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Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: A state of the art survey

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GAMM Mitteilungen 2013年第1期36卷 32-52页

作者： Benner, Peter Saak, Jens Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Sandtorstr. 1 D-39106 Magdeburg Germany Chemnitz University of Technology Department of Mathematics Mathematics in Industry and Technology D-09107 Chemnitz Germany

Efficient numerical algorithms for the solution of large and sparse matrix Riccati and Lyapunov equations based on the low rank alternating directions implicit (ADI) iteration have become available around the year 2000. Over the decade that passed since then, additional methods based on extended and rational Krylov subspace projection have entered the field and proved to be competitive alternatives. In this survey we sketch both types of methods and discuss their advantages and drawbacks. We focus on the continuous time case here, but corresponding results for discrete time problems can for most results be found in the available literature. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Surveys

Parallel algorithms for tensor train arithmetic

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arXiv 2020年

作者： Al Daas, Hussam Ballard, Grey Benner, Peter Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Computer Science Department Wake Forest University Winston SalemNC United States

We present efficient and scalable parallel algorithms for performing mathematical operations for low-rank tensors represented in the tensor train (TT) format. We consider algorithms for addition, elementwise multiplication, computing norms and inner products, orthonormalization, and rounding (rank truncation). These are the kernel operations for applications such as iterative Krylov solvers that exploit the TT structure. The parallel algorithms are designed for distributed- memory computation, and we propose a data distribution and strategy that parallelizes computations for individual cores within the TT format. We analyze the computation and communication costs of the proposed algorithms to show their scalability, and we present numerical experiments that demonstrate their efficiency on both shared-memory and distributed-memory parallel systems. For example, we observe better single-core performance than the existing MATLAB TT-Toolbox in rounding a 2GB TT tensor, and our implementation achieves a 34× speedup using all 40 cores of a single node. We also show nearly linear parallel scaling on larger TT tensors up to over 10,000 cores for all mathematical operations. Copyright © 2020, The Authors. All rights reserved.

关键词： Tensors

Variational Analysis of the Spectral Abscissa at a Matrix with a Nongeneric Multiple Eigenvalue

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Set-Valued and Variational Analysis

Set-Valued and Variational Analysis 2014年第1期22卷 19-43页

作者： Grundel, Sara Overton, Michael L. Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 39106 Madgeburg Germany Courant Institute of Mathematical Sciences New York University 251 Mercer Street NY NY 10012 United States

The spectral abscissa is a fundamental map from the set of complex matrices to the real numbers. Denoted α and defined as the maximum of the real parts of the eigenvalues of a matrix X, it has many applications in stability analysis of dynamical systems. The function α is nonconvex and is non-Lipschitz near matrices with multiple eigenvalues. Variational analysis of this function was presented in Burke and Overton (Math Program 90:317-352, 2001), including a complete characterization of its regular subgradients and necessary conditions which must be satisfied by all its subgradients. A complete characterization of all subgradients of α at a matrix X was also given for the case that all active eigenvalues of X (those whose real part equals α(X)) are nonderogatory (their geometric multiplicity is one) and also for the case that they are all nondefective (their geometric multiplicity equals their algebraic multiplicity). However, necessary and sufficient conditions for all subgradients in all cases remain unknown. In this paper we present necessary and sufficient conditions for the simplest example of a matrix X with a derogatory, defective multiple eigenvalue. © 2013 Springer Science+Business Media Dordrecht.

关键词： Matrix algebra

Peer methods for the solution of large-scale differential matrix equations

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arXiv 2018年

作者： Benner, Peter Lang, Norman Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory MagdeburgD-39106 Germany Technische Universitat Chemnitz Faculty of Mathematics Mathematics in Industry and Technology ChemnitzD-09126 Germany

We consider the application of implicit and linearly implicit (Rosenbrock-type) peer methods to matrix-valued ordinary differential equations. In particular the differential Riccati equation (DRE) is investigated. For the Rosenbrock-type schemes, a reformulation capable of avoiding a number of Jacobian applications is developed that, in the autonomous case, reduces the computational complexity of the algorithms. Dealing with large-scale problems, an efficient implementation based on low-rank symmetric indefinite factorizations is presented. The perfor- mance of both peer approaches up to order 4 is compared to existing implicit time integration schemes for matrix-valued differential *** Codes 15A23, 34A09, 65L99, 93A15 Copyright © 2018, The Authors. All rights reserved.

关键词： Ordinary differential equations

Inexact methods for the low rank solution to large scale Lyapunov equations

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arXiv 2018年

作者： Kürschner, Patrick Freitag, Melina A. Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg Germany Department of Mathematical Sciences University of Bath Claverton Down BA2 7AY United Kingdom

The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method the repeated solution to a shifted linear system is necessary. For very large systems this solve is usually implemented using iterative methods, leading to inexact solves within this inner iteration. We derive theory for a relaxation strategy within these inexact solves, both for the RKSM and the LR-ADI method. Practical choices for relaxing the solution tolerance within the inner linear system are then provided. The theory is supported by several numerical examples.65F10, 65F30 Copyright © 2018, The Authors. All rights reserved.

关键词： Lyapunov functions

The Fréchet derivative of the tensor t-function

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arXiv 2023年

作者： Lund, Kathryn Schweitzer, Marcel Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany School of Mathematics and Natural Sciences Bergische Universität Wuppertal Wuppertal42097 Germany

The tensor t-function, a formalism that generalizes the well-known concept of matrix functions to third-order tensors, is introduced in [K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numer. Linear Algebra Appl. 27 (3), e2288]. In this work, we investigate properties of the Fréchet derivative of the tensor t-function and derive algorithms for its efficient numerical computation. Applications in condition number estimation and nuclear norm minimization are explored. Numerical experiments implemented by the t-Frechet toolbox hosted at https://***/katlund/t-frechet illustrate properties of the t-function Fréchet derivative, as well as the efficiency and accuracy of the proposed *** Codes 15A69, 65F60, 65F35 Copyright © 2023, The Authors. All rights reserved.

关键词： Tensors

Reduced-Order Modeling of Index-1 Vibrational systems Using Interpolatory Projections

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Reduced-Order Modeling of Index-1 Vibrational Systems Using ...

International Conference on Computer and Information Technology

作者： Peter Benner Jens Saak M. Monir Uddin Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstra?e 1 D-39106 Magdeburg Germany Department of Mathematics and Physics North South University 15/B Bashundhara Dhaka-1239 Bangladesh

ISBN: (纸本)9781509040919

Large sparse second order index-1 descriptor systems arise in various disciplines of science and engineering, such as constraint mechanics or multibody dynamics, mechatronics (where mechanical and electrical elements are coupled), but also RLC circuit design. Simulation, controller design and design optimization are only some applications of such models. Either of these tasks, just like any other many-query situation becomes unfeasible when the system is high dimensional. This paper discusses an algorithm to obtain a reduced state space model of a large sparse second order index-1 system using an interpolatory projection method based on the iterative rational Krylov algorithm (IRKA). In each iteration of this algorithm, we need to solve a number of linear systems. The main contribution of this paper is to solve these linear systems by exploiting the sparsity of the original model, which reduces the computational cost drastically. The algorithm is applied to a micro-mechanical piezo-actuated structural FEM model of a certain building block of a machine tool. Numerical experiments with a complex 3d model of an adaptive spindle support (a piezo-mechanical multiphysics system) show the effectivity and efficiency of the techniques.

关键词： second order index-1 systems sparsity model reduction interpolatory projections

Low-rank-modified Galerkin methods for the Lyapunov equation

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arXiv 2023年

作者： Lund, Kathryn Palitta, Davide Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Dipartimento di Matematica Piazza di Porta S. Donato 5 Bologna40126 Italy

Of all the possible projection methods for solving large-scale Lyapunov matrix equations, Galerkin approaches remain much more popular than minimal-residual ones. This is mainly due to the different nature of the projected problems stemming from these two families of methods. While a Galerkin approach leads to the solution of a low-dimensional matrix equation per iteration, a matrix least-squares problem needs to be solved per iteration in a minimal-residual setting. The significant computational cost of these least-squares problems has steered researchers towards Galerkin methods in spite of the appealing properties of minimal-residual schemes. In this paper we introduce a framework that allows for modifying the Galerkin approach by low-rank, additive corrections to the projected matrix equation problem with the two-fold goal of attaining monotonic convergence rates similar to those of minimal-residual schemes while maintaining essentially the same computational cost of the original Galerkin method. We analyze the well-posedness of our framework and determine possible scenarios where we expect the residual norm attained by two low-rank-modified variants to behave similarly to the one computed by a minimal-residual technique. A panel of diverse numerical examples shows the behavior and potential of our new *** Codes 65F45, 65F50, 65F10, 65N22, 65J10 Copyright © 2023, The Authors. All rights reserved.

关键词： Galerkin methods

Sustainable research software hand-over

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arXiv 2019年

作者： Fehr, Jörg Himpe, Christian Rave, Stephan Saak, Jens Institute of Engineering and Computational Mechanics University of Stuttgart Pfaffenwaldring 9 Stuttgart70569 Germany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Institute for Computational and Applied Mathematics University of Münster Einsteinstrasse 62 Münster48149 Germany

Scientific software projects evolve rapidly in their initial development phase, yet at the end of a funding period, the completion of a research project, thesis, or publication, further engagement in the project may slow down or cease completely. To retain the invested effort for the sciences, this software needs to be preserved or handed over to a succeeding developer or team, such as the next generation of (PhD) students. Comparable guides provide top-down recommendations for project leads. This paper intends to be a bottom-up approach for sustainable hand-over processes from a developer's perspective. An important characteristic in this regard is the project's size, by which this guideline is structured. Furthermore, checklists are provided, which can serve as a practical guide for implementing the proposed *** Codes 68N30, 97N80, 97P99 Copyright © 2019, The Authors. All rights reserved.

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