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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

Distributed optimal control problems governed by coupled convection dominated PDEs with control constraints

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Lecture Notes in Computational Science and Engineering

Lecture Notes in computational Science and Engineering 2015年 103卷 469-478页

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

We study the numerical solution of control constrained optimal control problems governed by a system of convection diffusion equations with nonlinear reaction terms, arising from chemical processes. control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method or by adding a Moreau-Yosida-type penalty function to the cost functional. An adaptive mesh refinement indicated by a posteriori error estimates is applied for both approaches. © Springer International Publishing Switzerland 2015.

关键词： Optimal control systems

Model order reduction for gas and energy networks

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arXiv 2020年

作者： Himpe, Christian Grundel, Sara Benner, Peter Computational Methods in Systems and Control Theory Group The Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstraße 1 MagdeburgD-39106 Germany

To counter the volatile nature of renewable energy sources, gas networks take a vital role. But, to ensure fulfillment of contracts under these circumstances, a vast number of possible scenarios, incorporating uncertain supply and demand, has to be simulated ahead of time. This many-query gas network simulation task can be accelerated by model reduction, yet, large-scale, nonlinear, parametric, hyperbolic partial differential(-algebraic) equation systems, modeling natural gas transport, are a challenging application for model order reduction algorithms. For this industrial application, we bring together the scientific computing topics of: mathematical modeling of gas transport networks, numerical simulation of hyperbolic partial differential equation, and parametric model reduction for nonlinear systems. This research resulted in the morgen (Model Order Reduction for Gas and Energy Networks) software platform, which enables modular testing of various combinations of models, solvers, and model reduction methods. In this work we present the theoretical background on systemic modeling and structured, data-driven, system-theoretic model reduction for gas networks, as well as the implementation of morgen and associated numerical experiments testing model reduction adapted to gas network *** Codes 93-04, 76N15, 93B20 Copyright © 2020, The Authors. All rights reserved.

关键词： Nonlinear equations

Next-Gen gas network simulation

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arXiv 2021年

To overcome many-query optimization, control, or uncertainty quantification work loads in reliable gas and energy network operations, model order reduction is the mathematical technology of choice. To this end, we enhance the model, solver and reductor components of the morgen platform, introduced in HIMPE ET AL [J. Math. Ind. 11:13, 2021], and conclude with a mathematically, numerically and computationally favorable model-solver-reductor ensemble. Copyright © 2021, The Authors. All rights reserved.

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The LR Cholesky algorithm for symmetric hierarchical matrices

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The LR Cholesky algorithm for symmetric hierarchical matrice...

作者： Benner, Peter Mach, Thomas Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Sandtorstr. 1 39106 Magdeburg Germany Technische Universität Chemnitz Fakultät für Mathematik Reichenhainer Str. 41 09126 Chemnitz Germany

We investigate the application of the LR Cholesky algorithm to symmetric hierarchical matrices, symmetric simple structured hierarchical matrices and symmetric hierarchically semiseparable (HSS) matrices. The data-sparsity of these matrices make the otherwise expensive LR Cholesky algorithm applicable, as long as the data-sparsity is preserved. We will see in an example that the ranks of the low rank blocks grow and the data-sparsity gets lost. We will explain this behavior by applying a theorem on the structure preservation of diagonal plus semiseparable matrices under LR Cholesky transformations. Therefore we have to give a new more constructive proof for the theorem. We will show that the structure of H-matrices is almost preserved and so the LR Cholesky algorithm is of almost quadratic complexity for H-matrices. © 2013 Elsevier Inc. All rights reserved.

关键词： Matrix algebra

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

Fast low-rank empirical cross gramians

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Fast low-rank empirical cross gramians

88th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2017

作者： Himpe, Christian Leibner, Tobias Rave, Stephan Saak, Jens Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Applied Mathematics University of Münster Einsteinstr. 62 Münster48149 Germany

The cross Gramian matrix encodes the input-output coherence of linear control systems and is used in projection-based model reduction. The empirical cross Gramian is a data-driven variant of the cross Gramian which also extends to nonlinear systems. A drawback of the empirical cross Gramian for large-scale systems is its full order and dense structure;yet, it may be computed column-wise. Using the hierarchical approximate proper orthogonal decomposition (HAPOD), this partial computability can be exploited to obtain a truncated projection for model order reduction without assembling a full cross Gramian. © 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Linear control systems

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

On reduced input-output dynamic mode decomposition

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arXiv 2017年

作者： Benner, Peter Himpe, Christian Mitchell, Tim Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

The identification of reduced-order models from high-dimensional data is a challenging task, and even more so if the identified system should not only be suitable for a certain data set, but generally approximate the input-output behavior of the data source. In this work, we consider the input-output dynamic mode decomposition method for system identification. We compare excitation approaches for the data-driven identification process and describe an optimization-based stabilization strategy for the identified *** Codes 93B30, 90C99 Copyright © 2017, The Authors. All rights reserved.

关键词： Clustering algorithms