instead study an asymptotic expansion of the errors of the equispaced sampling grids $θ_{j,n}$, compared to the exact grids $ξ_{j,n}$ (where $λ_j(A_n)=f(ξ_{j,n})$), that is, $E_{j,n}$ in $ξ_{j,n}=θ_{j,n}+E_{j,n}$. We present an algorithm to approximate the expansion. Finally we show numerically that this type of expansion works for various kind of Toeplitz-like matrices (Toeplitz, preconditioned Toeplitz, low-rank corrections of them). We critically discuss several specific examples and we demonstrate the superior numerical behavior of the present approach with respect to the previous ones.

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arXiv2018

A Multigrid method for nonlocal problems: non-diagonally dominant Toeplitz-plus-tridiagonal systems

The nonlocal problems have been used to model very different applied scientific phenomena, which involve the fractional Laplacian when one looks at the Lévy processes and stochastic interfaces. This paper deals with the nonlocal problems on a bounded domain, where the stiffness matrices of the resulting systems are Toeplitz-plus-tridiagonal and far from being diagonally dominant, as it occurs when dealing with linear finite element approximations. By exploiting a weakly diagonally dominant Toeplitz property of the stiffness matrices, the optimal convergence of the two-grid method is well established [Fiorentino and Serra-Capizzano, {\em SIAM J. Sci. Comput.}, {17} (1996), pp. 1068--1081; Chen and Deng, {\em SIAM J. Matrix Anal. Appl.}, {38} (2017), pp. 869--890]; and there are still questions about best ways to define coarsening and interpolation operator when the stiffness matrix is far from being weakly diagonally dominant [Stüben, {\em J. Comput. Appl. Math.}, {128} (2001), pp. 281--309]. In this work, using spectral indications from our analysis of the involved matrices, the simple (traditional) restriction operator and prolongation operator are employed in order to handle general algebraic systems which are {\em neither Toeplitz nor weakly diagonally dominant} corresponding to the fractional Laplacian kernel and the constant kernel, respectively. We focus our efforts on providing the detailed proof of the convergence of the two-grid method for such situations. Moreover, the convergence of the full multigrid is also discussed with the constant kernel. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform, where $N$ is the number of the grid points.

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