arXiv subjects

Qiu-Yan Song

Publications and source records attributed to Qiu-Yan Song.

5 recordsLinked to original sources

arXiv2025

A Unified Low-rank ADI Framework with Shared Linear Solves for Simultaneously Solving Multiple Lyapunov, Sylvester, and Riccati Equations

The alternating direction implicit (ADI) methods are computationally efficient and numerically effective tools for computing low-rank solutions of large-scale linear matrix equations. It is known in the literature that the low-rank ADI method for Lyapunov equations is a Petrov-Galerkin projection algorithm that implicitly performs model order reduction. It recursively enforces interpolation at the mirror images of the ADI shifts and places the poles of the reduced-order models at the ADI shifts. In this paper, we show that the low-rank ADI methods for Sylvester and Riccati equations are also Petrov-Galerkin projection algorithms that implicitly perform model order reduction. These methods likewise enforce interpolation at the mirror images of the ADI shifts; however, they do not place the poles at the mirror images of the interpolation points. Instead, their pole placement ensures that the projected Sylvester and Riccati equations they implicitly solve admit a unique solution. By observing that the ADI methods for Lyapunov, Sylvester, and Riccati equations differ only in pole placement and not in their interpolatory nature, we show that the shifted linear solves-which constitute the bulk of the computational cost-can be shared. The pole-placement step involves only small-scale operations and is therefore inexpensive. We propose a unified ADI framework that requires only two shifted linear solves per iteration to simultaneously solve six Lyapunov equations, one Sylvester equation, and ten Riccati equations, thus substantially increasing the return on investment for the computational cost spent on the linear solves. All operations needed to extract the individual solutions from these shared linear solves are small-scale and inexpensive. Three numerical examples are presented to demonstrate the effectiveness of the proposed ADI framework.

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arXiv2025

Data-driven Implementations of Various Generalizations of Balanced Truncation

Quadrature-based approximation of Gramians in standard balanced truncation yields a non-intrusive, data-driven implementation that requires only transfer function samples on the imaginary axis, which can be measured experimentally. This idea has recently been extended to several generalizations of balanced truncation, including positive-real balanced truncation, bounded-real balanced truncation, and balanced stochastic truncation. However, these extensions require samples of some spectral factorizations on the imaginary axis, and no practical method exists to obtain such data experimentally. As a result, these non-intrusive implementations are mainly of theoretical interest at present. This paper shows that if the Gramians in these generalizations are approximated via rational interpolation rather than numerical integration, the resulting non-intrusive implementations do not require spectral factorization samples. Instead, they rely only on transfer function samples. Based on this idea, non-intrusive implementations are first developed for several variants of balanced truncation, wherein the Gramians are approximated implicitly using low-rank Alternating Direction Implicit (ADI) methods for Lyapunov and Riccati equations. These formulations require transfer function samples in the right half of the $s$-plane, which cannot be measured experimentally. Next, building on these results, novel data-driven non-intrusive implementations are proposed that require only transfer function samples on the imaginary axis. Hence, unlike the quadrature-based and ADI-based approaches, these non-intrusive formulations can be implemented using practically measurable data. Numerical results are presented for benchmark problems in model order reduction, which show that the proposed non-intrusive implementations achieve accuracy comparable to their intrusive counterparts.

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arXiv2024

$\mathcal{H}_2$-optimal Model Reduction of Linear Quadratic Output Systems in Finite Frequency Range

In frequency-limited model order reduction, the objective is to maintain the frequency response of the original system within a specified frequency range in the reduced-order model. In this paper, a mathematical expression for the frequency-limited $\mathcal{H}_2$ norm is derived, which quantifies the error within the desired frequency interval. Subsequently, the necessary conditions for a local optimum of the frequency-limited $\mathcal{H}_2$ norm of the error are derived. The inherent difficulty in satisfying these conditions within a Petrov-Galerkin projection framework is also discussed. Using the optimality conditions and the Petrov-Galerkin projection, a stationary point iteration algorithm is proposed, which approximately satisfies these optimality conditions upon convergence. The main computational effort in the proposed algorithm involves solving sparse-dense Sylvester equations. These equations are frequently encountered in $\mathcal{H}_2$ model order reduction algorithms and can be solved efficiently. Moreover, the algorithm bypasses the requirement of matrix logarithm computation, which is typically necessary for most frequency-limited reduction methods and can be computationally demanding for high-order systems. An illustrative example is provided to numerically validate the developed theory. The proposed algorithm's effectiveness in accurately approximating the original high-order model within the specified frequency range is demonstrated through the reduction of an advection-diffusion equation-based model, commonly used in model reduction literature for testing algorithms. Additionally, the algorithm's computational efficiency is highlighted by successfully reducing a flexible space structure model of order one million.

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arXiv2024

Time-limited H2-optimal Model Order Reduction of Linear Systems with Quadratic Outputs

An important class of dynamical systems with several practical applications is linear systems with quadratic outputs. These models have the same state equation as standard linear time-invariant systems but differ in their output equations, which are nonlinear quadratic functions of the system states. When dealing with models of exceptionally high order, the computational demands for simulation and analysis can become overwhelming. In such cases, model order reduction proves to be a useful technique, as it allows for constructing a reduced-order model that accurately represents the essential characteristics of the original high-order system while significantly simplifying its complexity. In time-limited model order reduction, the main goal is to maintain the output response of the original system within a specific time range in the reduced-order model. To assess the error within this time interval, a mathematical expression for the time-limited $\mathcal{H}_2$-norm is derived in this paper. This norm acts as a measure of the accuracy of the reduced-order model within the specified time range. Subsequently, the necessary conditions for achieving a local optimum of the time-limited $\mathcal{H}_2$ norm error are derived. The inherent inability to satisfy these optimality conditions within the Petrov-Galerkin projection framework is also discussed. After that, a stationary point iteration algorithm based on the optimality conditions and Petrov-Galerkin projection is proposed. Upon convergence, this algorithm fulfills three of the four optimality conditions. To demonstrate the effectiveness of the proposed algorithm, a numerical example is provided that showcases its ability to effectively approximate the original high-order model within the desired time interval.

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arXiv2024

Balanced Truncation of Linear Systems with Quadratic Outputs in Limited Time and Frequency Intervals

Model order reduction involves constructing a reduced-order approximation of a high-order model while retaining its essential characteristics. This reduced-order model serves as a substitute for the original one in various applications such as simulation, analysis, and design. Often, there's a need to maintain high accuracy within a specific time or frequency interval, while errors beyond this limit can be tolerated. This paper addresses time-limited and frequency-limited model order reduction scenarios for linear systems with quadratic outputs, by generalizing the recently introduced structure-preserving balanced truncation algorithm. To that end, limited interval system Gramians are defined, and the corresponding generalized Lyapunov equations governing their computation are derived. Additionally, low-rank solutions for these equations are investigated. Next, balanced truncation algorithms are proposed for time-limited and frequency-limited scenarios, each utilizing its corresponding limited-interval system Gramians. The proposed algorithms ensure accurate results within specified time and frequency intervals while preserving the quadratic-output structure. Two benchmark numerical examples are presented to demonstrate the effectiveness of the algorithms, showcasing their ability to achieve superior accuracy within the desired time or frequency interval.

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