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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

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

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

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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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

Machine learning for material characterization with an application for predicting mechanical properties

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GAMM Mitteilungen 2021年第1期44卷 e202100003-e202100003页

作者： Stoll, Anke Benner, Peter Data-Mining and Machine Learning Fraunhofer Institute for Machine Tools and Forming Technology IWU Chemnitz Germany Computational Methods in Systems and Control Theory The Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany

Currently, the growth of material data from experiments and simulations is expanding beyond processable amounts. This makes the development of new data-driven methods for the discovery of patterns among multiple lengthscales and time-scales and structure-property relationships essential. These data-driven approaches show enormous promise within materials science. The following review covers machine learning (ML) applications for metallic material characterization. Many parameters associated with the processing and the structure of materials affect the properties and the performance of manufactured components. Thus, this study is an attempt to investigate the usefulness of ML methods for material property prediction. Material characteristics such as strength, toughness, hardness, brittleness, or ductility are relevant to categorize a material or component according to their quality. In industry, material tests like tensile tests, compression tests, or creep tests are often time consuming and expensive to perform. Therefore, the application of ML approaches is considered helpful for an easier generation of material property information. This study also gives an application of ML methods on small punch test (SPT) data for the determination of the property ultimate tensile strength for various materials. A strong correlation between SPT data and tensile test data was found which ultimately allows to replace more costly tests by simple and fast tests in combination with ML. © 2021 The Authors. GAMM - Mitteilungen published by Wiley-VCH GmbH.

关键词： Machine learning

Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: A state of the art survey

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GAMM Mitteilungen 2013年第1期36卷 32-52页

作者： Benner, Peter Saak, Jens Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Sandtorstr. 1 D-39106 Magdeburg Germany Chemnitz University of Technology Department of Mathematics Mathematics in Industry and Technology D-09107 Chemnitz Germany

Efficient numerical algorithms for the solution of large and sparse matrix Riccati and Lyapunov equations based on the low rank alternating directions implicit (ADI) iteration have become available around the year 2000. Over the decade that passed since then, additional methods based on extended and rational Krylov subspace projection have entered the field and proved to be competitive alternatives. In this survey we sketch both types of methods and discuss their advantages and drawbacks. We focus on the continuous time case here, but corresponding results for discrete time problems can for most results be found in the available literature. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Surveys