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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

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The LR Cholesky algorithm for symmetric hierarchical matrices

The LR Cholesky algorithm for symmetric hierarchical matrice...

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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

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

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

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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

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Fast low-rank empirical cross gramians

Fast low-rank empirical cross gramians

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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

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

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

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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

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AFast Solver for an H_(1) Regularized PDE-Constrained Optimization Problem

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Communications in computational Physics 2016年第1期19卷 143-167页

作者： Andrew T.Barker Tyrone Rees Martin Stoll Center for Applied Scientific Computing Lawrence Livermore National LaboratoryMail Stop L-561LivermoreCA 94551USA Numerical Analysis Group Scientific Computing DepartmentRutherford Appleton LaboratoryChiltonDidcotOxfordshireOX110QXUnited Kingdom Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical SystemsSandtorstr.139106MagdeburgGermany

In this paperwe consider PDE-constrained optimization problemswhich incorporate an H_(1)regularization control *** focus on a time-dependent PDE,and consider both distributed and boundary *** problems we consider include bound constraints on the state,and we use a Moreau-Yosida penalty function to handle *** propose Krylov solvers and Schur complement preconditioning strategies for the different problems and illustrate their performance with numerical examples.

关键词： Preconditioning Krylov methods PDE-constrained optimization optimal control of PDEs

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On reduced input-output dynamic mode decomposition

arXiv

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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

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Adaptively restarted block Krylov subspace methods with low-synchronization skeletons

arXiv

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arXiv 2022年

作者： Lund, Kathryn Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

With the recent realization of exascale performace by Oak Ridge National Laboratory’s Frontier supercomputer, reducing communication in kernels like QR factorization has become even more imperative. Low-synchronization Gram-Schmidt methods, first introduced in [K. Swirydowicz, ´ J. Langou, S. Ananthan, U. Yang, and S. Thomas, Low Synchronization Gram-Schmidt and Generalized Minimum Residual Algorithms, Numer. Lin. Alg. Appl., Vol. 28(2), e2343, 2020)], have been shown to improve the scalability of the Arnoldi method in high-performance distributed computing. Block versions of low-synchronization Gram-Schmidt show further potential for speeding up algorithms, as column-batching allows for maximizing cache usage with matrix-matrix operations. In this work, low-synchronization block Gram-Schmidt variants from [E. Carson, K. Lund, M. Rozložník, and S. Thomas, Block Gram-Schmidt algorithms and their stability properties, Lin. Alg. Appl., 638, pp. 150–195, 2022] are transformed into block Arnoldi variants for use in block full orthogonalization methods (BFOM) and block generalized minimal residual methods (BGMRES). An adaptive restarting heuristic is developed to handle instabilities that arise with the increasing condition number of the Krylov basis. The performance, accuracy, and stability of these methods are assessed via a flexible benchmarking tool written in MATLAB. The modularity of the tool additionally permits generalized block inner products, like the global inner *** Codes 65F10, 65F25, 65F50, 15-04 Copyright © 2022, The Authors. All rights reserved.

关键词： Stability

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Model order reduction via substructuring for a nonlinear, differential-algebraic machine tool model with moving loads

Model order reduction via substructuring for a nonlinear, di...

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92nd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)

作者： Aumann, Quirin Benner, Peter Saak, Jens Vettermann, Julia Mathematics in Industry and Technology Faculty of Mathematics Technische Universitat Chemnitz Reichenhainer Str. 41 Chemnitz09126 Germany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

Machine tools are permanently exposed to complex static, dynamic and thermic loads. This often results in an undesired dis¬placement of the tool center point (TCP), causing errors in the production process and thus limiting the achievable workpiece quality. Recent research efforts try to counter these effects by process-parallel solutions utilizing machine internal data. Therefore, fast simulation models are a fundamental requirement. Here, model order reduction (MOR) becomes crucial. The resulting low-dimensional models are required for various applications, e.g., in digital twins, the correction of thermally in¬duced errors at the TCP during the production process as well as for lifetime calculations in predictive maintenance. In this contribution, we present a strategy of MOR for a coupled thermo-mechanical model with a nonlinear subsystem and moving loads, using the example of a feed axis with nonlinear machine components. Applying tailored substructuring tech¬niques, we are able to separate the linear and nonlinear system components. This allows to apply classic linear MOR methods to the much larger linear part, whereas the small nonlinear part is kept, and thus enables drastically reduced computing times. The relative movement is considered by a switched system approach. Transient thermo-mechanical interactions of the feed axis are calculated in a final investigation, comparing the performance of the resulting reduced-order model and the original one. © 2023 The Authors.

关键词： Machine tools

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Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms

arXiv

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arXiv 2020年

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

In this work, the empirical-Gramian-based model reduction methods: Empirical poor man’s truncated balanced realization, empirical approximate balancing, empirical dominant subspaces, empirical balanced truncation, and empirical balanced gains are compared in a non-parametric and two parametric variants, via ten error measures: Approximate Lebesgue L0, L1, L2, L∞, Hardy H2, H∞, Hankel, Hilbert-Schmidt-Hankel, modified induced primal, and modified induced dual norms, for variants of the thermal block model reduction benchmark. This comparison is conducted via a new meta-measure for model reducibility called MORscore. inMSC Codes 93A15, 93B11, 65Y20 Copyright © 2020, The Authors. All rights reserved.

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