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Revisiting IRKA: Connections with pole placement and backward stability

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arXiv 2019年

作者： Beattie, C. Drmač, Z. Gugercin, S. Department of Mathematics Virginia Tech BlacksburgVA24061 United States Faculty of Science Department of Mathematics University of Zagreb Zagreb10000 Croatia Department of Mathematics Division of Computational Modeling and Data Analytics Virginia Tech BlacksburgVA24061 United States

The iterative rational Krylov algorithm (IRKA) is a popular approach for producing locally optimal reduced-order H2-approximations to linear time-invariant (LTI) dynamical systems. Overall, IRKA has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, IRKA has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear dynamical systems. Convergence of the basic IRKA iteration is generally observed to be rapid | but not always;and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic IRKA iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the IRKA iteration that are revealed when the iteration is framed with respect to a primitive *** find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability. Copyright © 2019, The Authors. All rights reserved.

关键词： Iterative methods

Pseudospectra of Loewner Matrix Pencils

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arXiv 2019年

作者： Embree, Mark Cosmin Ioniţă, A. Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States MathWorks Inc. NatickMA01760 United States

Loewner matrix pencils play a central role in the system realization theory of Mayo and Antoulas, an important development in data-driven modeling. The eigenvalues of these pencils reveal system poles. How robust are the poles recovered via Loewner realization? With several simple examples, we show how pseudospectra of Loewner pencils can be used to investigate the influence of interpolation point location and partitioning on pole stability, the transient behavior of the realized system, and the effect of noisy measurement data. We include an algorithm to efficiently compute such pseudospectra by exploiting Loewner structure. Copyright © 2019, The Authors. All rights reserved.

关键词： Poles

Structure-preserving interpolation of bilinear control systems

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arXiv 2020年

作者： Benner, Peter Gugercin, Serkan Werner, Steffen W.R. Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Otto von Guericke University Faculty of Mathematics Universitätsplatz 2 Magdeburg39106 Germany Department of Mathematics and Computational Modeling Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

In this paper, we extend the structure-preserving interpolatory model reduction framework, originally developed for linear systems, to structured bilinear control systems. Specifically, we give explicit construction formulae for the model reduction bases to satisfy different types of interpolation conditions. First, we establish the analysis for transfer function interpolation for single-input single-output structured bilinear systems. Then, we extend these results to the case of multi-input multi-output structured bilinear systems by matrix interpolation. The effectiveness of our structure-preserving approach is illustrated by means of various numerical examples. Copyright © 2020, The Authors. All rights reserved.

关键词： Interpolation

Structure-preserving interpolation for model reduction of parametric bilinear systems

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arXiv 2020年

作者： Benner, Peter Gugercin, Serkan Werner, Steffen W.R. Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Otto von Guericke University Faculty of Mathematics Universitätsplatz 2 Magdeburg39106 Germany Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

In this paper, we present an interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis. Copyright © 2020, The Authors. All rights reserved.

关键词： Interpolation

Sampled limited memory methods for massive linear inverse problems

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arXiv 2019年

作者： Chung, Julianne Chung, Matthias Tanner Slagel, J. Tenorio, Luis Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Department of Mathematics Virginia Tech BlacksburgVA United States Department of Applied Mathematics and Statistics Colorado School of Mines GoldenCO United States

In many modern imaging applications the desire to reconstruct high resolution images, coupled with the abundance of data from acquisition using ultra-fast detectors, have led to new challenges in image reconstruction. A main challenge is that the resulting linear inverse problems are massive. The size of the forward model matrix exceeds the storage capabilities of computer memory, or the observational dataset is enormous and not available all at once. Row-action methods that iterate over samples of rows can be used to approximate the solution while avoiding memory and data availability constraints. However, their overall convergence can be slow. In this paper, we introduce a sampled limited memory row-action method for linear least squares problems, where an approximation of the global curvature of the underlying least squares problem is used to speed up the initial convergence and to improve the accuracy of iterates. We show that this limited memory method is a generalization of the damped block Kaczmarz method, and we prove linear convergence of the expectation of the iterates and of the error norm up to a convergence horizon. Numerical experiments demonstrate the benefits of these sampled limited memory row-action methods for massive 2D and 3D inverse problems in tomography applications. Copyright © 2019, The Authors. All rights reserved.

关键词： Tomography

Comparison and Evaluation of Methods for a Predict+Optimize Problem in Renewable Energy

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arXiv 2022年

作者： Bergmeir, Christoph de Nijs, Frits Sriramulu, Abishek Abolghasemi, Mahdi Bean, Richard Betts, John Bui, Quang Dinh, Nam Trong Einecke, Nils Esmaeilbeigi, Rasul Ferraro, Scott Galketiya, Priya Genov, Evgenii Glasgow, Robert Godahewa, Rakshitha Kang, Yanfei Limmer, Steffen Magdalena, Luis Montero-Manso, Pablo Peralta, Daniel Kumar, Yogesh Pipada Sunil Rosales-Pérez, Alejandro Ruddick, Julian Stratigakos, Akylas Stuckey, Peter Tack, Guido Triguero, Isaac Yuan, Rui Department of Data Science and Artificial Intelligence Monash University Melbourne Australia School of Mathematics and Physics University of Queensland Brisbane Australia Centre for Energy Data Innovation School of Information Technology and Electrical Engineering University of Queensland Brisbane Australia School of Electrical and Electronics Engineering University of Adelaide Adelaide Australia Honda Research Institute Europe GmbH Offenbach am Main63073 Germany School of Information Technology Deakin University Melbourne Australia Building and Property Division Monash University Melbourne Australia EVERGi MOBI Vrije Universiteit Brussel Brussels Belgium School of Economics and Management Beihang University Beijing China E.T.S. Ingenieros Informáticos Universidad Politécnica de Madrid Madrid28660 Spain Disciple of Business Analytics University of Sydney Australia IDLab Department of Information Technology Ghent University - imec Belgium Department of Computer Science Centro de Investigación en Matem áticas Monterrey66629 Mexico Mines Paris PSL University Sophia Antipolis06904 France DaSCI Andalusian Institute in Data Science and Computational Intelligence Granada Spain Department of Computer Science and Artificial Intelligence University of Granada Granada Spain

Algorithms that involve both forecasting and optimization are at the core of solutions to many difficult real-world problems, such as in supply chains (inventory optimization), traffic, and in the transition towards carbon-free energy generation in battery/load/production scheduling in sustainable energy systems. Typically, in these scenarios we want to solve an optimization problem that depends on unknown future values, which therefore need to be forecast. As both forecasting and optimization are difficult problems in their own right, relatively few research has been done in this area. This paper presents the findings of the "IEEE-CIS Technical Challenge on Predict+Optimize for Renewable Energy Scheduling," held in 2021. We present a comparison and evaluation of the seven highest-ranked solutions in the competition, to provide researchers with a benchmark problem and to establish the state of the art for this benchmark, with the aim to foster and facilitate research in this area. The competition used data from the Monash Microgrid, as well as weather data and energy market data. It then focused on two main challenges: forecasting renewable energy production and demand, and obtaining an optimal schedule for the activities (lectures) and on-site batteries that lead to the lowest cost of energy. The most accurate forecasts were obtained by gradient-boosted tree and random forest models, and optimization was mostly performed using mixed integer linear and quadratic programming. The winning method predicted different scenarios and optimized over all scenarios jointly using a sample average approximation method. Copyright © 2022, The Authors. All rights reserved.

关键词： Forecasting

Assessing Trustworthy AI in Times of COVID-19: Deep Learning for Predicting a Multiregional Score Conveying the Degree of Lung Compromise in COVID-19 Patients

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IEEE Transactions on Technology and Society

IEEE Transactions on Technology and Society 2022年第4期3卷 272-289页

作者： Allahabadi, Himanshi Amann, Julia Balot, Isabelle Beretta, Andrea Binkley, Charles Bozenhard, Jonas Bruneault, Frederick Brusseau, James Candemir, Sema Cappellini, Luca Alessandro Chakraborty, Subrata Cherciu, Nicoleta Cociancig, Christina Coffee, Megan Ek, Irene Espinosa-Leal, Leonardo Farina, Davide Fieux-Castagnet, Genevieve Frauenfelder, Thomas Gallucci, Alessio Giuliani, Guya Golda, Adam Van Halem, Irmhild Hildt, Elisabeth Holm, Sune Kararigas, Georgios Krier, Sebastien A. Kuhne, Ulrich Lizzi, Francesca Madai, Vince I. Markus, Aniek F. Masis, Serg Mathez, Emilie Wiinblad Mureddu, Francesco Neri, Emanuele Osika, Walter Ozols, Matiss Panigutti, Cecilia Parent, Brendan Pratesi, Francesca Moreno-Sanchez, Pedro A. Sartor, Giovanni Savardi, Mattia Signoroni, Alberto Sormunen, Hanna-Maria Spezzatti, Andy Srivastava, Adarsh Stephansen, Annette F. Theng, Lau Bee Tithi, Jesmin Jahan Tuominen, Jarno Umbrello, Steven Vaccher, Filippo Vetter, Dennis Westerlund, Magnus Wurth, Renee Zicari, Roberto V. Ey Netherlands Enterprise Intelligence Department Amsterdam1083 HP Netherlands Eth Zurich Health Ethics and Policy Lab Department of Health Sciences and Technology Zürich8092 Switzerland Center for Diplomatic and Strategic Studies Postgraduate Studies in Diplomacy and International Relations Paris75015 France Pisa56124 Italy Hackensack Meridian Health Bioethics Center EdisonNJ08820 United States University of Oxford Faculty of Philosophy OxfordOX2 6GG United Kingdom Collège André- Laurendeau Philosophie Department MontrealQCH8N 2J4 Canada Université du Québec À Montréal École des Médias MontrealQCH2L 2C4 Canada Pace University Philosophy Department New YorkNY10038 United States The Ohio State University Wexner Medical Center Department of Radiology ColumbusOH43210 United States Humanitas Research Hospital Department of Radiology Milan20089 Italy Humanitas University Department of Biomedical Sciences Milan20089 Italy University of New England Faculty of Science Agriculture Business and Law ArmidaleNSW2351 Australia University of Technology Sydney Faculty of Engineering and Information Technology SydneyNSW2007 Australia Scuola Superiore Sant'Anna European Centre of Excellence on the Regulation of Robotics and Ai Pisa56127 Italy University of Bremen Group of Computer Architecture Bremen28359 Germany New York University Grossman School of Medicine Division of Infectious Diseases and Immunology Department of Medicine New YorkNY10016 United States Digital Institute Ai Research Section Stockholm16731 Sweden Arcada University of Applied Sciences Department of Business Management and Analytics Helsinki00550 Finland University of Brescia Radiological Sciences and Public Health Department of Medical and Surgical Specialties Brescia25121 Italy Sncf Reseau Sa Ethique Groupe La Plaine93418 France Institute of Diagnostic and Interventional Radiology University Hospital Zurich Zürich8091 Switzerland Eindhoven University of Tech

This article's main contributions are twofold: 1) to demonstrate how to apply the general European Union's High-Level Expert Group's (EU HLEG) guidelines for trustworthy AI in practice for the domain of healthcare and 2) to investigate the research question of what does 'trustworthy AI' mean at the time of the COVID-19 pandemic. To this end, we present the results of a post-hoc self-assessment to evaluate the trustworthiness of an AI system for predicting a multiregional score conveying the degree of lung compromise in COVID-19 patients, developed and verified by an interdisciplinary team with members from academia, public hospitals, and industry in time of pandemic. The AI system aims to help radiologists to estimate and communicate the severity of damage in a patient's lung from Chest X-rays. It has been experimentally deployed in the radiology department of the ASST Spedali Civili clinic in Brescia, Italy, since December 2020 during pandemic time. The methodology we have applied for our post-hoc assessment, called Z-Inspection®, uses sociotechnical scenarios to identify ethical, technical, and domain-specific issues in the use of the AI system in the context of the pandemic. © 2020 IEEE.

关键词： COVID-19

Flexible krylov methods for pregularization

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arXiv 2018年

作者： Chung, Julianne Gazzola, Silvia Department of Mathematics Computational Modeling Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States

In this paper we develop flexible Krylov methods for efficiently computing regular- ized solutions to large-scale linear inverse problems with an 2fit-to-data term and an .p penalization term, for p ≥ 1. First we approximate the p-norm penalization term as a sequence of 2-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion, and then we exploit flexible preconditioning tech- niques to efficiently incorporate the weight updates. To handle general (non-square) .p-regularized least-squares problems, we introduce a flexible Golub-Kahan approach and exploit it within a Krylov-Tikhonov hybrid framework. The key benefits of our approach compared to existing optimization methods for .p regularization are that effi- cient projection methods replace inner-outer schemes and that expensive regularization parameter selection techniques can be avoided. Theoretical insights are provided, and numerical results from image deblurring and tomographic reconstruction illustrate the benefits of this approach, compared to well-established methods. Furthermore, we show that our approach for p = 1 can be used to efficiently compute solutions that are sparse with respect to some transformations. Copyright © 2018, The Authors. All rights reserved.

关键词： Image enhancement

Polynomial preconditioned arnoldi

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arXiv 2018年

作者： EMBREE, MARK LOE, JENNIFER A. MORGAN, RONALD B. Department of Mathematics and Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA24061 United States Department of Mathematics Baylor University WacoTX76798-7328 United States

Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization;for difficult problems, it can also reduce the number of matrix-vector products. Parallel computations can particularly benefit from the reduction of communication-intensive operations. The GMRES algorithm provides a simple and effective way of generating the preconditioning polynomial. For some problems high degree polynomials are especially effective, but they can lead to stability problems that must be mitigated. A two-level "double polynomial preconditioning" strategy provides an effective way to generate high-degree preconditioners.65F15, 15A18 Copyright © 2018, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions