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Efficient Solution of Sequences of Parametrized Lyapunov Equations With applications

numerical linear algebra WITH applications 2025年第1期32卷

作者： Palitta, Davide Tomljanovic, Zoran Nakic, Ivica Saak, Jens Univ Bologna Dipartimento Matemat Ctr AM2 Bologna Italy JJ Strossmayer Univ Osijek Sch Appl Math & Informat Osijek Croatia Univ Zagreb Fac Sci Dept Math Zagreb Croatia Max Planck Inst Dynam Complex Tech Syst Computat Methods Syst & Control Theory CSC Magdeburg Germany

Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of trace(EX)$$ \mathrm{trace}(EX) $$, where X$$ X $$ denotes the solution of a Lyapunov equation and E$$ E $$ is a given matrix. We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low-rank modification to a seed, fixed matrix. We propose two novel numerical procedures that fully exploit such a common structure. The first one builds upon the Sherman-Morrison-Woodbury (SMW) formula and recycling Krylov techniques, and it is well-suited for small dimensional problems as it makes use of dense numerical linear algebra tools. The second algorithm can instead address large-scale problems by relying on state-of-the-art projection techniques based on the extended Krylov subspace. We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi-agent systems. Our results show that the algorithms we propose are superior to state-of-the-art techniques as they are able to remarkably speed up the computation of accurate solutions.

关键词： extended Krylov methods multi-agent systems parametrized Lyapunov equations recycling Krylov methods vibrational systems

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A subspace constrained randomized Kaczmarz method for structure or external knowledge exploitation

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linear algebra AND ITS applications 2024年 698卷 220-260页

作者： Lok, Jackie Rebrova, Elizaveta Princeton Univ Dept Operat Res & Financial Engn Princeton NJ 08544 USA

We study a version of the randomized Kaczmarz algorithm for solving systems of linear equations where the iterates are confined to the solution space of a selected subsystem. We show that the subspace constraint leads to an accelerated convergence rate, especially when the system has approximately low-rank structure. On Gaussian-like random data, we show that it results in a form of dimension reduction that effectively increases the aspect ratio of the system. Furthermore, this method serves as a building block for a second, quantile-based algorithm for solving linear systems with arbitrary sparse corruptions, which is able to efficiently utilize external knowledge about corruption- free equations and achieve convergence in difficult settings. numerical experiments on synthetic and realistic data support our theoretical results and demonstrate the validity of the proposed methods for even more general data models than guaranteed by the theory. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Kaczmarz algorithm Stochastic iterative methods Quantile methods Corrupted linear systems

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Acceleration and restart for the randomized Bregman-Kaczmarz method

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linear algebra AND ITS applications 2024年 699卷 508-538页

作者： Tondji, Lionel Necoara, Ion Lorenz, Dirk A. TU Braunschweig Inst Anal & Algebra D-38092 Braunschweig Germany Univ Bremen Ctr Ind Math Postfach 330440 D-28334 Bremen Germany Natl Univ Sci & Technol Politehn Bucharest Automat Control & Syst Engn Dept Bucharest 060042 Romania Romanian Acad Gheorghe Mihoc Caius Iacob Inst Math Stat & Appl M Bucharest 050711 Romania

Optimizing strongly convex functions subject to linear constraints is a fundamental problem with numerous applications. In this work, we propose a block (accelerated) randomized Bregman-Kaczmarz method that only uses a block of constraints in each iteration to tackle this problem. We consider a dual formulation of this problem in order to deal in an efficient way with the linear constraints. Using convex tools, we show that the corresponding dual function satisfies the Polyak-Lojasiewicz (PL) property, provided that the primal objective function is strongly convex and verifies additionally some other mild assumptions. However, adapting the existing theory on coordinate descent methods to our dual formulation can only give us sublinear convergence results in the dual space. In order to obtain convergence results in some criterion corresponding to the primal (original) problem, we transfer our algorithm to the primal space, which combined with the PL property allows us to get linear convergence rates. More specifically, we provide a theoretical analysis of the convergence of our proposed method under different as- sumptions on the objective and demonstrate in the numerical experiments its superior efficiency and speed up compared to existing methods for the same problem. (c) 2024 Published by Elsevier Inc.

关键词： linear systems Sparse solutions method Acceleration Restart Randomized Bregman-Kaczmarz

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On the uniqueness and computation of commuting extensions

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linear algebra AND ITS applications 2024年 703卷 645-666页

作者： Koiran, Pascal Univ Claude Bernard Lyon 1 ENS Lyon CNRS INRIALIPUMR 5668 F-69342 Lyon 07 France

A tuple (Z(1) , ... , Z( p) ) of matrices of size r x r is said to be a commuting extension of a tuple (A(1), ... , A(p) ) of matrices of size n x n if the Z i pairwise commute and each A i sits in the upper left corner of a block decomposition of Z i (here, r and n are two arbitrary integers with n < r ). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called "quantum Zeno dynamics." Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results: (i) Theorems on the uniqueness of commuting extensions for three matrices or more. (ii) Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r = 4n/3, n/ 3, and are apparently the first provably efficient algorithms for this problem applicable beyond r = n + 1. (iii) A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices. (c) 2024 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// *** /licenses /by /4 .0/).

关键词： Commuting matrices Algorithms algebraic complexity Tensor rank Quantum information numerical analysis

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A two-step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations

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numerical linear algebra WITH applications 2024年第3期31卷 e2510-e2510页

作者： Bai, Zhong-Zhi Chinese Acad Sci Inst Computat Math & Sci Engn Comp Acad Math & Syst Sci State Key Lab Sci Engn Comp Beijing Peoples R China Chinese Acad Sci Inst Computat Math & Sci Engn Comp Acad Math & Syst Sci State Key Lab Sci Engn Comp POB 2719 Beijing 100190 Peoples R China

For solving large sparse systems of linear equations, we construct a paradigm of two-step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive-definite matrix class. This two-step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two-step matrix splitting iteration methods. This result provides systematic treatment for the two-step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems.

关键词： convergence property iteration method matrix splitting system of linear equations two-step scheme

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Computation of quasiseparable representations of Green matrices

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linear algebra and Its applications 2025年 721卷 47-80页

作者： Boito, P. Eidelman, Y. Dipartimento di Matematica Università di Pisa Largo Bruno Pontecorvo 5 Pisa56127 Italy School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University Ramat-Aviv69978 Israel

The well-known Asplund theorem states that the inverse of a (possibly one-sided) band matrix A is a Green matrix. In accordance with quasiseparable theory, such a matrix admits a quasiseparable representation in its rank-structured part. Based on this idea, we derive algorithms that compute a quasiseparable representation of A−1 with linear complexity. Many inversion algorithms for band matrices exist in the literature. However, algorithms based on a computation of the rank structure performed theoretically via the Asplund theorem appear for the first time in this paper. numerical experiments confirm complexity estimates and offer insight into stability properties. © 2024 The Author(s)

关键词： Matrix algebra

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Invariants of quadratic forms and applications in design theory

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linear algebra AND ITS applications 2024年 682卷 1-27页

作者： Gnilke, Oliver W. Cathain, Padraig o Olmez, Oktay Ponasso, Guillermo Nunez Aalborg Univ Dept Math Sci DK-9220 Aalborg Denmark Dublin City Univ Fiontar & Scoil Na Gaeilge Dublin Ireland Afiniti Hamilton Bermuda Worcester Polytech Inst Worcester MA USA

The study of regular incidence structures, such as projective planes and symmetric block designs, is a well established topic in discrete mathematics. The work of Bruck, Ryser and Chowla in the mid-twentieth century applied the HasseMinkowski lo cal -global theory for quadratic forms to derive non-existence results for certain design parameters. Several combinatorialists have provided alternative proofs of this result, replacing conceptual arguments with algorithmic ones. In this paper, we show that the methods required are purely linear-algebraic and are no more difficult conceptually than the theory of the Jordan Canonical Form. Computationally, they are rather easier. We conclude with some classical and recent applications to design theory, including a novel application to the decomposition of incidence matrices of symmetric designs. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

关键词： Quadratic forms Incidence matrix Symmetric design

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Recent and Upcoming Developments in Randomized numerical linear algebra for Machine Learning 24

Recent and Upcoming Developments in Randomized Numerical Lin...

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30th ACM SIGKDD conference on Knowledge Discovery and Data Mining

作者： Derezinski, Michal Mahoney, Michael W. Univ Michigan Ann Arbor MI 48109 USA LBNL ICSI Berkeley CA USA Univ Calif Berkeley Berkeley CA USA

ISBN: (纸本)9798400704901

Large matrices arise in many machine learning and data analysis applications, including as representations of datasets, graphs, model weights, and first and second-order derivatives. Randomized numerical linear algebra (RandNLA) is an area which uses randomness to develop improved algorithms for ubiquitous matrix problems. The area has reached a certain level of maturity;but recent hardware trends, efforts to incorporate RandNLA algorithms into core numerical libraries, and advances in machine learning, statistics, and random matrix theory, have lead to new theoretical and practical challenges. This article provides a self-contained overview of RandNLA, in light of these developments.

关键词： Matrix computations Randomization Optimization Statistics

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Binary linear Codes via Zeon and Sym-Clifford algebras 1

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1st International conference of Advanced Computational applications of Geometric algebra (ICACGA)

作者： Price, Melissa McLeod Staples, G. Stacey Southern Illinois Univ Edwardsville Dept Math & Stat Edwardsville IL 62026 USA

ISBN: (数字)9783031340314

ISBN: (纸本)9783031340307;9783031340314

Zeon algebras have proven to be useful for enumerating structures in graphs, such as paths, trails, cycles, matchings, cliques, and independent sets. Sym-Clifford algebras have been used to enumerate walks on hypercubes without the need for adjacency matrices. In the current work, zeon ("nil-Clifford") and "sym-Clifford" methods are used to reformulate essential concepts of binary linear coding theory. In particular, zeon and sym-Clifford methods are used to generate linear codes and to illustrate Clifford-algebraic formulations of encoding, decoding and error-correction.

关键词： Binary codes Clifford algebras zeons

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Boundedness and stability of nonlinear hybrid stochastic differential delay equation disturbed by Levy noise

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IET CONTROL theory AND applications 2023年第1期17卷 109-122页

作者： Yang, Ning Chen, Dongyan Hu, Jun Harbin Univ Sci & Technol Dept Math Harbin 150080 Peoples R China Harbin Univ Sci & Technol Sch Automat Harbin Peoples R China

This study focuses on the boundedness and stability of the nonlinear hybrid stochastic differential delay equation (NHSDDE) disturbed by Levy noise (LN). Different from the previous systems with Markovian switching, not only different parameters under different modes, but also different structures under different modes are considered, where a novel Lyapunov function is constructed to handle the impacts of different structures under different modes. Without the linear growth condition, the existence of the global solution is proved for the NHSDDE with LN. Then, the M-matrix technique is applied to build some reasonable theorems upon the vth moment asymptotic boundedness, the vth moment exponential stability and the almost surely exponential stability for the NHSDDE driven by LN. An approach to solving the admissible upper bound of the time-delay derivative is given for guaranteeing the asymptotic boundedness and the exponential stability of the global solution. Finally, two numerical examples and homologous simulated outcomes are provided to indicate the usefulness of the presented theoretical findings.

关键词： Nonlinear control systems asymptotic stability moment asymptotic boundedness moment exponential stability Lyapunov methods Time-varying control systems almost surely exponential stability M-matrix technique Markovian switching time-delay derivative nonlinear systems Lévy noise homologous simulated outcomes Distributed parameter control systems delays matrix algebra NHSDDE stochastic systems Lyapunov function Differential equations (numerical analysis) nonlinear differential equations linear algebra (numerical analysis) Stability in control theory nonlinear hybrid stochastic differential delay equation Markov processes

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