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On an invariance principle for the solution space of the differential riccati equation

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On an invariance principle for the solution space of the dif...

89th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2018

作者： Behr, Maximilian Benner, Peter Heiland, Jan Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

The differential Riccati equation appears in different fields of applied mathematics like control theory and systems theory. For large-scale systems the numerical solution comes with a large amount of storage requirements. This motivates the use of Krylov subspace and projection based methods [1–3]. In the present paper we apply an invariance theorem for ODEs to the differential Riccati Equation. We show that the solution is contained in a Krylov like subspace and extend our results to certain time-varying cases. © 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Riccati equations

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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Exact Optimization: Part I

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arXiv 2019年

作者： Lin, Li-Gang Liang, Yew-Wen The Department of Mechanical Engineering National Central University Taoyuan32001 Taiwan The Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany The Institute of Electrical Control Engineering National Yang Ming Chiao Tung University Hsinchu30010 Taiwan

Nonlinear programming is explicitly analyzed via a novel perspective/method and from a bottom-up manner. The philosophy is based on the recent findings on convex quadratic equation (CQE), which help clarify a geometric interpretation that relates CQE to convex quadratic function (CQF). More specifically, regarding the solvability of CQE, its necessary and sufficient condition as well as a unified parameterization of all the solutions has recently been analytically formulated. Moving forward, the understanding of CQE is utilized to describe the geometric structure of CQF, and the CQE-CQF relation. All these results are shown closely related to a basis in the optimization literature, namely quadratic programming (QP). For the first time from this viewpoint, the QPs subject to equality, inequality, equality-and-inequality, and extended constraints can be algebraically solved in derivative-free closed formulae, respectively. All the results are derived without knowing a feasible point, a priori and any time during the *** Codes Primary: 15A18, 52A41, 90C20, 90C25, and 90C46 Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra

High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter Eigenvalue Problem

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arXiv 2019年

作者： Penke, Carolin Marek, Andreas Vorwerk, Christian Draxl, Claudia Benner, Peter Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Germany Max Planck Computing and Data Facility Garching Germany Institut für Physik and IRIS Adlershof Humboldt-Universität zu Berlin Berlin Germany

We present a high-performance solver for dense skew-symmetric matrix eigenvalue problems. Our work is motivated by applications in computational quantum physics, where one solution approach to solve the so-called Bethe-Salpeter equation involves the solution of a large, dense, skew-symmetric eigenvalue problem. The computed eigenpairs can be used to compute the optical absorption spectrum of molecules and crystalline systems. One state-of-the art high-performance solver package for symmetric matrices is the ELPA (Eigenvalue SoLvers for Petascale Applications) library. We exploit a link between tridiagonal skew-symmetric and symmetric matrices in order to extend the methods available in ELPA to skew-symmetric matrices. This way, the presented solution method can benefit from the optimizations available in ELPA that make it a well-established, efficient and scalable library. The solution strategy is to reduce a matrix to tridiagonal form, solve the tridiagonal eigenvalue problem and perform a back-transformation for eigenvectors of interest. ELPA employs a one-step or a two-step approach for the tridiagonalization of symmetric matrices. We adapt these to suit the skew-symmetric case. The two-step approach is generally faster as memory locality is exploited better. If all eigenvectors are required, the performance improvement is compensated by the additional back transformation step. We exploit the symmetry in the spectrum of skew-symmetric matrices, such that only half of the eigenpairs need to be computed, making the two-step approach the favorable method. We compare performance and scalability of our method to the only available high-performance approach for skew-symmetric matrices, an indirect route involving complex arithmetic. In total, we achieve a performance that is up to 3.67 higher than the reference method using Intel’s ScaLAPACK implementation. Our method is freely available in the current release of the ELPA *** Codes 65Y05, 15B57 Copyright © 2019

关键词： Eigenvalues and eigenfunctions

Residual minimizing shift parameters for the low-rank ADI iteration

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arXiv 2018年

作者： Kürschner, Patrick Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg Germany

The low-rank alternating directions implicit (LR-ADI) iteration is a frequently employed method for efficiently computing low-rank approximate solutions of large-scale Lyapunov equations. In order to achieve a rapid error reduction, the iteration requires shift parameters whose selection and generation is often a difficult task, especially for nonsymmetric coefficients in the Lyapunov equation. This article represents a follow up of Benner et al. [ETNA, 43 (2014–2015), pp. 142–162] and investigates self-generating shift parameters based on a minimization principle for the Lyapunov residual norm. Since the involved objective functions are too expensive to evaluate and, hence, intractable, compressed objective functions are introduced which are efficiently constructed from the available data generated by the LR-ADI iteration. Several numerical experiments indicate that these residual minimizing shifts using approximated objective functions outperform existing precomputed and dynamic shift parameter selection techniques, although their generation is more involved.15A06, 65F10, 65F30 Copyright © 2018, The Authors. All rights reserved.

关键词： Lyapunov functions

Improving the performance of numerical algorithms for the bethe-salpeter eigenvalue problem

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Improving the performance of numerical algorithms for the be...

89th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2018

作者： Benner, Peter Marek, Andreas Penke, Carolin Max Planck Institute for Dynamics of Complex Technical Systems: Computational Methods in Systems and Control Theory Sandtorstr. 1 Magdeburg39106 Germany Max Planck Computing and Data Facility Gießenbachstraße 2 Garching85748 Germany

The Bethe-Salpeter eigenvalue problem arises in the computation of the electronic structure of many-body physical systems. The resulting matrix is complex, admits a certain block structure and can become extremely large. This raises the need for structure-preserving algorithms running in parallel on high performance compute clusters. In this paper we examine how a recently proposed direct method given in the BSEPACK library can be improved using eigenvalue solvers from the ELPA library. For large matrices a runtime reduction of up to 80% is achieved. © 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Electronic structure

Linear low-rank parameter-dependent fluid-structure interaction discretization in 2d

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Linear low-rank parameter-dependent fluid-structure interact...

89th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2018

作者： Weinhandl, Roman Benner, Peter Richter, Thomas Max Planck Institute for Dynamics of Complex Technical Systems: Computational Methods in Systems and Control Theory Sandtorstr. 1 Magdeburg39106 Germany Otto von Guericke University Magdeburg Faculty of Mathematics Universitätsplatz 2 Magdeburg39106 Germany

Fluid-structure interaction problems involve material parameters such as the shear modulus of the solid and the dynamic fluid viscosity. In order to examine the behaviors of various solids and adapt the problems to different fluid configurations there is a need to vary such material parameters flexibly. This can be done by a parameter-dependent fluid-structure interaction discretization which yields a system matrix that has block diagonal structure. As also discussed in [2], the resulting equation is equivalent to a matrix equation which allows for a low-rank approach where the iterate is represented by a tensor. A low-rank GMRES variant similar to what was discussed in [1] can then be applied to such parameter-dependent systems. © 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Fluid structure interaction

A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations

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arXiv 2018年

作者： Benner, Peter Bujanović, Zvonimir Kürschner, Patrick Saak, Jens Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg Germany University of Zagreb Department of Mathematics Zagreb Croatia

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance. Copyright © 2018, The Authors. All rights reserved.

关键词： Riccati equations

Cross-gramian-based dominant subspaces

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arXiv 2018年

作者： Benner, Peter Himpe, Christian Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Faculty of Mathematics Otto von Guericke University Magdeburg Universitätsplatz 2 Magdeburg39106 Germany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings. in93A15, 93B11, 93B20 Copyright © 2018, The Authors. All rights reserved.

关键词： Observability