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Generalizations of data-driven balancing: what to sample for different balancing-based reduced models

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arXiv 2023年

作者： Reiter, Sean Gosea, Ion Victor Gugercin, Serkan Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

The Quadrature-based Balanced Truncation (QuadBT) framework of [14] is a "non-intrusive" reformulation of balanced truncation;a classical projection-based model-order reduction technique for linear systems. QuadBT is non-intrusive in the sense that it builds approximate balanced reduced-order models entirely from system response data (e.g., transfer function measurements) without the need to reference an explicit state-space realization of the underlying full-order model. In this work, we generalize and extend QuadBT to other types of balanced truncation model reduction. Namely, we develop non-intrusive implementations for balanced stochastic truncation, positive-real balanced truncation, and bounded-real balanced truncation. We show that the data-driven construction of these balanced reduced-order models requires sampling certain spectral factors associated with the system of interest. Numerical examples are included in each case to validate our *** Codes 93B15 (Primary) 93A15, 37M99, 65D30, 65K99, 15A24 (Secondary) Copyright © 2023, The Authors. All rights reserved.

关键词： Linear systems

H2 optimal model reduction of linear systems with multiple quadratic outputs

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arXiv 2024年

作者： Reiter, Sean Duff, Igor Pontes Gosea, Ion Victor Gugercin, Serkan Department of Mathematics Virginia Tech BlacksburgVA24061 United States Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

In this work, we consider the H2 optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared H2 system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the H2 optimal approximation of a linear quadratic output (LQO) system. The resulting H2 optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection;the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the H2 model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing *** Codes 15A24, 46N10, 49K15, 93A15, 93C10, 93C80 Copyright © 2024, The Authors. All rights reserved.

关键词： Linear systems

Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback controller design

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arXiv 2024年

作者： Heiland, Jan Kim, Yongho Werner, Steffen W.R. Department of Mathematics and Natural Sciences Institute of Mathematics Technische Universität Ilmenau Germany Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstraße 1 Magdeburg39106 Germany Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

Polytopic autoencoders provide low-dimensional parametrizations of states in a polytope. For nonlinear PDEs, this is readily applied to low-dimensional linear parameter-varying (LPV) approximations as they have been exploited for efficient nonlinear controller design via series expansions of the solution to the state-dependent Riccati equation. In this work, we develop a polytopic autoencoder for control applications and show how it improves on standard linear approaches in view of LPV approximations of nonlinear systems. We discuss how the particular architecture enables exact representation of target states and higher order series expansions of the nonlinear feedback law at little extra computational effort in the online phase and how the linear though high-dimensional and nonstandard Lyapunov equations are efficiently computed during the offline phase. In a numerical study, we illustrate the procedure and how this approach can reliably outperform the standard linear-quadratic regulator *** Codes 34H05, 37N35, 65F45, 76D55, 93B52 Copyright © 2024, The Authors. All rights reserved.

关键词： Riccati equations

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

IRKA is a Riemannian Gradient Descent Method

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arXiv 2023年

作者： Mlinarić, Petar Beattie, Christopher A. Drmač, Zlatko Gugercin, Serkan Department of Mathematics Virginia Tech BlacksburgVA24061 United States Faculty of Natural Sciences University of Zagreb Zagreb10000 Croatia Department of Mathematics Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech 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, 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 IRKA and this extension on a few examples demonstrate significant benefits. Copyright © 2023, The Authors. All rights reserved.

关键词： Rational functions

L2-OPTIMAL REDUCED-ORDER modeling USING PARAMETER-SEPARABLE FORMS

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arXiv 2022年

作者： Mlinarić, Petar Gugercin, Serkan Department of Mathematics Virginia Tech BlacksburgVA24061 United States Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States

We provide a unifying framework for L2-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the L2 cost function with respect to the reduced matrices, which then allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection. Copyright © 2022, The Authors. All rights reserved.

关键词： Optimization

A UNIFYING FRAMEWORK FOR INTERPOLATORY L2-OPTIMAL REDUCED-ORDER modeling

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arXiv 2022年

作者： Mlinarić, Petar 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

We develop a unifying framework for interpolatory L2-optimal reduced-order modeling for a wide classes of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for H2-optimal model order reduction and leads to the interpolatory conditions for H2 ⨂ L2-optimal model order reduction of multi-input/multi-output parametric dynamical systems. Moreover, we derive novel interpolatory optimality conditions for rational discrete least-squares minimization and for L2-optimal model order reduction of a class of parametric stationary models. We show that bitangential Hermite interpolation appears as the main tool for optimality across different domains. The theoretical results are illustrated on two numerical examples. Copyright © 2022, The Authors. All rights reserved.

关键词： Linear time-invariant system

MAPPING INCIDENCE AND PREVALENCE PEAK data FOR SIR FORECASTING APPLICATIONS

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arXiv 2024年

作者： Murph, Alexander C. Casey Gibson, G. Beesley, Lauren J. Panda, Nishant Castro, Lauren A. Del Valle, Sara Osthus, Dave Computer Computational and Statistical Sciences Division Los Alamos National Laboratory Los AlamosNM87545 United States Data Analytics & Forecasting A-1: Information Systems & Modeling Los Alamos National Laboratory Los AlamosNM87545 United States Theoretical Division Los Alamos National Laboratory Los AlamosNM87545 United States

Infectious disease modeling and forecasting have played a key role in helping assess and respond to epidemics and pandemics. Recent work has leveraged data on disease peak infection and peak hospital incidence to fit compartmental models for the purpose of forecasting and describing the dynamics of a disease outbreak. Incorporating these data can greatly stabilize a compartmental model fit on early observations, where slight perturbations in the data may lead to model fits that project wildly unrealistic peak infection. We introduce a new method for incorporating historic data on the value and time of peak incidence of hospitalization into the fit for a Susceptible-Infectious-Recovered (SIR) model by formulating the relationship between an SIR model's starting parameters and peak incidence as a system of two equations that can be solved computationally. This approach is assessed for practicality in terms of accuracy and speed of computation via simulation. To exhibit the modeling potential, we update the Dirichlet-Beta State Space modeling framework to use hospital incidence data, as this framework was previously formulated to incorporate only data on total infections. © 2024, CC BY.

关键词： Hospitals

Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems

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arXiv 2023年

作者： Gosea, Ion Victor Gugercin, Serkan Werner, Steffen W.R. Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Department of Mathematics and Division of Computational Modeling and Data Analytics Academy of Data Science Virginia Tech BlacksburgVA24061 United States Courant Institute of Mathematical Sciences New York University New YorkNY10012 United States

An essential tool in data-driven modeling of dynamical systems from frequency response measurements is the barycentric form of the underlying rational transfer function. In this work, we propose structured barycentric forms for modeling dynamical systems with second-order time derivatives using their frequency domain input-output data. By imposing a set of interpolation conditions, the systems’ transfer functions are rewritten in different barycentric forms using different parametrizations. Loewner-like algorithms are developed for the explicit computation of second-order systems from data based on the developed barycentric forms. Numerical experiments show the performance of these new structured data driven modeling methods compared to other interpolation-based data-driven modeling techniques from the *** Codes 41A20, 65D05, 93B15, 93C05, 93C80 Copyright © 2023, The Authors. All rights reserved.

关键词： Interpolation