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作者： Köhler, Martin Saak, Jens Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems MagdeburgD-39104 Germany

The Gauss-Jordan Elimination scheme is an alternative to the LU decomposition for solving linear systems or computing the inverse of a matrix. We develop a multi-GPU aware implementation of this algorithm on an OpenPOWER 8 system with application to the Matrix Sign Function. Thereby, we analyze the influence of the CPU clock frequency scaling on the overall energy consumption. The results show possible energy saving of 14.2% without a noteworthy increase of the runtime. © 2018 John Wiley & Sons, Ltd.

关键词： Energy efficiency

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

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

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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

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Numerical solution to low rank perturbed lyapunov equations by the sign function method

Numerical solution to low rank perturbed lyapunov equations ...

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Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

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

We investigate the numerical solution to a low rank perturbed Lyapunov equation AT X + XA = W via the sign function method (SFM). The sign function method has been proposed to solve Lyapunov equations, see e.g. [1], but here we focus on a framework where the matrix A has a special structure, i.e. A = B + UCVT, where B is a blockdiagonal matrix and UCVT is a low rank perturbation. We show that this structure can be kept throughout the sign function iteration but the rank of the perturbation doubles per iteration. Therefore, we apply a low rank approximation to the perturbation in order to keep its numerical rank small. We compare the standard SFM with its structure preserving variant presented in this paper by means of numerical examples from viscously damped mechanical systems. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Matrix algebra

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Towards riccati-feedback control of complex flows with moving interfaces

Towards riccati-feedback control of complex flows with movin...

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

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

Problems featuring moving interfaces appear in many applications. They can model solidification and melting of pure materials, crystal growth and other multi-phase problems. The control of the moving interface enables to, for example, influence production processes and, thus, the product material quality. We consider the two-phase Stefan problem that models a solid and a liquid phase separated by the moving interface. In the liquid phase, the heat distribution is characterized by a convection-diffusion equation. The fluid flow in the liquid phase is described by the Navier–Stokes equations which introduces a differential algebraic structure to the system. The interface movement is coupled with the temperature through the Stefan condition, which adds additional algebraic constraints. Our formulation uses a sharp interface representation and we define a quadratic tracking-type cost functional as a target of a control input. We compute an open loop optimal control for the Stefan problem using an adjoint system. For a feedback representation, we linearize the system about the trajectory defined by the open loop control. This results in a linear-quadratic regulator problem, for which we formulate the differential Riccati equation with time varying coefficients. This Riccati equation defines the corresponding feedback gain. Further, we present the feedback formulation that takes into account the structure and the differential algebraic components of the problem. Also, we discuss how the complexities that come, for example, with mesh movements, can be handled in a feedback setting. © 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Feedback

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Adjoint-based optimal boundary control of a stefan problem system fully coupled with navier-stokes equations

Adjoint-based optimal boundary control of a stefan problem s...

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Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

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

Free boundary and moving boundary problems, that can be used to model crystal growth or the solidification and melting of pure materials, receive growing attention in science and technology. The optimal control of these problems appear even more interesting since certain desired shapes of the boundaries improve, e.g., the material quality in the case of crystal growth. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface. In the work presented, we take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state at a final time as well as the control costs. Following the "optimize-then-discretize" paradigm, we formulate a first order optimality system using the formal Lagrange approach and derive the adjoint PDE system that provides the needed gradient of the cost functional. By means of an example setup of a container with an in-and outflow of water and a cooling unit at the bottom, we illustrate how the derived formulations can be used to achieve a desired interface between the solid and the fluid phase by controlling the flow at the inlet. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Crystal growth

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An artificial neural network for surrogate modeling of stress fields in viscoplastic polycrystalline materials

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npj computational Materials 2023年第1期9卷 1981-1990页

作者： Mohammad S.Khorrami Jaber R.Mianroodi Nima H.Siboni Pawan Goyal Bob Svendsen Peter Benner Dierk Raabe Microstructure Physics and Alloy Design Max-Planck-Institut für EisenforschungDüsseldorfGermany Ergodic Labs Lohmühlenstraße 6512435 BerlinGermany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany Material Mechanics RWTH Aachen UniversityAachenGermany

The purpose of this work is the development of a trained artificial neural network for surrogate modeling of the mechanical response of elasto-viscoplastic grain *** this end,a U-Net-based convolutional neural network(CNN)is trained using results for the von Mises stress field from the numerical solution of initial-boundary-value problems(IBVPs)for mechanical equilibrium in such microstructures subject to quasi-static uniaxial *** resulting trained CNN(tCNN)accurately reproduces the von Mises stress field about 500 times faster than numerical solutions of the corresponding IBVP based on spectral *** of the tCNN to test cases based on microstructure morphologies and boundary conditions not contained in the training dataset is also investigated and discussed.

关键词： methods boundary faster

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Solving differential matrix equations using parareal

Solving differential matrix equations using parareal

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Joint 87th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2016 and Deutsche Mathematiker‐Vereinigung, DMV 2016

作者： Köhler, Martin Lang, Norman Saak, Jens Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Computational Methods in Systems and Control Theory Germany Technische Universität Chemnitz Faculty of Mathematics Germany

Differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time-varying systems and many more. Here, we will focus on differential Riccati equations (DRE). Solving such matrix-valued ordinary differential equations (ODE) is a highly time consuming process. We present a Parareal based algorithm applied to Rosenbrock methods for the solution of the matrix-valued differential Riccati equations. Considering problems of moderate size, direct matrix equation solvers for the solution of the algebraic Lyapunov equations arising inside the time intgration methods are used. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Ordinary differential equations

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On stability, passivity and reciprocity preservation of ESVDMOR

On stability, passivity and reciprocity preservation of ESVD...

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Workshop on Model Reduction for Circuit Simulation

作者： Benner, Peter Schneider, André 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 09107 Chemnitz Germany

ISBN: (纸本)9789400700888

The reduction of parasitic linear subcircuits is one of many issues in model order reduction (MOR) for VLSI design. This issue is well explored, but recently the incorporation of subcircuits from different modelling sources into the circuit model has led to new structural aspects: so far, the number of elements in the subcircuits was significantly larger than the number of connections to the whole circuit, the so called pins or terminals. This assumption is no longer valid in all cases such that the simulation of these circuits or rather the reduction of the model requires new methods. In Feldmann and Liu (ICCAD '04: Proceedings of the 2004 IEEE/ACM International Conference on Computer-Aided Design, 88-92, 2004), Liu et al. (ICCAD '05: Proceedings of the 2005 IEEE/ACM International Conference on Computer-Aided Design, 821-826, 2004;Integr. VLSI J. 41(2):210-218, 2008) the extended singular value decomposition based model order reduction (ESVDMOR) algorithm is introduced as a way to handle this kind of circuits with a massive number of terminals. Unfortunately, the ESVDMOR approach has some drawbacks because it uses the SVD for matrix factorizations. In Benner (Mathematics in Industry 14, 2009), Schneider (Matrix decomposition based approaches for model order reduction of linear systems with a large number of terminals, 2008) the truncated SVD (TSVD) as an alternative to the SVD within the ESVDMOR is introduced. In this paper we show that ESVDMOR as well as the modified approach is stability, passivity, and reciprocity preserving under reasonable assumptions. © 2011 Springer Science+Business Media B.V.

关键词： Singular value decomposition

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PD controllers to solve single-input, index-1 DAE based LQR problems

PD controllers to solve single-input, index-1 DAE based LQR ...

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European control Conference (ECC)

作者： Chayan Bhawal Jan Heiland Peter Benner Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg

ISBN: (数字)9783907144022

ISBN: (纸本)9781728188133

In this paper, we present a method to compute the optimal trajectories of a linear quadratic regulator (LQR) problem corresponding to a single input, index-1 constant coefficient DAE system. Further, we also show that using a suitably designed proportional-derivative (PD) state-feedback controller one can force the trajectories of the corresponding DAE system to its optimal trajectories. Unlike the results present in the literature, the results in this paper are applicable for any arbitrary initial condition of the system.

关键词： Trajectory Indexes Matrix converters PD control Optimal control Regulators controllability

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Residual minimizing shift parameters for the low-rank ADI iteration

arXiv

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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

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