TUHH Open Research (TORE)https://tore.tuhh.deTORE captures, stores, indexes, preserves, and distributes digital research material.Thu, 30 Mar 2023 12:41:28 GMT2023-03-30T12:41:28Z50461- An Intrusive PCE Extension of the Contour Integral Method and its Application in Electrical Engineeringhttp://hdl.handle.net/11420/4186Title: An Intrusive PCE Extension of the Contour Integral Method and its Application in Electrical Engineering
Authors: Frick, Eduard; Dahl, David; Seifert, Christian; Lindner, Marko; Schuster, Christian
Wed, 02 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11420/41862019-01-02T00:00:00Z
- An affirmative answer to a core issue on limit operatorshttp://hdl.handle.net/11420/9524Title: An affirmative answer to a core issue on limit operators
Authors: Lindner, Marko; Seidel, Markus
Abstract: An operator on an lᵖ-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. It is known that a rich band-dominated operator is ��-Fredholm (which is a generalization of the classical Fredholm property) if and only if all of its so-called limit operators are invertible and their inverses are uniformly bounded. We show that the condition on uniform boundedness is redundant in this statement.
Thu, 03 Apr 2014 00:00:00 GMThttp://hdl.handle.net/11420/95242014-04-03T00:00:00Z
- Pseudospectrumhttp://hdl.handle.net/11420/10585Title: Pseudospectrum
Authors: Lindner, Marko; Böttcher, Albrecht
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/11420/105852008-01-01T00:00:00Z
- Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operatorhttp://hdl.handle.net/11420/10584Title: Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator
Authors: Chandler-Wilde, Simon N.; Chonchaiya, Ratchanikorn; Lindner, Marko
Abstract: The purpose of this paper is to prove that the spectrum of the non-self-adjoint one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433--6443) has interior points. We do this by first recalling that the spectrum of this random operator is the union of the set of ℓ∞ eigenvalues of all infinite matrices with the same structure. We then construct an infinite matrix of this structure for which every point of the open unit disk is an ℓ∞ eigenvalue, this following from the fact that the components of the eigenvector are polynomials in the spectral parameter whose non-zero coefficients are ± 1's, forming the pattern of an infinite discrete Sierpinski triangle.
Sat, 20 Mar 2010 00:00:00 GMThttp://hdl.handle.net/11420/105842010-03-20T00:00:00Z
- Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section methodhttp://hdl.handle.net/11420/10578Title: Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method
Authors: Chandler-Wilde, Simon N.; Lindner, Marko
Abstract: We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form D = (x, z) ∈ Rn × R: z > f(x) where f: Rn → R is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example, acoustic scattering problems, problems involving elastic waves and problems in potential theory, have been reformulated as second kind integral equations u + Ku = v in the space BC of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator A = I + K under consideration, with an emphasis on the function space setting BC. Firstly, under which conditions is A a Fredholm operator, and, secondly, when is the finite section method applicable to A?. © 2008 Rocky Mountain Mathematics Consortium.
Mon, 01 Dec 2008 00:00:00 GMThttp://hdl.handle.net/11420/105782008-12-01T00:00:00Z
- Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisationhttp://hdl.handle.net/11420/10582Title: Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation
Authors: Betcke, Timo; Chandler-Wilde, Simon N.; Graham, Ivan G.; Langdon, Stephen; Lindner, Marko
Abstract: We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the L 2 condition numbers for these formulations and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number k, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like k1/3 as k →∞, when the scatterer is a circle or sphere, it can grow as fast as k7/5 for a class of "trapping" obstacles. In this article, we prove further bounds, sharpening and extending our previous results. In particular, we show that there exist trapping obstacles for which the condition numbers grow as fast as exp(γk), for some γ > 0, as k →∞ through some sequence. This result depends on exponential localization bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low k. In the second part of the article, we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper. © 2010 Wiley Periodicals, Inc.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/11420/105822011-01-01T00:00:00Z
- Limit operators, collective compactness, and the spectral theory of infinite matriceshttp://hdl.handle.net/11420/10579Title: Limit operators, collective compactness, and the spectral theory of infinite matrices
Authors: Chandler-Wilde, Simon N.; Lindner, Marko
Abstract: In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator A (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space l p(Z N, U), where p ∈ [1, ∞] and U is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility invertibility at infinity and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator A is a locally compact perturbation of the identity. Especially we obtain stronger results than previously known for the subtle limiting cases of p = 1 and ∞. Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between e 1 and l infin; and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrodinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on R. © 2010 American Mathematical Society.
Tue, 01 Mar 2011 00:00:00 GMThttp://hdl.handle.net/11420/105792011-03-01T00:00:00Z
- Approximating the inverse of banded matrices by banded matrices with applications to probability and statisticshttp://hdl.handle.net/11420/10574Title: Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics
Authors: Bickel, Peter J.; Lindner, Marko
Abstract: In the first part of this paper we give an elementary proof of the fact that if an infinite matrix A, which is invertible as a bounded operator on ℓ 2, can be uniformly approximated by banded matrices, then so can the inverse of A. We give explicit formulas for the banded approximations of A -1 as well as bounds on their accuracy and speed of convergence in terms of their bandwidth. We then use these results to prove that the so-called Wiener algebra is inverse closed. In the second part of the paper we apply these results to covariance matrices ∑ of Gaussian processes and study mixing and beta mixing of processes in terms of properties of ∑. Finally, we note some applications of our results to statistics. © by SIAM.
Mon, 28 May 2012 00:00:00 GMThttp://hdl.handle.net/11420/105742012-05-28T00:00:00Z
- Classes of multiplication operators and their limit operatorshttp://hdl.handle.net/11420/10586Title: Classes of multiplication operators and their limit operators
Authors: Lindner, Marko
Abstract: Limit operators have proven to be a device for the study of several properties of an operator including Fredholmness and invertibility at infinity, but also the applicability of approximation methods. For band-dominated operators, the question of existence and structure of their limit operators essentially reduces to the study of multiplication operators and their limit operators, which is the topic of this paper.
Wed, 31 Mar 2004 00:00:00 GMThttp://hdl.handle.net/11420/105862004-03-31T00:00:00Z
- A note on the spectrum of bi-infinite bi-diagonal random matriceshttp://hdl.handle.net/11420/10573Title: A note on the spectrum of bi-infinite bi-diagonal random matrices
Authors: Lindner, Marko
Abstract: The purpose of this paper is to demonstrate the use of the results from [5, 6] for the explicit computation of the spectrum of two-sided infinite matrices with random diagonals. Here we consider the case of two random diagonals, one of them the main diagonal. Our result is a generalization of [24, Theorem 8.1] by Trefethen, Contedini and Embree from the case of one random and one constant diagonal to the case of two random diagonals.
Thu, 23 Oct 2008 00:00:00 GMThttp://hdl.handle.net/11420/105732008-10-23T00:00:00Z
- Finite sections of band operators with slowly oscillating coefficientshttp://hdl.handle.net/11420/10580Title: Finite sections of band operators with slowly oscillating coefficients
Authors: Lindner, Marko; Rabinovich, Vladimir S.; Roch, Steffen
Abstract: The purpose of this note is to show that the finite section method for a band operator with slowly oscillating coefficients is stable if and only if the operator is invertible. This result generalizes the classical stability criterion for the finite section method for band Toeplitz operators (= the case of constant coefficients). © 2004 Elsevier Inc. All rights reserved.
Fri, 01 Oct 2004 00:00:00 GMThttp://hdl.handle.net/11420/105802004-10-01T00:00:00Z
- Fredholmness and index of operators in the Wiener algebra are independent of the underlying spacehttp://hdl.handle.net/11420/10576Title: Fredholmness and index of operators in the Wiener algebra are independent of the underlying space
Authors: Lindner, Marko
Abstract: The purpose of this paper is to demonstrate the so-called Fredholm-inverse closedness
of the Wiener algebra W and to deduce independence of the Fredholm property and index of
the underlying space. More precisely, we look at operators A ∈ W as acting on a family of
vector valued p spaces and show that the Fredholm regularizer of A for one of these spaces
can always be chosen in W as well and therefore regularizes A (modulo compact operators) on
all of the p spaces under consideration. We conclude that both Fredholmness and the index of
A do not depend on the p space that A is considered as acting on.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/11420/105762008-01-01T00:00:00Z
- Wave problems in unbounded domains: Fredholmness and the finite section methodhttp://hdl.handle.net/11420/10589Title: Wave problems in unbounded domains: Fredholmness and the finite section method
Authors: Chandler-Wilde, Simon N.; Lindner, Marko
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/11420/105892007-01-01T00:00:00Z
- Invertibility at Infinity of Band-Dominated Operators on the Space of Essentially Bounded Functionshttp://hdl.handle.net/11420/10575Title: Invertibility at Infinity of Band-Dominated Operators on the Space of Essentially Bounded Functions
Authors: Lindner, Marko; Silbermann, Bernd
Abstract: he topic of this paper is band operators and the norm limits of such — so-called band-dominated operators, both classes acting on L∞(ℝn). Invertibility at infinity is closely related to Fredholmness. In fact, in the discrete case ℓp(ℤn), 1 ≤ p ≤ ∞, both properties coincide. For many applications, e.g., the question of applicability of certain approximation methods, in the situation at hand, Lp(ℝn), 1 ≤ p ≤ ∞, it has however proved to be useful to study invertibility at infinity rather than Fredholmness.
We will present a criterion for a band-dominated operator’s invertibility at infinity in terms of the invertibility of its limit operators. It is the same criterion that was found for ℓp(ℤn), 1 < p < ∞ in [21] and for the C*- algebra L 2 (ℝn in [22]. Our investigations concentrate on one of the most unvolved cases, being L ∞(ℝn). With the techniques presented here it is clear now how the remaining cases ℓ1, ℓ∞ and L p, (p≠2) have to be treated.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11420/105752004-01-01T00:00:00Z
- Condition number estimates for combined potential boundary integral operators in acoustic scatteringhttp://hdl.handle.net/11420/10570Title: Condition number estimates for combined potential boundary integral operators in acoustic scattering
Authors: Chandler-Wilde, Simon N.; Graham, Ivan G.; Langdon, Stephen; Lindner, Marko
Abstract: We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators. © 2009 Rocky Mountain Mathematics Consortium.
Tue, 01 Dec 2009 00:00:00 GMThttp://hdl.handle.net/11420/105702009-12-01T00:00:00Z
- The finite section method in the space of essentially bounded functions: An approach using limit operatorshttp://hdl.handle.net/11420/10581Title: The finite section method in the space of essentially bounded functions: An approach using limit operators
Authors: Lindner, Marko
Abstract: We present an approach to the finite section method for band-dominated operators - the norm-limits of band operators on L∞(ℝn). We hereby show that the sequence of finite sections is stable if and only if some associated operator is invertible at infinity. By means of the theory in Lindner and Silbermann (Lindner, M., Silbermann, B. (2003). Invertibility at infinity of band-dominated operators in the space of essentially bounded functions, (accepted at) Integral Equations and Operator Theory.) and Lindner (Lindner, M. (2003). Classes of multiplication operators and their limit operators (submitted to) Zeitschrift für Analysis und ihre Anwendungen), we study this invertibility at infinity using limit operators. Having the mentioned criterion at our disposal, we will give some applications in an algebra of convolution and multiplication operators: one for the usual finite section method and one for an approximation method of operators on the space of continuous functions.
Fri, 07 Feb 2003 00:00:00 GMThttp://hdl.handle.net/11420/105812003-02-07T00:00:00Z
- On the integer points in a lattice polytope: n-fold Minkowski sum and boundaryhttp://hdl.handle.net/11420/10572Title: On the integer points in a lattice polytope: n-fold Minkowski sum and boundary
Authors: Lindner, Marko; Roch, Steffen
Abstract: In this article we compare the set of integer points in the homothetic copy nΠ of a lattice polytope Π⊆ᵈ with the set of all sums x₁+⋯+xn with x₁,...,xn∈ Π∩ᵈ and n∈. We give conditions on the polytope Π under which these two sets coincide and we discuss two notions of boundary for subsets of ᵈ or, more generally, subsets of a finitely generated discrete group.
Thu, 10 Jun 2010 00:00:00 GMThttp://hdl.handle.net/11420/105722010-06-10T00:00:00Z
- Sufficiency of Favard's condition for a class of band-dominated operators on the axishttp://hdl.handle.net/11420/10583Title: Sufficiency of Favard's condition for a class of band-dominated operators on the axis
Authors: Chandler-Wilde, Simon N.; Lindner, Marko
Abstract: The purpose of this paper is to show that, for a large class of band-dominated operators on ℓ∞ (Z, U), with U being a complex Banach space, the injectivity of all limit operators of A already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of A, which, on the other hand, is often equivalent to the Fredholmness of A. As a consequence, for operators A in the Wiener algebra, we can characterize the essential spectrum of A on ℓp (Z, U), regardless of p ∈ [1, ∞], as the union of point spectra of its limit operators considered as acting on ℓ∞ (Z, U). © 2007 Elsevier Inc. All rights reserved.
Fri, 15 Feb 2008 00:00:00 GMThttp://hdl.handle.net/11420/105832008-02-15T00:00:00Z
- The finite section method and stable subsequenceshttp://hdl.handle.net/11420/10588Title: The finite section method and stable subsequences
Authors: Lindner, Marko
Abstract: The purpose of this paper is to prove a sufficient and necessary criterion on the stability of a subsequence of the finite section method for a so-called band-dominated operator on ℓp (ZN, X). We hereby generalize previous results into several directions: We generalize the subsequence theorem from dimension N = 1 (see Rabinovich, Roch and Silbermann (2008) [18]) to arbitrary dimensions N ≥ 1; and even for the case of the full sequence, our result is new in dimensions N > 2 and it corrects a mistake in the literature for N = 2. Moreover, we allow the truncations to be taken by homothetic copies of very general starlike geometries Ω ∈ RN rather than convex polytopes. © 2009 IMACS.
Thu, 01 Apr 2010 00:00:00 GMThttp://hdl.handle.net/11420/105882010-04-01T00:00:00Z
- Two stable modifications of the finite section methodhttp://hdl.handle.net/11420/10600Title: Two stable modifications of the finite section method
Authors: Lindner, Marko
Abstract: In this article we demonstrate and compare two modified versions of the classical finite section method for band-dominated operators in case the latter is not stable. For both methods we give explicit criteria for their applicability.
Wed, 24 Feb 2010 00:00:00 GMThttp://hdl.handle.net/11420/106002010-02-24T00:00:00Z
- Spectral approximation of banded laurent matrices with localized random perturbationshttp://hdl.handle.net/11420/10577Title: Spectral approximation of banded laurent matrices with localized random perturbations
Authors: Böttcher, A.; Embree, M.; Lindner, Marko
Abstract: This paper explores the relationship between the spectra of perturbed infinite banded Laurent matrices L(a) + K and their approximations by perturbed circulant matrices Cn(a) + PnKPn for large n. The entries Kjk of the perturbation matrices assume values in prescribed sets Ωjk at the sites (j, k) of a fixed finite set E, and are zero at the sites (j, k) outside E. With KΩE denoting the ensemble of these perturbation matrices, it is shown that limn→∞ ∪K∈κΩE sp(Cn(a) + PnKPn) = ∪K∈κΩE sp(L(a) + K) under several fairly general assumptions on E and Ω.
Tue, 03 Dec 2002 00:00:00 GMThttp://hdl.handle.net/11420/105772002-12-03T00:00:00Z
- Analysis Code for Finite Sections of Periodic Schrödinger Operatorshttp://hdl.handle.net/11420/10544Title: Analysis Code for Finite Sections of Periodic Schrödinger Operators
Authors: Gabel, Fabian Nuraddin Alexander; Gallaun, Dennis; Großmann, Julian Peter; Lindner, Marko; Ukena, Riko
Abstract: Simulation code and supplementary material for the article "Finite Sections of Periodic Schrödinger Operators".
Mon, 18 Oct 2021 00:00:00 GMThttp://hdl.handle.net/11420/105442021-10-18T00:00:00Z
- Convergence and numerics of a multisection method for scattering by three-dimensional rough surfaceshttp://hdl.handle.net/11420/10590Title: Convergence and numerics of a multisection method for scattering by three-dimensional rough surfaces
Authors: Heinemeyer, Eric; Lindner, Marko; Potthast, Roland
Abstract: We introduce a novel multisection method for the solution of integral equations on unbounded domains. The method is applied to the rough surface scattering problem in three dimensions, in particular to a Brakhage-Werner-type integral equation for acoustic scattering by an unbounded rough surface with Dirichlet boundary condition, where the fundamental solution is replaced by some appropriate half-space Green's function. The basic idea of the multisection method is to solve an integral equation Aφ = f by approximately solving the equation PρAPτφ = Pρf for some positive constants ρ, τ. Here Pρ is a projection operator that truncates a function to a ball with radius ρ > 0. For a very general class of operators A, for which the Brakhage-Werner equation from acoustic scattering is a particular example, we will show existence of approximate solutions to the multisection equation and show that approximate solutions to the multisection equation approximate the true solution φ0 of the operator equation Aφ = f. Finally, we describe a numerical implementation of the multisection algorithm and provide numerical examples for the case of rough surface scattering in three dimensions. © 2008 Society for Industrial and Applied Mathematics.
Mon, 10 Nov 2008 00:00:00 GMThttp://hdl.handle.net/11420/105902008-11-10T00:00:00Z
- Finite sections of random Jacobi operatorshttp://hdl.handle.net/11420/10568Title: Finite sections of random Jacobi operators
Authors: Lindner, Marko; Roch, Steffen
Abstract: This article is about a problem in the numerical analysis of random operators. We study a version of the finite section method for the approximate solution of equations Ax = b in infinitely many variables, where A is a random Jacobi (i.e., tridiagonal) operator. In other words, we approximately solve infinite second order difference equations with stochastic coefficients by reducing the infinite volume case to the (large) finite volume case via a particular truncation technique. For most of the paper we consider non-self-adjoint operators A, but we also comment on the self-adjoint case when simplifications occur. © 2012 Society for Industrial and Applied Mathematics.
Mon, 28 May 2012 00:00:00 GMThttp://hdl.handle.net/11420/105682012-05-28T00:00:00Z
- Note on Spectra of Non-Selfadjoint Operators over Dynamical Systemshttp://hdl.handle.net/11420/2942Title: Note on Spectra of Non-Selfadjoint Operators over Dynamical Systems
Authors: Beckus, Siegfried; Lenz, Daniel; Lindner, Marko; Seifert, Christian
Abstract: We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.
Tue, 01 May 2018 00:00:00 GMThttp://hdl.handle.net/11420/29422018-05-01T00:00:00Z
- Novel Method for Error Estimation in Applications of Polynomial Chaos Expansion to Stochastic Modeling of Multi-Resonant Systemshttp://hdl.handle.net/11420/4192Title: Novel Method for Error Estimation in Applications of Polynomial Chaos Expansion to Stochastic Modeling of Multi-Resonant Systems
Authors: Frick, Eduard; Dahl, David; Seifert, Christian; Lindner, Marko; Schuster, Christian
Abstract: A novel technique for the estimation of polynomial chaos expansion (PCE) errors in multi-resonant systems is derived and demonstrated on the example of the non-intrusive PCE application to the contour integral method (CM).; A novel technique for the estimation of polynomial chaos expansion (PCE) errors in multi-resonant systems is derived and demonstrated on the example of the non-intrusive PCE application to the contour integral method (CM).
Mon, 01 Oct 2018 00:00:00 GMThttp://hdl.handle.net/11420/41922018-10-01T00:00:00Z
- Multiscale Simulation of 2-D Photonic Crystal Structures Using a Contour Integral Methodhttp://hdl.handle.net/11420/2370Title: Multiscale Simulation of 2-D Photonic Crystal Structures Using a Contour Integral Method
Authors: Dahl, David; Frick, Eduard; Seifert, Christian; Lindner, Marko; Schuster, Christian
Abstract: This paper presents a multiscale method for the numerically efficient electromagnetic analysis of two-dimensional (2-D) photonic and electromagnetic crystals. It is based on a contour integral method and a segmented analysis of more complex structures in terms of building blocks which are models for essential components. The scattering properties of essential photonic crystal components, such as waveguide sections, bends, and junctions, can be expressed independent of the electromagnetic wave launch parts which are used for the excitation by de-embedding of the network parameters. To enable this, the launch properties are extracted by a calibration technique using several calibration standards analog to a measurement. The de-embedding can be applied both to the proposed integral method and to the reference results from other full-wave methods. The extracted scattering parameters of the components can be used in a multiscale analysis for the efficient simulation of very large 2-D photonic and microwave structures with circular inclusions as the concatenation is performed only in terms of the network parameters. The proposed approach is about one to two orders of magnitude faster than the conventional unsegmented analysis with the contour integral method and several orders of magnitude faster than the full-wave reference method. © 2016 IEEE.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11420/23702019-01-01T00:00:00Z
- Finite sections of the Fibonacci Hamiltonianhttp://hdl.handle.net/11420/3456Title: Finite sections of the Fibonacci Hamiltonian
Authors: Lindner, Marko; Söding, Hagen
Abstract: We study finite but growing principal square submatrices An of the one- or two-sided infinite Fibonacci Hamiltonian A. Our results show that such a sequence (An), no matter how the points of truncation are chosen, is always stable – implying that An is invertible for sufficiently large n and A–1n → A–1 pointwise.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11420/34562018-01-01T00:00:00Z
- Circulant matrices: Norm, powers, and positivityhttp://hdl.handle.net/11420/3028Title: Circulant matrices: Norm, powers, and positivity
Authors: Lindner, Marko
Abstract: In their recent paper “The spectral norm of a Horadam circulant matrix”, Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix C equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the above problem to positivity of sufficiently high powers of the matrix C>C. We then generalize the result to complex circulant matrices.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11420/30282018-01-01T00:00:00Z
- Finite sections: A functional analytic perspective on approximation methodshttp://hdl.handle.net/11420/2838Title: Finite sections: A functional analytic perspective on approximation methods
Authors: Lindner, Marko; Seifert, Christian
Abstract: We review approximation methods and their stability and applicability. We then focus on the finite section method and Galerkin methods and show that on separable Hilbert spaces either one can be interpreted as the other. In the end we demonstrate that well-known methods such as the finite element method and polynomial chaos expansion are particular examples of the finite section method; their applicability can therefore be studied via the latter.
Sat, 01 Sep 2018 00:00:00 GMThttp://hdl.handle.net/11420/28382018-09-01T00:00:00Z
- Feasibility of uncertainty quantification for power distribution network modeling using PCE and a contour integral methodhttp://hdl.handle.net/11420/2576Title: Feasibility of uncertainty quantification for power distribution network modeling using PCE and a contour integral method
Authors: Dahl, David; Yildiz, Ömer Faruk; Frick, Eduard; Seifert, Christian; Lindner, Marko; Schuster, Christian
Abstract: This work presents the modeling of the printed circuit board part of power distribution networks (PDNs) and example results for the uncertainty quantification for the magnitude of the corresponding impedance. Variability is considered for several parameters, including geometry, material properties, and the models of the decoupling capacitors. For the computation of the parallel plate impedance an efficient and accurate two-dimensional contour integral method (CIM) is applied together with models for the wave number for the complete frequency range of interest. Polynomial chaos expansion (PCE) is used in the non-intrusive form of stochastic testing for the uncertainty quantification and Monte Carlo simulations are used for the validation of these results. To our knowledge this combination of methods represents the first application of CIM and PCE to the modeling of PDNs. The PCE is found to be numerically more efficient than Monte Carlo in cases where parameters are varied that have an influence on the parallel plate impedance. It can be less efficient for variation of only the models of decoupling capacitors. It is applicable if not too many parameters are varied at a time and accurate if resonance effects due to low-loss substrate materials and components are not too pronounced at the considered frequency.
Fri, 22 Jun 2018 00:00:00 GMThttp://hdl.handle.net/11420/25762018-06-22T00:00:00Z
- Efficient Simulation of Substrate-Integrated Waveguide Antennas Using a Hybrid Boundary Element Methodhttp://hdl.handle.net/11420/3390Title: Efficient Simulation of Substrate-Integrated Waveguide Antennas Using a Hybrid Boundary Element Method
Authors: Dahl, David; Brüns, Heinz-Dietrich; Wang, Lei; Frick, Eduard; Seifert, Christian; Lindner, Marko; Schuster, Christian
Abstract: This paper presents a hybrid boundary element method for the efficient simulation of substrate-integrated waveguide (SIW) horn antennas. It is applicable with good accuracy to relatively thin structures with conventional circular ground vias. In the multiscale simulations, a 2-D contour integral method is used for the modeling of the fields inside the structure with numerous vias and a method of moments is used for the radiated fields outside. The contour integral method is extended in this paper by a new waveguide port of finite size based on the unit cell analysis of an SIW segment. Several SIW horn antennas are studied to validate the proposed method in terms of the input impedance, the field distribution on the aperture, and the radiation diagram. The proposed method shows good to reasonable accuracy and has a numerical efficiency which is about 2-3 orders higher than FEM-based full-wave simulations. It is therefore well suited for fast optimizations.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11420/33902019-01-01T00:00:00Z
- Variability analysis of via crosstalk using polynomial chaos expansionhttp://hdl.handle.net/11420/3515Title: Variability analysis of via crosstalk using polynomial chaos expansion
Authors: Frick, Eduard; Preibisch, Jan; Seifert, Christian; Lindner, Marko; Schuster, Christian
Abstract: In this work we use the method of polynomial chaos expansion (PCE) for statistical analysis of a via interconnect in a printed circuit board (PCB) in presence of geometrical uncertainties. A physics-based via (PBV) model is applied to the case of two signal vias surrounded by ground vias. This model consist of a near-field part and a far-field part. Here, the focus is on the far-field part modeled by the contour integral method (CIM). In order to account for variability in the geometry, namely a variation of the pitch, PCE is applied to the CIM in an intrusive manner. The system matrix of the deterministic case is replaced by an augmented version. The method is validated with Monte Carlo Sampling (MC) and shows excellent agreement for a simple geometry. We observe the same agreement in more complicated geometric configurations- A t least for frequencies below the first resonance.
Mon, 03 Jul 2017 00:00:00 GMThttp://hdl.handle.net/11420/35152017-07-03T00:00:00Z
- Approximation of pseudospectra on a Hilbert spacehttp://hdl.handle.net/11420/6249Title: Approximation of pseudospectra on a Hilbert space
Authors: Schmidt, Torge; Lindner, Marko
Abstract: The study of spectral properties of linear operators on an infinite-dimensional Hilbert space is of great interest. This task is especially difficult when the operator is non-selfadjoint or even non-normal. Standard approaches like spectral approximation by finite sections generally fail in that case. In this talk we present an algorithm which rigorously computes upper and lower bounds for the spectrum and pseudospectrum of such operators using finite-dimensional approximations. One of our main fields of research is an efficient implementation of this algorithm. To this end we will demonstrate and evaluate methods for the computation of the pseudospectrum of finite-dimensional operators based on continuation techniques.
Wed, 08 Jun 2016 00:00:00 GMThttp://hdl.handle.net/11420/62492016-06-08T00:00:00Z
- Essential pseudospectra and essential norms of band-dominated operatorshttp://hdl.handle.net/11420/5948Title: Essential pseudospectra and essential norms of band-dominated operators
Authors: Hagger, Raffael; Lindner, Marko; Seidel, Markus
Abstract: An operator A on an lp-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of A in the Calkin algebra determines, for example, the Fredholmness of A, the Fredholm index, the essential spectrum, the essential norm and the so-called essential pseudospectrum of A. This coset can be identified with the collection of all so-called limit operators of A. It is known that this identification preserves invertibility (hence spectra). We now show that it also preserves norms and in particular resolvent norms (hence pseudospectra). In fact we work with a generalization of the ideal of compact operators, so-called P-compact operators, allowing for a more flexible framework that naturally extends to lp-spaces with p∈(1, ∞) and/or vector-valued lp-spaces.
Sun, 01 May 2016 00:00:00 GMThttp://hdl.handle.net/11420/59482016-05-01T00:00:00Z
- On the spectrum of operator families on discrete groups over minimal dynamical systemshttp://hdl.handle.net/11420/3772Title: On the spectrum of operator families on discrete groups over minimal dynamical systems
Authors: Beckus, Siegfried; Lenz, Daniel; Lindner, Marko; Seifert, Christian
Abstract: It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in ℓp(Z). Here, we generalize this to a large class of bounded linear operator families on Banach-space valued ℓp-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.
Fri, 01 Dec 2017 00:00:00 GMThttp://hdl.handle.net/11420/37722017-12-01T00:00:00Z
- Infinite matrices and their finite sectionshttp://hdl.handle.net/11420/10587Title: Infinite matrices and their finite sections
Authors: Lindner, Marko
Abstract: In this book we are concerned with the study of a certain class of in?nite matrices and two important properties of them: their Fredholmness and the stability of the approximation by their ?nite truncations. Let us take these two properties as a starting point for the big picture that shall be presented in what follows. Stability Fredholmness We think of our in?nite matrices as bounded linear operators on a Banach space E of two-sided in?nite sequences. Probably the simplest case to start with 2 +? is the space E = of all complex-valued sequences u=(u ) for which m m=?? 2 |u | is summable over m? Z. m Theclassofoperatorsweareinterestedinconsistsofthoseboundedandlinear operatorsonE whichcanbeapproximatedintheoperatornormbybandmatrices. We refer to them as band-dominated operators. Of course, these considerations 2 are not limited to the space E = . We will widen the selection of the underlying space E in three directions: p • We pass to the classical sequence spaces with 1? p??. n • Our elements u=(u )? E have indices m? Z rather than just m? Z. m • We allow values u in an arbitrary ?xed Banach spaceX rather than C.
Wed, 07 Jun 2006 00:00:00 GMThttp://hdl.handle.net/11420/105872006-06-07T00:00:00Z
- Finite section method for aperiodic Schrödinger operatorshttp://hdl.handle.net/11420/13414Title: Finite section method for aperiodic Schrödinger operators
Authors: Gabel, Fabian Nuraddin Alexander; Gallaun, Dennis; Großmann, Julian Peter; Lindner, Marko; Ukena, Riko
Abstract: We consider discrete Schrödinger operators with aperiodic potentials given by a Sturmian word, which is a natural generalisation of the Fibonacci Hamiltonian. We introduce the finite section method, which is often used to solve operator equations approximately, and apply it first to periodic Schrödinger operators. It turns out that the applicability of the method is always guaranteed for integer-valued potentials provided that the operator is invertible. By using periodic approximations, we find a necessary and sufficient condition for the applicability of the finite section method for aperiodic Schrödinger operators and a numerical method to check it.
Thu, 01 Apr 2021 00:00:00 GMThttp://hdl.handle.net/11420/134142021-04-01T00:00:00Z
- Finite sections of periodic Schrödinger operatorshttp://hdl.handle.net/11420/10707Title: Finite sections of periodic Schrödinger operators
Authors: Gabel, Fabian Nuraddin Alexander; Gallaun, Dennis; Großmann, Julian Peter; Lindner, Marko; Ukena, Riko
Abstract: We study discrete Schrödinger operators H with periodic potentials as they are typically used to approximate aperiodic Schrödinger operators like the Fibonacci Hamiltonian. We prove an efficient test for applicability of the finite section method, a procedure that approximates H by growing finite square submatrices Hn. For integer-valued potentials, we show that the finite section method is applicable as soon as H is invertible. This statement remains true for {0, λ}-valued potentials with fixed rational λ and period less than nine as well as for arbitrary real-valued potentials of period two.
Mon, 18 Oct 2021 00:00:00 GMThttp://hdl.handle.net/11420/107072021-10-18T00:00:00Z
- Minimal Families of Limit Operatorshttp://hdl.handle.net/11420/11125Title: Minimal Families of Limit Operators
Authors: Lindner, Marko
Abstract: We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both abstract settings are illustrated by practically relevant examples, including discrete Schrödinger operators with periodic, quasiperiodic, almost-periodic, Sturmian and pseudo-ergodic potential. The main tool is the method of limit operators, known from studies of Fredholm operators and convergence of projection methods. We close by connecting this tool to the study of subwords of the operator potential.; We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both abstract settings are illustrated by practically relevant examples, including discrete Schrödinger operators with periodic, quasiperiodic, almost-periodic, Sturmian and pseudo-ergodic potential. The main tool is the method of limit operators, known from studies of Fredholm operators and convergence of projection methods. We close by connecting this tool to the study of subwords of the operator potential.
Wed, 01 Jun 2022 00:00:00 GMThttp://hdl.handle.net/11420/111252022-06-01T00:00:00Z
- Coburn's lemma and the finite section method for random Jacobi operatorshttp://hdl.handle.net/11420/5951Title: Coburn's lemma and the finite section method for random Jacobi operators
Authors: Chandler-Wilde, Simon N.; Lindner, Marko
Abstract: © 2015 Elsevier Inc. principal submatrices, in the case where each of the three diagonals varies over a separate compact set, say U,V,W⊂C. Such matrices are sometimes termed stochastic Toeplitz matrices A+ in the semi-infinite case and stochastic Laurent matrices A in the bi-infinite case. Their spectra, σ=specA and σ+=specA+, are independent of A and A+ as long as A and A+ are pseudoergodic (in the sense of Davies (2001) [20]), which holds almost surely in the random case. This was shown in Davies (2001) [20] for A; that the same holds for A+ is one main result of this paper. Although the computation of σ and σ+ in terms of U, V and W is intrinsically difficult, we give upper and lower spectral bounds, and we explicitly compute a set G that fills the gap between σ and σ+ in the sense that σ∪G=σ+. We also show that the invertibility of one (and hence all) operators A+ implies the invertibility - and uniform boundedness of the inverses - of all finite tridiagonal square matrices with diagonals varying over U, V and W. This implies that the so-called finite section method for the approximate solution of a system A+x=b is applicable as soon as A+ is invertible, and that the finite section method for estimating the spectrum of A+ does not suffer from spectral pollution. Both results illustrate that tridiagonal stochastic Toeplitz operators share important properties of (classical) Toeplitz operators. Indeed, one of our main tools is a new stochastic version of the Coburn lemma for classical Toeplitz operators, saying that a stochastic tridiagonal Toeplitz operator, if Fredholm, is always injective or surjective. In the final part of the paper we bound and compare the norms, and the norms of inverses, of bi-infinite, semi-infinite and finite tridiagonal matrices over U, V and W. This, in particular, allows the study of the resolvent norms, and hence the pseudospectra, of these operators and matrices.
Fri, 15 Jan 2016 00:00:00 GMThttp://hdl.handle.net/11420/59512016-01-15T00:00:00Z
- A note on Hausdorff convergence of pseudospectrahttp://hdl.handle.net/11420/12951Title: A note on Hausdorff convergence of pseudospectra
Authors: Lindner, Marko; Schmeckpeper, Dennis
Abstract: For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities.
Sun, 01 Jan 2023 00:00:00 GMThttp://hdl.handle.net/11420/129512023-01-01T00:00:00Z
- Spectral approximation of generalized Schrödinger operators via approximation of subwordshttp://hdl.handle.net/11420/13693Title: Spectral approximation of generalized Schrödinger operators via approximation of subwords
Authors: Gabel, Fabian Nuraddin Alexander; Gallaun, Dennis; Großmann, Julian Peter; Lindner, Marko; Ukena, Riko
Abstract: We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schrödinger operators on the discrete line or half-line. In fact, our results are neither limited to Schrödinger or self-adjoint operators, nor to Hilbert space or 1D.
Fri, 23 Sep 2022 00:00:00 GMThttp://hdl.handle.net/11420/136932022-09-23T00:00:00Z
- Example Potentials for Spectral Approximation of Generalized Schrödinger Operators via Approximation of Subwordshttp://hdl.handle.net/11420/14560Title: Example Potentials for Spectral Approximation of Generalized Schrödinger Operators via Approximation of Subwords
Authors: Ukena, Riko; Gabel, Fabian Nuraddin Alexander; Gallaun, Dennis; Großmann, Julian Peter; Lindner, Marko
Abstract: Potentials used for the generation of Figures 2-7 in the article "Spectral Approximation of Generalized Schrödinger Operators via Approximation of Subwords"
Tue, 17 Jan 2023 00:00:00 GMThttp://hdl.handle.net/11420/145602023-01-17T00:00:00Z
- Half-line compressions and finite sections of discrete Schrödinger operators with integer-valued potentialshttp://hdl.handle.net/11420/13692Title: Half-line compressions and finite sections of discrete Schrödinger operators with integer-valued potentials
Authors: Lindner, Marko; Ukena, Riko
Abstract: We study 1D discrete Schrödinger operators H with integer-valued potential and show that, (i), invertibility (in fact, even just Fredholmness) of H always implies invertibility of its half-line compression H₊ (zero Dirichlet boundary condition, i.e. matrix truncation). In particular, the Dirichlet eigenvalues avoid zero -- and all other integers. We use this result to conclude that, (ii), the finite section method (approximate inversion via finite and growing matrix truncations) is applicable to H as soon as H is invertible. The same holds for H₊.
Mon, 08 Aug 2022 00:00:00 GMThttp://hdl.handle.net/11420/136922022-08-08T00:00:00Z
- The main diagonal of a permutation matrixhttp://hdl.handle.net/11420/3166Title: The main diagonal of a permutation matrix
Authors: Lindner, Marko; Strang, Gilbert
Abstract: By counting 1's in the "right half" of 2w consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth w. Then the matrix can be correctly centered and factored into block-diagonal permutation matrices. Part II of the paper discusses the same questions for the much larger class of band-dominated matrices. The main diagonal is determined by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is determined "at infinity" in general, but from only 2w rows for banded permutations.
Thu, 03 May 2012 00:00:00 GMThttp://hdl.handle.net/11420/31662012-05-03T00:00:00Z