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FREQUENCY-BASED REDUCED MODELS FROM PURELY TIME-DOMAIN data VIA data INFORMATIVITY

SIAM Journal on Scientific Computing 2025年第2期47卷 A1225-A1250页

作者： Ackermann, Michael S. Gugercin, Serkan Department of Mathematics Virginia Tech BlacksburgVA24061 United States Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

Frequency-based methods have been successfully employed in creating high-fidelity data-driven reduced order models (DDROMs) for linear dynamical systems. These methods require access to values (and sometimes derivatives) of the frequency-response function (transfer function) in the complex plane. These frequency domain values can at times be costly or difficult to obtain (especially if the method of choice requires resampling);instead one may have access to only time-domain input-output data. The data informativity approach to moment matching provides a powerful new framework for recovering the required frequency data from a single time-domain trajectory. In this work, we analyze and extend upon this framework, resulting in significantly improved conditioning of the associated linear systems, an error indicator, and the removal of an assumption that the system order is known. This analysis leads to a robust algorithm for recovering frequency information from time-domain data, suitable for large-scale systems. We demonstrate the effectiveness of our algorithm by forming frequency-based DDROMs from time-domain data of several dynamical systems. © 2025 Society for Industrial and Applied mathematics.

关键词： data assimilation

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IRKA Is a Riemannian Gradient Descent Method

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IEEE Transactions on Automatic Control 2025年第5期70卷 2979-2991页

作者： Mlinaric, Petar Beattie, Christopher A. Drmac, Zlatko Gugercin, Serkan Virginia Tech Department of Mathematics BlacksburgVA24061 United States University of Zagreb Faculty of Natural Sciences Zagreb10000 Croatia Virginia Tech Department of Mathematics and Division of Computational Modeling and Data Analytics Academy of Data Science BlacksburgVA24061 United States

The iterative rational Krylov algorithm (IRKA) is a commonly used fixed point iteration developed to minimize the H2 model order reduction error. In this work, the IRKA is recast as a Riemannian gradient descent method with a fixed step size over the manifold of rational functions having fixed degree. This interpretation motivates the development of a Riemannian gradient descent method utilizing as a natural extension variable step size and line search. Comparisons made between the IRKA and this extension on a few examples demonstrate significant benefits. © 1963-2012 IEEE.

关键词： Rational functions

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Interpolatory model reduction of dynamical systems with root mean squared error 11

Interpolatory model reduction of dynamical systems with root...

引用

11th Vienna International Conference on Mathematical Modelling, MATHMOD 2025

作者： Reiter, Sean Werner, Steffen W.R. Department of Mathematics Virginia Tech BlacksburgVA24061 United States Department of Mathematics and Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

The root mean squared error is an important measure used in a variety of applications like structural dynamics and acoustics to model averaged deviations from standard behavior. For large-scale systems, simulations of this quantity quickly become computationally prohibitive. Model order reduction techniques resolve this issue via the construction of surrogate models that emulate the root mean squared error measure using an intermediate linear system. However, classical approaches require a large number of system outputs, which is disadvantageous in the design of reduced-order models. In this work, we consider directly the root mean squared error as the quantity of interest using the concept of quadratic-output models and propose several new model reduction techniques for the construction of appropriate surrogates. Numerical tests are performed on a model of a plate with tuned vibration absorbers. Copyright © 2025 The Authors.

关键词： Mean square error

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Interpolatory model reduction of dynamical systems with root mean squared error

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IFAC-PapersOnLine 2025年第1期59卷 385-390页

作者： Sean Reiter Steffen W.R. Werner Department of Mathematics Virginia Tech Blacksburg VA 24061 USA Department of Mathematics and Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech Blacksburg VA 24061 USA

关键词： model order reduction root mean squared error rational interpolation quadratic terms nonlinear systems

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TIME-DOMAIN ITERATIVE RATIONAL KRYLOV METHOD

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SIAM Journal on Scientific Computing 2025年第3期47卷 A1628-A1651页

作者： Ackermann, Michael S. Gugercin, Serkan Department of Mathematics Virginia Tech Blacksburg 24061 VA United States Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech Blacksburg 24061 VA United States

The Realization Independent Iterative Rational Krylov Algorithm (TF-IRKA) is a frequency-based data-driven reduced order modeling (DDROM) method that constructs H2 optimal DDROMs. However, as the H2 optimal approximation theory dictates, TF-IRKA requires repeated sampling of frequency data, that is, values of the system transfer function and its derivative, outside the unit circle. This repeated evaluation of frequency data requires repeated full model computations and may not be feasible. The data informativity framework for moment matching provides a method for obtaining such frequency data from a single time-domain simulation. However, this framework usually requires solving linear systems with prohibitively ill-conditioned matrices, especially when recovering frequency data from off the unit circle as required for optimality. In this paper, building upon our previous work with the data-informativity framework for moment matching, we provide a formula for the nonzero extreme eigenvalues of a symmetric rank-1 perturbation to an orthogonal projection, which then leads to an optimal scaling of the aforementioned linear systems. We also establish connections between the underlying dynamical system and conditioning of these linear systems. This analysis then leads to our algorithmic development, time-domain IRKA, which allows us to implement a time-domain variant of TF-IRKA, constructing H2 optimal DDROMs from a single time-domain simulation without requiring repeated frequency data evaluations. The numerical examples illustrate the effectiveness of the proposed algorithm. © 2025 Society for Industrial and Applied mathematics.

关键词： data informativity eigenvalues optimal data-driven modeling rational interpolation transfer function

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KLAP: KYP LEMMA BASED LOW-RANK APPROXIMATION FOR H2-OPTIMAL PASSIVATION

arXiv

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arXiv 2025年

作者： Nicodemus, Jonas Voigt, Matthias Gugercin, Serkan Unger, Benjamin University of Stuttgart Universitätsstr. 32 Stuttgart70569 Germany Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States Faculty of Mathematics and Computer Science UniDistance Suisse Schinerstr. 18 Brig3900 Switzerland

We present a novel passivity enforcement (passivation) method, called KLAP, for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma and the closely related Lur’e equations. The passivation problem in our framework corresponds to finding a perturbation to a given non-passive system that renders the system passive while minimizing the H2 or frequency-weighted H2 distance between the original non-passive and the resulting passive system. We show that this problem can be formulated as an unconstrained optimization problem whose objective function can be differentiated efficiently even in large-scale settings. We show that any minimizer of the unconstrained problem yields the same passive system. Furthermore, we prove that, in the absence of a feedthrough term, every local minimizer is also a global minimizer. For cases involving a non-trivial feedthrough term, we analyze global minimizers in relation to the extremal solutions of the Lur’e equations, which can serve as tools for identifying local minima. To solve the resulting numerical optimization problem efficiently, we propose an initialization strategy based on modifying the feedthrough term and a restart strategy when it is likely that the optimization has converged to a local minimum. Numerical examples illustrate the effectiveness of the proposed *** Codes 37J06, 37M99, 65P10, 93A30, 93B30, 93C05 © 2025, CC BY.

关键词： Convergence of numerical methods

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Biomedical data and AI

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Science China Life Sciences 2025年第5期68卷 1536-1540页

作者： Hao Xu Shibo Zhou Zefeng Zhu Vincenzo Vitelli Liangyi Chen Ziwei Dai Ning Yang Luhua Lai Shengyong Yang Sergey Ovchinnikov Zhuoran Qiao Sirui Liu Chen Song Jianfeng Pei Han Wen Jianfeng Feng Yaoyao Zhang Zhengwei Xie Yang-Yu Liu Zhiyuan Li Fulai Jin Hao Li Mohammad Lotfollahi Xuegong Zhang Ge Yang Shihua Zhang Ge Gao Pulin Li Qi Liu Jing-Dong Jackie Han Peking-Tsinghua Center for Life Sciences (CLS) Academy for Advanced Interdisciplinary StudiesPeking University Center for Quantitative Biology (CQB) Academy for Advanced Interdisciplinary StudiesPeking University Peking University-Tsinghua University-National Institute of Biological Sciences Joint Graduate Program Academy for Advanced Interdisciplinary StudiesPeking University Department of Physics University of Chicago School of Life Sciences Southern University of Science and Technology Peking University Chengdu Academy for Advanced Interdisciplinary Biotechnologies College of Chemistry and Molecular Engineering Peking University Department of Biotherapy Cancer Center and State Key Laboratory of BiotherapyWest China HospitalSichuan University Department of Biology Massachusetts Institute of Technology Lambic Therapeutics Inc. Changping Laboratory Al for Science Institute Key Laboratory of Computational Neuroscience and Brain-Inspired Intelligence Fudan University Department of Obstetrics and Gynecology West China Second University HospitalSichuan University Peking University International Cancer Institute and Peking University-Yunnan Baiyao International Medical Institute and State Key Laboratory of Natural and Biomimetic Drugs Department of Molecular and Cellular PharmacologySchool of Pharmaceutical SciencesPeking University Health Science CenterPeking University Channing Division of Network Medicine Department of MedicineBrigham and Women's Hospital and Harvard Medical School Center for Artificial Intelligence and Modeling the Carl R.Woese Institute for Genomic BiologyUniversity of Illinois Urbana-Champaign Department of Genetics and Genome Sciences School of Medicine and Department of Computer and Data Sciences and Department of Population and Quantitative Health SciencesCase Western Reserve University Department of Biochemistry and Biophysics University of California Sanger Institute Department of Automation Tsinghua University State Key Laboratory of Multimodal Artificial Intelligence Systems I

The development of artificial intelligence(AI) and the mining of biomedical data complement each other. From the direct use of computer vision results to analyze medical images for disease screening, to now integrating biological knowledge into models and even accelerating the development of new AI based on biological discoveries, the boundaries of both are constantly expanding, and their connections are becoming closer.

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Flexible krylov methods for pregularization

arXiv

引用

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

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Localisation of regularised and multiview support vector machine learning

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2024年第1期25卷 18029-18075页

作者： Aurelian Gheondea Cankat Tilki Institute of Mathematics of the Romanian Academy Bucharest Romania and Department of Mathematics Bilkent University Bilkent Ankara Turkey Department of Mathematics and Division of Computational Modeling and Data Analytics Virginia Polytechnic Institute and State University Blacksburg Virginia

We prove some representer theorems for a localised version of a semisupervised, manifold regularised and multiview support vector machine learning problem introduced by H.Q. Minh, L. Bazzani, and V. Murino, Journal of Machine Learning Research, 17(2016) 1-72, that involves operator valued positive semidefinite kernels and their reproducing kernel Hilbert spaces. The results concern general cases when convex or nonconvex loss functions and finite or infinite dimensional underlying Hilbert spaces are considered. We show that the general framework allows infinite dimensional Hilbert spaces and nonconvex loss functions for some special cases, in particular in case the loss functions are Gâteaux differentiable. Detailed calculations are provided for the exponential least squares loss functions that lead to systems of partially nonlinear equations for which some Newton's approximation methods based on the interior point method can be used. Some numerical experiments are performed on a toy model that illustrate the tractability of the methods that we propose.

关键词： operator valued reproducing kernel Hilbert spaces manifold co-regularised and multiview learning support vector machine learning loss functions representer theorem

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Hybrid projection methods with recycling for inverse problems

arXiv

引用

arXiv 2020年

作者： Chung, Julianne Sturler, Eric D.E. Jiang, Jiahua Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States 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

Iterative hybrid projection methods have proven to be very effective for solving large linear inverse problems due to their inherent regularizing properties as well as the added flexibility to select regularization parameters adaptively. In this work, we develop Golub-Kahan-based hybrid projection methods that can exploit compression and recycling techniques in order to solve a broad class of inverse problems where memory requirements or high computational cost may otherwise be prohibitive. For problems that have many unknown parameters and require many iterations, hybrid projection methods with recycling can be used to compress and recycle the solution basis vectors to reduce the number of solution basis vectors that must be stored, while obtaining a solution accuracy that is comparable to that of standard methods. If reorthogonalization is required, this may also reduce computational cost substantially. In other scenarios, such as streaming data problems or inverse problems with multiple datasets, hybrid projection methods with recycling can be used to efficiently integrate previously computed information for faster and better reconstruction. Additional benefits of the proposed methods are that various subspace selection and compression techniques can be incorporated, standard techniques for automatic regularization parameter selection can be used, and the methods can be applied multiple times in an iterative fashion. Theoretical results show that, under reasonable conditions, regularized solutions for our proposed recycling hybrid method remain close to regularized solutions for standard hybrid methods and reveal important connections among the resulting projection matrices. Numerical examples from image processing show the potential benefits of combining recycling with hybrid projection methods. Copyright © 2020, The Authors. All rights reserved.

关键词： Inverse problems

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