Criss-Cross Type Algorithms for Computing the Real Pseudospectral Abscissa
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On the Linear Stability of Plane Couette Flow for an Oldroyd-B Fluid and Its
J. Non-Newtonian Fluid Mech. 127 (2005) 169–190 On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation Raz Kupferman Institute of Mathematics, The Hebrew University, Jerusalem, 91904, Israel Received 14 December 2004; received in revised form 10 February 2005; accepted 7 March 2005 Abstract It is well known that plane Couette flow for an Oldroyd-B fluid is linearly stable, yet, most numerical methods predict spurious instabilities at sufficiently high Weissenberg number. In this paper we examine the reasons which cause this qualitative discrepancy. We identify a family of distribution-valued eigenfunctions, which have been overlooked by previous analyses. These singular eigenfunctions span a family of non- modal stress perturbations which are divergence-free, and therefore do not couple back into the velocity field. Although these perturbations decay eventually, they exhibit transient amplification during which their “passive" transport by shearing streamlines generates large cross- stream gradients. This filamentation process produces numerical under-resolution, accompanied with a growth of truncation errors. We believe that the unphysical behavior has to be addressed by fine-scale modelling, such as artificial stress diffusivity, or other non-local couplings. © 2005 Elsevier B.V. All rights reserved. Keywords: Couette flow; Oldroyd-B model; Linear stability; Generalized functions; Non-normal operators; Stress diffusion 1. Introduction divided into two main groups: (i) numerical solutions of the boundary value problem defined by the linearized system The linear stability of Couette flow for viscoelastic fluids (e.g. [2,3]) and (ii) stability analysis of numerical methods is a classical problem whose study originated with the pi- for time-dependent flows, with the objective of understand- oneering work of Gorodtsov and Leonov (GL) back in the ing how spurious solutions emerge in computations. -
Proquest Dissertations
RICE UNIVERSITY Magnetic Damping of an Elastic Conductor by Jeffrey M. Hokanson A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Master of Arts APPROVED, THESIS COMMITTEE: IJU&_ Mark Embread Co-chair Associate Professor of Computational and Applied M/jmematics Steven J. Cox, (Co-chair Professor of Computational and Applied Mathematics David Damanik Associate Professor of Mathematics Houston, Texas April, 2009 UMI Number: 1466785 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI® UMI Microform 1466785 Copyright 2009 by ProQuest LLC All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ii ABSTRACT Magnetic Damping of an Elastic Conductor by Jeffrey M. Hokanson Many applications call for a design that maximizes the rate of energy decay. Typical problems of this class include one dimensional damped wave operators, where energy dissipation is caused by a damping operator acting on the velocity. Two damping operators are well understood: a multiplication operator (known as viscous damping) and a scaled Laplacian (known as Kelvin-Voigt damping). Paralleling the analysis of viscous damping, this thesis investigates energy decay for a novel third operator known as magnetic damping, where the damping is expressed via a rank-one self-adjoint operator, dependent on a function a. -
Reproducing Kernel Hilbert Space Compactification of Unitary Evolution
Reproducing kernel Hilbert space compactification of unitary evolution groups Suddhasattwa Dasa, Dimitrios Giannakisa,∗, Joanna Slawinskab aCourant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA bFinnish Center for Artificial Intelligence, Department of Computer Science, University of Helsinki, Helsinki, Finland Abstract A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator Wτ on a reproducing kernel Hilbert space (RKHS). A key element of this approach is that Wτ is skew-adjoint (unlike regularization approaches based on the addition of diffusion), and thus can be characterized by a unique projection- valued measure, discrete by compactness, and an associated orthonormal basis of eigenfunctions. These eigenfunctions can be ordered in terms of a Dirichlet energy on the RKHS, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, the regularized generator has a well-defined Borel functional calculus allowing tWτ the construction of a unitary evolution group fe gt2R on the RKHS, which approximates the unitary Koopman evolution group of the original system. We establish convergence results for the spectrum and Borel functional calculus of the regularized generator to those of the original system in the limit τ ! 0+. Convergence results are also established for a data-driven formulation, where these operators are approximated using finite-rank operators obtained from observed time series. An advantage of working in spaces of observables with an RKHS structure is that one can perform pointwise evaluation and interpolation through bounded linear operators, which is not possible in Lp spaces. -
Banach J. Math. Anal. 6 (2012), No. 1, 45–60 LINEAR MAPS
Banach J. Math. Anal. 6 (2012), no. 1, 45–60 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/BJMA/ LINEAR MAPS PRESERVING PSEUDOSPECTRUM AND CONDITION SPECTRUM G. KRISHNA KUMAR1 AND S. H. KULKARNI2 Communicated by K. Jarosz Abstract. We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its -perturbation. Sev- eral results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let A, B be complex unital Banach algebras and > 0. Let Φ : A → B be an -pseudospectrum preserving linear onto map. Then Φ preserves spectrum. If A and B are uniform algebras, then, Φ is an isometric isomorphism. (2) Let A, B be uniform algebras and 0 < < 1. Let Φ : A → B be 0 an -condition spectrum preserving linear map. Then Φ is an -almost 0 multiplicative map, where , tend to zero simultaneously. 1. Introduction Let A be a complex Banach algebra with unit 1. We shall identify λ.1 with λ. We recall that the spectrum of an element a ∈ A is defined as −1 σ(a) = λ ∈ C : λ − a∈ / A , where A−1 is the set of all invertible elements of A. The spectral radius of an element a is defined as r(a) = sup{|λ| : λ ∈ σ(a)}. Date: Received: 20June 2011; Revised: 25 August 2011; Accepted: 8 September 2011. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 47B49; Secondary 46H05, 46J05, 47S48. Key words and phrases. Pseudospectrum, condition spectrum, almost multiplicative map, linear preserver, perturbation. 45 46 G. -
Computation of Pseudospectral Abscissa for Large-Scale Nonlinear
IMA Journal of Numerical Analysis (2017) 37, 1831–1863 doi: 10.1093/imanum/drw065 Advance Access publication on January 11, 2017 Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems Downloaded from https://academic.oup.com/imajna/article-abstract/37/4/1831/2894467 by Koc University user on 20 August 2019 Karl Meerbergen Department of Computer Science, KU Leuven, University of Leuven, Heverlee 3001, Belgium Emre Mengi∗ Department of Mathematics, Ko¸c University, Rumelifeneri Yolu, Sarıyer-Istanbul˙ 34450, Turkey ∗Corresponding author: [email protected] and Wim Michiels and Roel Van Beeumen Department of Computer Science, KU Leuven, University of Leuven, Heverlee 3001, Belgium [Received on 26 June 2015; revised on 21 October 2016] We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. -
Lecture Notes on Spectra and Pseudospectra of Matrices and Operators
Lecture Notes on Spectra and Pseudospectra of Matrices and Operators Arne Jensen Department of Mathematical Sciences Aalborg University c 2009 Abstract We give a short introduction to the pseudospectra of matrices and operators. We also review a number of results concerning matrices and bounded linear operators on a Hilbert space, and in particular results related to spectra. A few applications of the results are discussed. Contents 1 Introduction 2 2 Results from linear algebra 2 3 Some matrix results. Similarity transforms 7 4 Results from operator theory 10 5 Pseudospectra 16 6 Examples I 20 7 Perturbation Theory 27 8 Applications of pseudospectra I 34 9 Applications of pseudospectra II 41 10 Examples II 43 1 11 Some infinite dimensional examples 54 1 Introduction We give an introduction to the pseudospectra of matrices and operators, and give a few applications. Since these notes are intended for a wide audience, some elementary concepts are reviewed. We also note that one can understand the main points concerning pseudospectra already in the finite dimensional case. So the reader not familiar with operators on a separable Hilbert space can assume that the space is finite dimensional. Let us briefly outline the contents of these lecture notes. In Section 2 we recall some results from linear algebra, mainly to fix notation, and to recall some results that may not be included in standard courses on linear algebra. In Section 4 we state some results from the theory of bounded operators on a Hilbert space. We have decided to limit the exposition to the case of bounded operators. -
Resolvent Conditions for the Control of Parabolic Equations Thomas Duyckaerts, Luc Miller
Resolvent conditions for the control of parabolic equations Thomas Duyckaerts, Luc Miller To cite this version: Thomas Duyckaerts, Luc Miller. Resolvent conditions for the control of parabolic equations. Jour- nal of Functional Analysis, Elsevier, 2012, 263 (11), pp.3641-3673. 10.1016/j.jfa.2012.09.003. hal- 00620870v2 HAL Id: hal-00620870 https://hal.archives-ouvertes.fr/hal-00620870v2 Submitted on 5 Sep 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. RESOLVENT CONDITIONS FOR THE CONTROL OF PARABOLIC EQUATIONS THOMAS DUYCKAERTS1 AND LUC MILLER2 Abstract. Since the seminal work of Russell and Weiss in 1994, resolvent conditions for various notions of admissibility, observability and controllability, and for various notions of linear evolution equations have been investigated in- tensively, sometimes under the name of infinite-dimensional Hautus test. This paper sets out resolvent conditions for null-controllability in arbitrary time: necessary for general semigroups, sufficient for analytic normal semigroups. For a positive self-adjoint operator A, it gives a sufficient condition for the null-controllability of the semigroup generated by −A which is only logarithmi- cally stronger than the usual condition for the unitary group generated by iA. -
Pseudospectrum of an Element of a Banach Algebra
Tusi Advances in Operator Theory Mathematical Research https://doi.org/10.1007/s43036-019-00016-x Group ORIGINAL PAPER Decomposition of the (n, )-pseudospectrum of an element of a Banach algebra Kousik Dhara1 · S. H. Kulkarni2 Received: 20 June 2019 / Accepted: 9 October 2019 © Tusi Mathematical Research Group (TMRG) 2019 Abstract Let A be a complex Banach algebra with unit. For an integer n ≥ 0 and >0, the (n,)-pseudospectrum of a ∈ A is defined by −2n 1/2n 1 Λ ,(A, a) := λ ∈ C : (λ − a) A (λ − a) ≥ . n is not invertible in or Let p ∈ A be a nontrivial idempotent. Then pAp ={pbp : b ∈ A} is a Banach subalgebra of A with unit p, known as a reduced Banach algebra. Suppose ap = pa. We study the relationship of Λn,(A, a) and Λn,(pAp, pa). We extend this by considering first a finite family, and then an at most countable family of idempotents satisfying some conditions. We establish that under suitable assumptions, the (n,)- pseudospectrum of a can be decomposed into the union of the (n,)-pseudospectra of some elements in reduced Banach algebras. Keywords Banach algebra · Spectrum · Pseudospectrum · (n,)-Pseudospectrum Mathematics Subject Classification 47A10 · 46H05 · 47A12 Communicated by Chi-Keung Ng. B Kousik Dhara [email protected] S. H. Kulkarni [email protected] 1 Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre, 8th Mile, Mysore Road, Bangalore 560059, India 2 Department of Mathematics, Indian Institute of Technology Palakkad, Palakkad, Kerala 678557, India K. Dhara and S. H. Kulkarni 1 Introduction It is known that the spectrum of the direct sum of a finite number of operators on the direct sum of Hilbert spaces is the union of their spectra (Problem 98 of [9]). -
International Conference on Linear Algebra and Its Applications REPORT
International Conference on Linear Algebra and its Applications December 11-15, 2017 REPORT The Fourth DAE-BRNS Theme Meeting on Generation and Use of Covariance Matrices in the Applications of Nuclear Data December 09-13, 2017 Department of Statistics, PSPH Level VI, Health Sciences Library Building Manipal Academy of Higher Education Manipal–576104, Karnataka, INDIA Editor Shreemathi S. Mayya Compiler David Raj Micheal January 15, 2018 MAHE honours save nature policy and limited number of this report are printed for internal circulation and for the communications with sponsors. Table of Contents 1 Overview of ICLAA 2017 & Theme Meeting on Covariance Matrix 5 1.1 Invited Delegates: ICLAA 20176 1.2 Delegates Contributing Paper: ICLAA 20178 1.3 Delegates Presenting Poster: ICLAA 2017 10 1.4 Fourth DAE-BRNS Theme Meeting on Covariance Matrix 10 1.5 Speakers in DAE-BRNS Theme Meeting 10 1.6 Acknowledgements 12 1.7 Special Issues 12 2 Committees 15 3 Message 19 4 From the Desk of Chairman (DAE-BRNS Theme Meeting) 21 5 Technical Committee: DAE-BRNS Theme Meeting 23 6 Program: DAE-BRNS Theme Meeting 25 7 Abstracts: DAE-BRNS Theme Meeting 29 8 List of Delegates: Theme Meeting 51 9 Program: ICLAA 2017 55 10 Abstracts: ICLAA 2017 61 10.1 Special Lectures & Plenary Talks 61 10.2 Invited Talks 67 10.3 Contributory Talks 85 10.4 Posters 131 11 List of Delegates: ICLAA 2017 137 3 Overview of ICLAA 2017 & Theme Meeting on Covariance Matrix International conference on Linear Al- gebra and its Applications–ICLAA 2017, third in its sequence, following CMT- GIM 2012 and ICLAA 2014, held in Manipal Academy of Higher Educa- tion, Manipal, India in December 11-15, 2017. -
Spectrum and Pseudospectrum of a Tensor
Spectrum and Pseudospectrum of a Tensor Lek-Heng Lim University of California, Berkeley July 11, 2008 L.-H. Lim (Berkeley) Spectrum and Pseudospectrum of a Tensor July 11, 2008 1 / 27 Matrix eigenvalues and eigenvectors One of the most important ideas ever invented. I R. Coifman et. al.: \Eigenvector magic: eigenvectors as an extension of Newtonian calculus." Normal/Hermitian A I Invariant subspace: Ax = λx. > > I Rayleigh quotient: x Ax=x x. > 2 I Lagrange multipliers: x Ax − λ(kxk − 1). > I Best rank-1 approximation: minkxk=1kA − λxx k. Nonnormal A −1 −1 I Pseudospectrum: σ"(A) = fλ 2 C j k(A − λI ) k > " g. ∗ I Numerical range: W (A) = fx Ax 2 C j kxk = 1g. ∗ I Irreducible representations of C (A) with natural Borel structure. ∗ I Primitive ideals of C (A) with hull-kernel topology. How can one define these for tensors? L.-H. Lim (Berkeley) Spectrum and Pseudospectrum of a Tensor July 11, 2008 2 / 27 DARPA mathematical challenge eight One of the twenty three mathematical challenges announced at DARPA Tech 2007. Problem Beyond convex optimization: can linear algebra be replaced by algebraic geometry in a systematic way? Algebraic geometry in a slogan: polynomials are to algebraic geometry what matrices are to linear algebra. Polynomial f 2 R[x1;:::; xn] of degree d can be expressed as > > f (x) = a0 + a1 x + x A2x + A3(x; x; x) + ··· + Ad (x;:::; x): n n×n n×n×n n×···×n a0 2 R; a1 2 R ; A2 2 R ; A3 2 R ;:::; Ad 2 R . Numerical linear algebra: d = 2. -
Mathematical Analysis of Quantum Mechanics with Non-Self-Adjoint Operators Radek Novak
Mathematical analysis of quantum mechanics with non-self-adjoint operators Radek Novak To cite this version: Radek Novak. Mathematical analysis of quantum mechanics with non-self-adjoint operators. Func- tional Analysis [math.FA]. Université de Nantes; Czech Technical University in Prague, 2018. English. tel-01903520 HAL Id: tel-01903520 https://hal.archives-ouvertes.fr/tel-01903520 Submitted on 24 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE DE DOCTORAT DE L'UNIVERSITE DE NANTES ET L’UNIVERSITE TECHNIQUE DE PRAGUE COMUE UNIVERSITE BRETAGNE LOIRE ECOLE DOCTORALE N° 601 Mathématiques et Sciences et Technologies de l'Information et de la Communication Spécialité : Mathématiques et leurs Interactions Par Radek NOVÁK Mathematical analysis of quantum mechanics with non-self-adjoint operators Thèse présentée et soutenue à Nantes, le 19 octobre 2018 Unité de recherche : Laboratoire de Mathématiques Jean Leray Rapporteurs avant soutenance : Michael LEVITIN Professeur, Université de Reading Thierry RAMOND Maître de Conférences, Université -
ON the OPEN BALL CENTERED at an INVERTIBLE ELEMENT of a BANACH ALGEBRA 1. Introduction Let a Be a Complex Unital Banach Algebra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Research Archive of Indian Institute of Technology Hyderabad O perators a nd & M atrices www.ele-math.com ON THE OPEN BALL CENTERED AT AN INVERTIBLE ELEMENT OF A BANACH ALGEBRA GEETHIKA SEBASTIAN AND SUKUMAR DANIEL Submitted to Operators and Matrices Abstract. Let A be a complex unital Banach algebra. Since the set of invertible elements is open, there is an open ball around every invertible element. In this article, we investigate the Banach algebras for which the radius given by the Neumann series is optimal. 1. Introduction Let A be a complex unital Banach algebra with unit e. G(A) and Sing(A) denote the set of invertible and singular elements of A respectively. The spectrum, spectral radius and the resolvent of an element a in A, are denoted by s(a), r(a) and r(a) respectively. It is well known that G(A) is open in A, as the open ball centered at any invert- a 1 B a; 1 G(A) ible element with radius ka−1k , denoted by ka−1k , is contained in ([2] Theorem 2.11). It is a natural to ask if there is a bigger open ball centered at a inside G(A)? It will be convenient to have the following definition. DEFINITION 1. An element a 2 G(A) is said to satisfy condition (B) (or belongs to the B class) if 1 B a; \ Sing(A) 6= f: ka−1k We say a Banach algebra A satisfies condition (B) if every a 2 G(A) satisfies condition (B).