Implicitly Restarted Krylov Subspace Methods for Stable Partial Realizations

Balanced truncation is a well-known technique for model-order reduction with a known uniform reduction error bound. However, its practical application to large-scale problems is hampered by its cubic computational complexity. While model-order reduction by projection to approximate dominant subspaces without balancing has produced encouraging experimental results, the approximation error bound has not been fully analyzed. In this paper, a square-integral reduction error bound is derived for unbalanced dominant subspace projection by using a frequency-domain solution of the Lyapunov equation. Such an error bound is valid in both the frequency and time domains. Then, a dominant subspace computation scheme together with three Krylov subspace options is introduced. It is analytically justified that the Krylov subspace for moment matching at low frequencies is able to provide a better dominant subspace approximation than the Krylov subspace at high frequencies, while a rational Krylov subspace with a proper real shift parameter is capable of achieving superior approximation than the Krylov subspace at low frequency. A heuristic method of choosing a real shift parameter is also introduced based on its new connection to the discretization of a continuous-time model. The computation algorithm and theoretical analysis are then examined by several numerical examples to demonstrate the effectiveness. Finally, the dominant subspace computation scheme is applied to the model-order reduction of two large-scale interconnect circuit examples.

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output projection method, which allows tractable computation even when the number of outputs is large. The output projection method requires minimal additional computation, and has a priori error bounds that can guide the choice of rank of the projection. Connections between POD and balanced truncation are also illuminated: in particular, balanced truncation may be viewed as POD of a particular dataset, using the observability Gramian as an inner product. The three methods are illustrated on a numerical example, the linearized flow in a plane channel.

Balanced truncation (BT) model order reduction (MOR) is known for its superior accuracy and computable error bounds. Balanced stochastic truncation (BST) is a particular BT procedure that provides a general, structure-independent MOR framework to preserve both passivity and stability of original models. Its application toward large scale systems, however, has been limited by the complexity of solving large size continuous time algebraic Riccati equations (CAREs). This paper introduces a novel quadratic extension of the alternating direction implicit (ADI) iteration, called QADI, that efficiently solves a CARE. A Cholesky factor variant of QADI, called CFQADI, further exploits low rank matrices and and produces solution in factor form that greatly accelerates BST. Remarkable efficiency of the proposed BST/(CF)QADI integration is demonstrated with numerical examples.

Zusammenfassung Die Dissertation stellt rigorose Fehlerschranken und Verfahren zur automatischen Entwicklungspunktwahl bei der Modellordnungsreduktion linearer, zeitinvarianter Systeme mittels Krylow-Unterraum-Methoden vor. Die örtliche Diskretisierung partieller Differentialgleichungen, welche zur Beschreibung dynamischer Systeme in diversen ingenieurwissenschaftlichen Bereichen zum Einsatz kommen, führt meist zu sehr großen Systemen gewöhnlicher Differentialgleichungen, deren Anzahl mit steigenden Ansprüchen an die Modellgenauigkeit zunimmt. Zur Erfüllung von Simulations-, Regelungs- oder Optimierungsaufgaben ist eine Vereinfachung des Modells daher oft unumgänglich; hierzu wurden zahlreiche Methoden mit spezifischen Vor- und Nachteilen beschrieben. Krylow-Unterraum-Methoden, die im Zentrum dieser Arbeit stehen, erfordern verhältnismäßig geringen numerischen Aufwand und sind daher zur Reduktion auch sehr großer Modelle geeignet. Allerdings erhalten sie nicht zwangsläufig die Stabilität des Modells, bieten keine Information über die Reduktionsgüte und erfordern die günstige Wahl gewisser Parameter, der sogenannten Entwicklungspunkte (,,Shifts“) sowie der Ordnung des reduzierten Modells. Ausgehend von einer neuen Formulierung des Fehlersystems werden neue Zugänge zu diesen Problemstellungen aufgezeigt. Ein kumulatives Reduktionsvorgehen, währenddessen das reduzierte Modell iterativ aufgebaut wird, ermöglicht die adaptive Wahl der reduzierten Ordnung und der Entwicklungspunkte. Letztere erfolgt mittels Optimierung in einem Abstiegsverfahren, das oft nur wenige Schritte benötigt. Schließlich werden globale Fehlerschranken für eine Klasse von Zustandsraummodellen eingeführt; der verursachten Überschätzung wird durch Umformulierung des Optimierungsproblems begegnet. Die vorgestellten Methoden können z.B. effizient auf viele Systeme zweiter Ordnung angewandt werden. Fallstudien anhand von Modellen aus der Strukturmechanik, Elektrothermik, Akustik u. a. belegen ihre Effektivität.

We propose a numerical method of frequency-weighted model reduction. The model to be reduced (an original model) is a stable SISO discrete-time model described by high-order state-space equations. We design the reduced-order model so that it can interpolate 1st- and 2nd-order information of the original model at complex frequency points (interpolation points) in the unit circle. The characteristics of the reduced-order model greatly depend on the choice of the interpolation points. The proposed model reduction method is a numerical one that chooses the interpolation points by searching in the unit circle to find the reduced-order model such that L∞-norm of the reduction error is less than a prescribed value. This method has the following features that show that it is an effective numerical method of the frequency-weighted model reduction. i) The reduced-order model is guaranteed to be stable. ii) The procedure for finding the reduced-order model is simple and requires a relatively small amount of computation. iii) The order of the reduced-order model can be controlled by choosing the number of interpolation points. © 1998 Scripta Technica, Electr Eng Jpn, 126(2): 31–39, 1999

A multiplicity of simulations is required to optimize systems with thermal transient processes in the presence of uncertain parameters. That is why model order reduction is applied to minimize the numerical effort. The consideration of heat radiation and convection with parameter-dependent heat transfer coefficients results in a nonlinear system with many inputs as these loads are distributed over the whole surface limiting the attainable reduced dimension. Therefore, a new input reduction method is presented approximating the input matrix based on load vector snapshots using singular value decomposition. Afterward, standard reduction methods like the Krylov subspace method or balanced truncation can be applied. Compared to proper orthogonal decomposition, the number of training simulations decreases significantly and the reduced-order model provides a high accuracy within a broad parameter range. In a second step, the discrete empirical interpolation method is used to limit the evaluation of the nonlinearity to a few degrees of freedom and proper orthogonal decomposition allows the fast adaptation of the emissivity. As a result, the reduced system becomes independent of the original dimensions and the computation time is reduced drastically. This approach enables an optimal method combination depending on the number of simulations performed with the reduced model.

In this paper a novel model order reduction method for nonlinear models, based on decoupled multi-model, via trajectory piecewise-linearization is proposed. Through different strategies in trajectory piecewise-linear model reduction, model order reduction of a trajectory piecewise-linear model based on output weighting (TPWLOW), has been developed by authors of current work. The structure of mentioned work was founded based on Krylov subspace method, which is appropriate for high order models. Indeed the size of the Krylov subspaces may increase with the number of inputs of the system. As a result, the use of Krylov subspace method may become deficient the case for multi-input systems of order few decades. This paper aims to generalize the idea of model reduction of TPWLOW model by concentrating on balanced truncation technique which is appropriate for medium size systems. In addition, the proposed method either guarantees or provides guaranteed stability in some mentioned conditions. Finally, applicability of the reduced model, instead of high-order decoupled multi-model in weighting multi-model controllers, is investigated through the control of a nonlinear Alstom gasifier benchmark.

The behavioral simulation of large-scale analog circuits is very expensive. Instead of the circuit complex model, reduction of the computing effort implies using a macromodel capable to capture the behaviors of the original circuit while preserving its essential properties. Building such a model is based on a simplified and unitary description of the circuit, by establishing a simple relationship between the input/output variables. In this paper, the model order reduction techniques based on method projection in Krylov subspace are presented in two implementation: on state equations and on semi-sate equations of the analog circuits. Illustrative simulations are performed that allow a comparison of the standard methods based on Padé approximation with Krylov subspace methods, and some important conclusions.

A system identification (SI) and Prony's model based algorithm for modelling and prediction of a time‐domain transient response is introduced and applied to time‐domain transmission line matrix (TLM) transient responses in order to extrapolate parameters of microwave circuits and accelerate the simulation process.Application of moment matching model order reduction (MOR) techniques to the TLM method is presented with emphasis placed on Krylov subspace methods based on the Lanczos process and its modifications. The Krylov subspace methods are attractive for MOR in the TLM framework because the construction of orthonormal and biorthogonal bases for correspondent Krylov subspaces can be carried out through a direct application of the implicit TLM solver. MOR applied to TLM allows one to generate a compact macromodel of a TLM‐system and significantly reduce the computational time.The efficiency and advantages of the SI and MOR techniques are demonstrated through their applications to the full‐wave analysis of different electromagnetic structures. In addition, a brief comparison of SI and MOR involved for modelling of a planar patch antenna is provided. Copyright © 2006 John Wiley & Sons, Ltd.

Convergence rates for inverse-free rational approximation of matrix functions

In the present work, multiple non existing model order reduction (MOR) techniques for unstable second‐order form systems (SOSs) are proposed. For unstable SOSs, continuous‐time algebraic Lyapunov equations get unsolvable that halt the reduction process. To avoid this problem, unstable SOS is first decomposed into stable and unstable portions and balanced truncation is applied to the stable part. The obtained reduced order model (ROM) for the stable portion is augmented with the unstable portion to obtain the overall reduced system. It is observed that the second‐order structure in ROM for the first technique gets lost as well as augmented unstable dynamics degrade the ROM performance. To cater to these constraints, two structure‐preserving second‐order balanced truncation techniques for unstable SOSs are proposed in second part. System is first stabilized using the Bernoulli feedback stabilization procedure and then gramians are computed for the stabilized system. Further, gramians are partitioned into position and velocity portions to achieve structure preservation in ROM and balanced truncation is applied. As second technique retains second‐order structure as well as involves stabilized system dynamics, this technique far closely approximates original system behavior. Proposed techniques are tested on multiple systems and results certify the correct development of proposed techniques that can be utilized for MOR applications of unstable SOSs. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

We consider the computation of a few eigenvectors and corresponding eigenvalues of a large sparse nonsymmetric matrix using shift-invert Arnoldi's method with and without implicit restarts. For the inner iterations we use preconditioned GMRES as the inexact iterative solver. The costs of the solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. We first extend the relaxation strategy developed by Simoncini [SIAM J. Numer. Anal., 43 (2005), pp. 1155–1174] to implicitly restarted Arnoldi's method, which yields an improvement in the overall costs of the method. Secondly, we apply a new preconditioning strategy to the inner solver. We show that small rank changes to the preconditioner can produce significant savings in the total number of iterations. The combination of the new preconditioner with the relaxation strategy in implicitly restarted Arnoldi produces enhancement in the overall costs of around 50 percent in the examples considered here. Numerical experiments illustrate the theory throughout the paper.

Heat conduction in an electronic device is commonly modeled as a discretized thermal system (e.g., finite element or finite difference models) that typically uses large matrices for solving complex problems. The large size of electronic-system heat transfer models can be reduced using model reduction methods and the resulting reduced-order models can yield accurate results with far less computational costs. Electronic devices are typically composed of components, like chips, printed circuit boards, and heat sinks that are coupled together. There are two ways of creating reduced-order models for devices that have many coupled components. The first way is to create a single reduced-order model of the entire device. The second way is to interconnect reduced-order models of the components that constitute the device. The second choice (which we call the "reduce then interconnect" approach) allows the heat transfer specialist to perform quick simulations of different architectures of the device by using a library of reduced-order models of the different components that make up the device. However, interconnecting reduced-order models in a straightforward manner can result in unstable behavior. The purpose of this paper is two-fold: creating reduced-order models of the components using a Krylov subspace algorithm and interconnecting the reduced-order models in a stable manner using concepts from control theory. In this paper, we explain the logic behind the "reduce then interconnect" approach, formulate a control-theoretic method for it, and finally exhibit the whole process numerically, by applying it to an example heat conduction problem

Model order reduction for nonlinear systems has been a challenging problem with limited methods and tools available. One approach is balanced truncation of empirically obtained Gramians (Lall et al., 1999) or controllability/ observability covariance matrices (Hahn et al., 2002). Empirical Gramians provide information about the controllability/ observability which is then used to transform a system so that only the subspaces that are most important to the input-to-output behavior are retained. While this provides an open-loop reduced order model, it may not be the best choice for robust controller and estimator (compensator) design. In this paper, an approach for obtaining empirical control and filter Riccati covariance matrices is proposed, and the empirical Riccati covariance matrices are used to obtain a reduced order model for systematic closed-loop compensator design. The proposed reduction method is demonstrated on a spatially discretized catalytic rod model, for which a constrained nonlinear model predictive controller and an extended Kalman filter are designed with the reduced order model. The efficacy of the proposed approach is established by comparing input and output responses of the resulting closed-loop system to those of the full order compensator and reduced compensators designed using other common balanced truncations methods.

This paper presents a framework for model order reduction based on Balanced Truncation (BT) method and Artificial Neural Networks (ANN). As social, architectural and technical models are witnessing rapid complexity increase, it has become essential to find solutions to obtain a lower complex version of models while still preserving the model integrity. The main idea of the proposed framework is to reduce uncertainty by cascading two different Model Order Reduction (MOR) approaches, the BT technique and the Jordan Recurrent Neural Network (JRNN). The reduced model obtained by the JRNN network is estimated using the Instrument Variable (IV) estimation approach. Finally, the BT reduced model and the estimated JRNN reduced model are cascaded to obtain the final reduced order version of the original plant. The cascading procedure takes the powerfulness of both techniques and present them in a single system capable to compensate for possible errors and uncertainties. Simulation results prove the efficiency of the proposed framework in terms of the deduction strength and the integrity level.