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Adaptively restarted block Krylov subspace methods with low-synchronization skeletons

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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

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

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

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

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

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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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GR decompositions and their relations to Cholesky-like factorizations

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arXiv 2020年

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

For a given matrix, we are interested in computing GR decompositions A = GR, where G is an isometry with respect to given scalar products. The orthogonal QR decomposition is the representative for the Euclidian scalar product. For a signature matrix, a respective factorization is given as the hyperbolic QR decomposition. Considering a skew-symmetric matrix leads to the symplectic QR decomposition. The standard approach for computing GR decompositions is based on the successive elimination of subdiagonal matrix entries. For the hyperbolic and symplectic case, this approach does in general not lead to a satisfying numerical accuracy. An alternative approach computes the QR decomposition via a Cholesky factorization, but also has bad stability. It is improved by repeating the procedure a second time. In the same way, the hyperbolic and the symplectic QR decomposition are related to the LDLT and a skew-symmetric Cholesky-like factorization. We show that methods exploiting this connection can provide better numerical stability than elimination-based *** Codes 65F99 Copyright © 2020, The Authors. All rights reserved.

关键词： Factorization

Fast and reliable reduced-order models for cardiac electrophysiology

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GAMM Mitteilungen 2023年第3-4期46卷 e202370014-e202370014页

作者： Chellappa, Sridhar Cansız, Barış Feng, Lihong Benner, Peter Kaliske, Michael Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Institute for Structural Analysis Technische Universität Dresden Dresden Germany

Mathematical models of the human heart increasingly play a vital role in understanding the working mechanisms of the heart, both under healthy functioning and during disease. The ultimate aim is to aid medical practitioners diagnose and treat the many ailments affecting the heart. Towards this, modeling cardiac electrophysiology is crucial as the heart's electrical activity underlies the contraction mechanism and the resulting pumping action. Apart from modeling attempts, the pursuit of efficient, reliable, and fast solution algorithms has been of great importance in this context. The governing equations and the constitutive laws describing the electrical activity in the heart are coupled, nonlinear, and involve a fast moving wave front, which is generally solved by the finite element method. The numerical treatment of this complex system as part of a virtual heart model is challenging due to the necessity of fine spatial and temporal resolution of the domain. Therefore, efficient surrogate models are needed to predict the electrical activity in the heart under varying parameters and inputs much faster than the finely resolved models. In this work, we develop an adaptive, projection-based reduced-order surrogate model for cardiac electrophysiology. We introduce an a posteriori error estimator that can accurately and efficiently quantify the accuracy of the surrogate model. Using the error estimator, we systematically update our surrogate model through a greedy search of the parameter space. Furthermore, using the error estimator, the parameter search space is dynamically updated such that the most relevant samples get chosen at every iteration. The proposed adaptive surrogate model is tested on three benchmark models to illustrate its efficiency, accuracy, and ability of generalization. © 2024 The Authors. GAMM - Mitteilungen published by Wiley-VCH GmbH.

关键词： Heart

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

Active learning enhanced deep-learning surrogate model for fast MEMS design with high-dimensional design parameter spaces

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Active learning enhanced deep-learning surrogate model for f...

IEEE International Conference on Nano/Micro Engineered and Molecular systems (NEMS)

作者： Chenzi Wang Lihong Feng Wenshuai Lu Wei Bian Zheng You Peter Benner Department of Precision Instrument Tsinghua University Beijing China Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany

ISBN: (数字)9798350359831

ISBN: (纸本)9798350359848

MEMS devices suffer from errors in manufacturing processes such as lithography, etching, thinning, etc. These errors cause non-negligible deviations between the actual device performance and the ideal performance obtained from finite element method (FEM) simulation in the design stage. Often, parametric sweeps via FEM simulations at heuristically selected geometrical and material parameter samples are implemented, in order to analyze the corresponding device performance deviations. This analysis is empirical and often fails due to large number of possible design parameter variations caused by manufacturing. Performing parametric sweeps for all possible parameter instances would result in an exponential increase in FEM simulations. In this paper, we propose a novel active learning enhanced deep-learning surrogate model, which replaces the FEM simulation and faithfully predicts the device performance deviations under numerous parameter variances. This provides a powerful tool for predicting device yield at the design stage. Furthermore, the proposed active learning technique could be applied to efficient training of deep learning models for other MEMS design problems.

关键词： Micromechanical devices Performance evaluation Training Nanoelectromechanical systems Training data Predictive models Data models

Efficient structure-preserving support tensor train machine

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2023年第1期24卷 98-119页

作者： Kirandeep Kour Peter Benner Sergey Dolgov Martin Stoll Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Department of Mathematical Sciences University of Bath Bath UK Faculty of Mathematics Technische Universität Chemnitz Chemnitz Germany

An increasing amount of the collected data are high-dimensional multi-way arrays (tensors), and it is crucial for efficient learning algorithms to exploit this tensorial structure as much as possible. The ever present curse of dimensionality for high dimensional data and the loss of structure when vectorizing the data motivates the use of tailored low-rank tensor classification methods. In the presence of small amounts of training data, kernel methods offer an attractive choice as they provide the possibility for a nonlinear decision boundary. We develop the Tensor Train Multi-way Multi-level Kernel (TT-MMK), which combines the simplicity of the Canonical Polyadic decomposition, the classification power of the Dual Structure-preserving Support Vector Machine, and the reliability of the Tensor Train (TT) approximation. We show by experiments that the TT-MMK method is usually more reliable computationally, less sensitive to tuning parameters, and gives higher prediction accuracy in the SVM classification when benchmarked against other state-of-the-art techniques.

关键词： tensor decomposition support vector machine kernel approximation high-dimensional data classification