# Sven-Erik EkströmUppsala University | UU · Department of Information Technology

Sven-Erik Ekström

Ph.D.

## About

44

Publications

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272

Citations

Introduction

Spectral analysis of matrix sequences.
www.2pi.se

**Skills and Expertise**

Additional affiliations

February 2020 - January 2021

June 2019 - February 2020

October 2018 - May 2019

Education

July 2006 - May 2018

September 1996 - June 2006

## Publications

Publications (44)

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 eigen-values of T n (f) in the non-Hermitian setting. In a recent paper, we assumed the following working hypothesis: if the eigenvalues of T n (f) are real for all n,

In a series of recent papers the spectral behavior of the matrix sequence {YnTn(f)} is studied in the sense of the spectral distribution, where Yn is the main antidiagonal (or flip matrix) and Tn(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 mo...

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 assumpti...

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Independently and under the milder hypothesis that f is even and monotone over [0,π], matrix-less algorithms have been developed for the fast eigenvalue computati...

It is well known that the discretization of fractional diffusion equations (FDEs) with fractional derivatives α ∈ (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 bel...

In a series of recent papers the spectral behavior of the matrix sequence {YnTn(f)} is studied in the sense of the spectral distribution, where Yn is the main antidiagonal (or flip matrix) and Tn(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 mot...

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 concep...

It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f )}n may not describe the asymptotic distribution of the eigenvalues of Tn(f ) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f ) are real for all n, then they admit an asymptotic expansion of the same type as consid...

A powerful tool for analyzing and approximating the singular values and eigen-values 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 assumpt...

Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Independently and under the milder hypothesis that f is even and monotonic over [0, π], matrix-less algorithms have been developed for the fast eigenvalue computa...

Sequences of structured matrices of increasing size arise in many scientific applications and especially in the numerical discretization of linear differential problems. We assume as a working hypothesis that the eigenvalues of a matrix X_n belonging to a sequence of this kind are given by a regular expansion. Based on the working hypothesis, which...

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 invol...

In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called τ_{ε,ϕ} algebra, a generalization of the well known τ algebra. We study the properties of eigenvalues and eigenvectors of the generator T_{n,ε,ϕ} of the τ_{ε,ϕ} algebra. In particular, we derive the asymptotics for the outliers of T_{n,ε,ϕ} and the asso...

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 theory of GLT sequences one can derive a function, called the symbol, which describes the singular value or eigenvalue distribution of the sequence. However, for small value...

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 or far fro...

In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called τ_{ε,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,ε,ϕ} of the τ_{ε,ϕ} algebra. In particular, we derive the asymptotics f...

In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called $\tau_{\varepsilon,\varphi}$ algebra, a generalization of the more known $\tau$ algebra originally proposed by Bini and Capovani. We study the properties of eigenvalues and eigenvectors of the generator $T_{n,\varepsilon,\varphi}$ of the $\tau_{\varepsi...

It is well known that the discretization of fractional diffusion equations (FDE) with fractional derivatives $\alpha\in(1,2)$, using the so-called weighted and shifted Gr{\"u}nwald formula, leads to linear systems whose coefficient matrices show a Toeplitz-like structure. More precisely, in the case of variable coefficients, the related matrix sequ...

We present an example-based exposition and review of recent advances in symbol-based spectral analysis. We consider constant- and variable-coefficient, second-order eigenvalue problems discretized through the (isogeometric) Galerkin method based on B-splines of degree p and smoothness C^k, 0<=k<=p-1. For each discretized problem,
we compute the so-...

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 t...

In the past few years, Bogoya, B\"ottcher, Grudsky, and Maximenko obtained the precise asymptotic expansion for the eigenvalues of a Toeplitz matrix $T_n(f)$, under suitable assumptions on the generating function $f$, as the matrix size $n$ goes to infinity. On the basis of several numerical experiments, it was conjectured by Serra-Capizzano that a...

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...

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, Ej,n in λj(An) = f (θj,n) + Ej,n, An = Tn(f), f real-valued cosine polynomial. In this paper we instead study an asymptotic exp...

We consider the B‐spline isogeometric analysis approximation of the Laplacian eigenvalue problem −Δu = λu over the d‐dimensional hypercube (0,1)d. By using tensor‐product arguments, we show that the eigenvalue–eigenvector structure of the resulting discretization matrix is completely determined by the eigenvalue–eigenvector structure of the matrix...

The nonlocal problems have been used to model very different applied scientific phenomena, which involve the fractional Laplacian when one looks at the L\'{e}vy 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...

We present an example-based exposition and review of recent advances in symbol-based spectral analysis. We consider constant- and variable-coefficient, second-order eigenvalue problems discretized through the (isogeometric) Galerkin method based on B-splines of degree p and smoothness C^k, 0 <= k <= p-1. For each discretized problem,
we compute the...

In such a short note we consider the spectral analysis of large matrices coming from the numerical approximation of the eigenvalue problem −(a(x)u (x)) = λ b(x)u(x), x ∈ (0, 1), where u(0) and u(1) are given, by using isogeometric methods based on B-splines. We give precise estimates for the extremal eigenvalues and global distributional results. T...

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix Tn ( f ), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f . A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where f is an s × s m...

Bogoya, B\"ottcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices $\{T_n(f)\}$, under suitable assumptions on the associated generating function $f$.
In this paper we provide numerical evidence that some of these assumptions can be relaxed and extended to the ca...

When modeling natural phenomena with linear partial differential equations, the discretized system of equations is in general represented by a matrix. To solve or analyze these systems, we are often interested in the spectral behavior of these matrices. Whenever the matrices of interest are Toeplitz, or Toeplitz-like, we can use the theory of Gener...

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained for the eigenvalues of a Toeplitz matrix, under suitable assumptions on the generating function, the precise asymptotic expansion as the matrix size goes to infinity. In this paper we provide numerical evidence that some of these assumptions can be relaxed. Moreover, based on the eigen...

It is known that for the tridiagonal Toeplitz matrix, having the main diagonal with constant a0=2 and the two first off-diagonals with constants a1=-1 (lower) and a-1=-1 (upper), there exists closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. The latter matrix corresponds to the well known case of the 1...

In this paper we consider the B-spline IgA approximation of the second-order eigenvalue problem -Delta u = lambda u on Omega =(0,1)d, with zero Dirichlet boundary conditions and with Delta = sumj=1dpartial2/partial xj2, dge 1. We use B-splines of degree p=(p1,...,pd) and maximal smoothness and we consider the natural Galerkin approach. By using ele...

In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained for the eigenvalues of a Toeplitz matrix Tn(f), under suitable assumptions on the generating function f, the precise asymptotic expansion as the matrix size n goes to infinity. On the basis of several numerical experiments, it was conjectured by Serra-Capizzano that a complete...

http://www.it.uu.se/research/publications/reports/2016-017/
It is well-known that the eigenvalues of (real) symmetric banded Toeplitz matrices of size n are approximately given by an equispaced sampling of the symbol f, up to an error which grows at most as h = 1/(n + 1), where the symbol is a real-valued cosine polynomial. Under the condition tha...

The understanding of flow problems, and finding their solution, has been important for most of human history, from the design of aqueducts to boats and airplanes. The use of physical miniature models and wind tunnels were, and still are, useful tools for design, but with the development of computers, an increasingly large part of the design process...

The discontinuous Galerkin (DG) method can be viewed as a generalization to higher orders of the finite volume method. At
lowest order, the standard DG method reduces to the cell-centered finite volume method.We introduce for the Euler equations
an alternative DG formulation that reduces to the vertex-centered version of the finite volume method at...

Agglomoration multigrid is used in many finite-volume codes for aerodynamic computations in order to reduce solution times.
We show that an existing agglomeration multigrid solver developed for equations discretized with a vertex-centered, edge-based
finite-volume scheme can be extended to accelerate convergence also for a vertex-centered discontin...

The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible flu- ids. The discontinuous Galerkin (DG) method has emerged as a promising high- accuracy alternative. The standardDG method reducesto a cell-centeredFV method at lowest order. However, many of t...