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Synchronization and aggregation of nonlinear power systems with consideration of bus network structures

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arXiv 2018年

作者： Mlinarić, Petar Ishizaki, Takayuki Chakrabortty, Aranya Grundel, Sara Benner, Peter Imura, Jun-Ichi Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany Department of Systems and Control Engineering School of Engineering Tokyo Institute of Technology 2-12-1 Meguro Tokyo Japan Electrical & Computer Engineering North Carolina State University RaleighNC27695

We study nonlinear power systems consisting of generators, generator buses, and non-generator buses. First, looking at a generator and its bus' variables jointly, we introduce a synchronization concept for a pair of such joint generators and buses. We show that this concept is related to graph symmetry. Next, we extend, in two ways, the synchronization from a pair to a partition of all generators in the networks and show that they are related to either graph symmetry or equitable partitions. Finally, we show how an exact reduced model can be obtained by aggregating the generators and associated buses in the network when the original system is synchronized with respect to a partition, provided that the initial condition respects the partition. Additionally, the aggregation-based reduced model is again a power *** Codes 93A15, 93C10, 94C15 Copyright © 2018, The Authors. All rights reserved.

关键词： Synchronization

An inexact Newton-Krylov method for stochastic eigenvalue problems

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arXiv 2017年

作者： Benner, Peter Onwunta, Akwum Stoll, Martin Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstrasse 1 Magdeburg39106 Germany Numerical Linear Algebra for Dynamical Systems Group Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstrasse 1 Magdeburg39106 Germany Technische Universität Chemnitz Faculty of Mathematics Professorship Scientific Computing Chemnitz09107 Germany

This paper aims at the efficient numerical solution of stochastic eigenvalue problems. Such problems often lead to prohibitively high dimensional systems with tensor product structure when discretized with the stochastic Galerkin method. Here, we exploit this inherent tensor product structure to develop a globalized low-rank inexact Newton method with which we tackle the stochastic eigenproblem. We illustrate the effectiveness of our solver with numerical experiments. Copyright © 2017, The Authors. All rights reserved.

关键词： Galerkin methods

A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solids

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arXiv 2024年

作者： Khorrami, Mohammad S. Goyal, Pawan Mianroodi, Jaber R. Svendsen, Bob Benner, Peter Raabe, Dierk Microstructure Physics and Alloy Design Max-Planck Institute for Sustainable Materials Düsseldorf Germany Computational Methods in Systems and Control Theory Max-Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Material Mechanics RWTH Aachen University Aachen Germany

The purpose of the current work is the development of a so-called physics-encoded Fourier neural operator (PeFNO) for surrogate modeling of the quasi-static equilibrium stress field in solids. Rather than accounting for constraints from physics in the loss function as done in the (now standard) physics-informed approach, the physics-encoded approach incorporates or"encodes" such constraints directly into the network or operator architecture. As a result, in contrast to the physics-informed approach in which only training is physically constrained, both training and output are physically constrained in the physics-encoded approach. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a"proof-of-concept" example application of the proposed PeFNO, a heterogeneous polycrystalline material consisting of isotropic elastic grains subject to uniaxial extension is considered. Stress field data for training are obtained from the numerical solution of a corresponding boundary-value problem for quasi-static mechanical equilibrium. This data is also employed to train an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO) for comparison. As confirmed by this comparison and as expected on the basis of their differences, the output of the trained PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the trained PgFNO or the trained PiFNO. Copyright © 2024, The Authors. All rights reserved.

关键词： Polycrystalline materials

INTERPOLATORY H2-OPTIMALITY CONDITIONS FOR STRUCTURED LINEAR TIME-INVARIANT systems

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arXiv 2023年

作者： Mlinarić, Petar Benner, Peter Gugercin, Serkan Department of Mathematics Virginia Tech BlacksburgVA24061 United States Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

Interpolatory necessary optimality conditions for H2-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on L2-optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for H2-optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. Under certain diagonalizability assumptions, we show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling. Copyright © 2023, The Authors. All rights reserved.

关键词： Linear time-invariant system

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

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

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arXiv 2017年

作者： Benner, Peter Dolgov, Sergey Onwunta, Akwum Stoll, Martin University of Bath Department of Mathematical Sciences Claverton Down BathBA2 7AY United Kingdom Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages. Copyright © 2017, The Authors. All rights reserved.

关键词： Stochastic systems

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

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arXiv 2022年

作者： Khorrami, Mohammad S. Mianroodi, Jaber Rezaei Siboni, Nima H. Goyal, Pawan Svendsen, Bob Benner, Peter Raabe, Dierk Microstructure Physics and Alloy Design Max-Planck-Institut für Eisenforschung Düsseldorf Germany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Material Mechanics RWTH Aachen University Aachen Germany

The purpose of this work is the development of an artificial neural network (ANN) for surrogate modeling of the mechanical response of viscoplastic grain microstructures. To this end, a U-Net-based convolutional neural network (CNN) is trained to account for the history dependence of the material behavior. The training data take the form of numerical simulation results for the von Mises stress field under quasi-static tensile loading. The trained CNN (tCNN) can accurately reproduce both the average response as well as the local von Mises stress field. The tCNN calculates the von Mises stress field of grain microstructures not included in the training dataset about 500 times faster than its calculation based on the numerical solution with a spectral solver of the corresponding initial-boundary-value problem. The tCNN is also successfully applied to other types of microstructure morphologies (e.g., matrix-inclusion type topologies) and loading levels not contained in the training dataset. Copyright © 2022, The Authors. All rights reserved.

关键词： Deep learning

Fast a posteriori state error estimation for reliable frequency sweeping in microwave circuits via the reduced-basis method

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arXiv 2021年

作者： de la Rubia, Valentín Chellappa, Sridhar Feng, Lihong Benner, Peter Departamento de Matemática Aplicada a las TIC ETSI de Telecomunicación Universidad Politécnica de Madrid Madrid28040 Spain Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany

We develop a compact, reliable model order reduction approach for fast frequency sweeps in microwave circuits by means of the reduced-basis method. Contrary to what has been previously done, special emphasis is placed on certifying the accuracy of the reduced-order model with respect to the original full-order model in an effective and efficient way. Previous works on model order reduction accuracy certification rely on costly a posteriori error estimators, which typically require expensive inf-sup constant evaluations of the underlying full-order model. This scenario is often too time-consuming and unaffordable in electromagnetic applications. As a result, less expensive and heuristic error estimators are commonly used instead. Very often, one is interested in knowing about the full state vector, instead of just some output quantities derived from the full state. Therefore, error estimators for the full state vector become relevant. In this work, we detail the frequency behavior of both the electric field and the state error when an approximation to the electric field solution is carried out. Both field quantities share the same frequency behavior. Based on this observation, we focus on the efficient estimation of the electric field state error and propose a fast evaluation of the reduced-order model state error in the frequency band of analysis, minimizing the number of full-order model evaluations. This methodology is of paramount importance to carry out a reliable fast frequency sweep in microwave circuits. Finally, real-life applications will illustrate the capabilities and efficiency of the proposed approach. © 2021, CC BY.

关键词： Computer aided design