Elias Jarlebring

Bio: Elias Jarlebring is an academic researcher from Royal Institute of Technology. The author has contributed to research in topics: Eigenvalues and eigenvectors & Inverse iteration. The author has an hindex of 21, co-authored 79 publications receiving 1195 citations. Previous affiliations of Elias Jarlebring include Braunschweig University of Technology & SERC Reliability Corporation.

Characterizing and Computing the ${\cal H}_{2}$ Norm of Time-Delay Systems by Solving the Delay Lyapunov Equation

Abstract: It is widely known that the solutions of Lyapunov equations can be used to compute the H2 norm of linear time-invariant (LTI) dynamical systems. In this paper, we show how this theory extends to dynamical systems with delays. The first result is that the H2 norm can be computed from the solution of a generalization of the Lyapunov equation, which is known as the delay Lyapunov equation. From the relation with the delay Lyapunov equation we can prove an explicit formula for the H2 norm if the system has commensurate delays, here meaning that the delays are all integer multiples of a basic delay. The formula is explicit and contains only elementary linear algebra operations applied to matrices of finite dimension. The delay Lyapunov equations are matrix boundary value problems. We show how to apply a spectral discretization scheme to these equations for the general, not necessarily commensurate, case. The convergence of spectral methods typically depends on the smoothness of the solution. To this end we describe the smoothness of the solution to the delay Lyapunov equations, for the commensurate as well as for the non-commensurate case. The smoothness properties allow us to completely predict the convergence order of the spectral method.

A linear eigenvalue algorithm for the nonlinear eigenvalue problem

Abstract: The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted $${\mathcal {B}}$$. We consider the Arnoldi method for the operator $${\mathcal {B}}$$ and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator $${\mathcal {B}}$$ in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.

Recent Advances in Optimization and its Applications in Engineering

Abstract: Mathematical optimization encompasses both a rich and rapidly evolving body of fundamental theory, and a variety of exciting applications in science and engineering. The present book contains a careful selection of articles on recent advances in optimization theory, numerical methods, and their applications in engineering. It features in particular new methods and applications in the fields of optimal control, PDE-constrained optimization, nonlinear optimization, and convex optimization. The authors of this volume took part in the 14th Belgian-French-German Conference on Optimization (BFG09) organized in Leuven, Belgium, on September 14-18, 2009. The volume contains a selection of reviewed articles contributed by the conference speakers as well as three survey articles by plenary speakers and two papers authored by the winners of the best talk and best poster prizes awarded at BFG09. Researchers and graduate students in applied mathematics, computer science, and many branches of engineering will find in this book an interesting and useful collection of recent ideas on the methods and applications of optimization.

A Krylov Method for the Delay Eigenvalue Problem

Abstract: The Arnoldi method is currently a very popular algorithm to solve large-scale eigenvalue problems. The main goal of this paper is to generalize the Arnoldi method to the characteristic equation of a delay-differential equation (DDE), here called a delay eigenvalue problem (DEP). The DDE can equivalently be expressed with a linear infinite-dimensional operator whose eigenvalues are the solutions to the DEP. We derive a new method by applying the Arnoldi method to the generalized eigenvalue problem associated with a spectral discretization of the operator and by exploiting the structure. The result is a scheme where we expand a subspace not only in the traditional way done in the Arnoldi method. The subspace vectors are also expanded with one block of rows in each iteration. More important, the structure is such that if the Arnoldi method is started in an appropriate way, it has the (somewhat remarkable) property that it is in a sense independent of the number of discretization points. It is mathematically equivalent to an Arnoldi method with an infinite matrix, corresponding to the limit where we have an infinite number of discretization points. We also show an equivalence with the Arnoldi method in an operator setting. It turns out that with an appropriately defined operator over a space equipped with scalar product with respect to which Chebyshev polynomials are orthonormal, the vectors in the Arnoldi iteration can be interpreted as the coefficients in a Chebyshev expansion of a function. The presented method yields the same Hessenberg matrix as the Arnoldi method applied to the operator.

Stability and Stabilization of Systems with Time Delay

Abstract: Time-delays are important components of many dynamical systems that describe coupling or interconnection between dynamics, propagation, or transport phenomena in shared environments, in heredity, and in competition in population dynamics. This monograph addresses the problem of stability analysis and the stabilisation of dynamical systems subjected to time-delays. It presents a wide and self-contained panorama of analytical methods and computational algorithms using a unified eigenvalue-based approach illustrated by examples and applications in electrical and mechanical engineering, biology, and complex network analysis.

NLEVP: A Collection of Nonlinear Eigenvalue Problems

Abstract: We present a collection of 52 nonlinear eigenvalue problems in the form of a MATLAB toolbox. The collection contains problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial eigenvalue problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear eigenvalue problems and to provide representative problems for testing, tuning, and benchmarking of algorithms and codes.