arXiv subjects

Serkan Gugercin

Publications and source records attributed to Serkan Gugercin.

86 records · Page 5Linked to original sources

arXiv2014

Model Reduction by Rational Interpolation

The last two decades have seen major developments in interpolatory methods for model reduction of large-scale linear dynamical systems. Advances of note include the ability to produce (locally) optimal reduced models at modest cost; refined methods for deriving interpolatory reduced models directly from input/output measurements; and extensions for the reduction of parametrized systems. This chapter offers a survey of interpolatory model reduction methods starting from basic principles and ranging up through recent developments that include weighted model reduction and structure-preserving methods based on generalized coprime representations. Our discussion is supported by an assortment of numerical examples.

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arXiv2014

Near-optimal Frequency-weighted Interpolatory Model Reduction

This paper develops an interpolatory framework for weighted-$\mathcal{H}_2$ model reduction of MIMO dynamical systems. A new representation of the weighted-$\mathcal{H}_2$ inner products in MIMO settings is introduced and used to derive associated first-order necessary conditions satisfied by optimal weighted-$\mathcal{H}_2$ reduced-order models. Equivalence of these new interpolatory conditions with earlier Riccati-based conditions given by Halevi is also shown. An examination of realizations for equivalent weighted-$\mathcal{H}_2$ systems leads then to an algorithm that remains tractable for large state-space dimension. Several numerical examples illustrate the effectiveness of this approach and its competitiveness with Frequency Weighted Balanced Truncation and an earlier interpolatory approach, the Weighted Iterative Rational Krylov Algorithm.

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arXiv2014

Model Reduction for DAEs with an Application to Flow Control

Direct numerical simulation of dynamical systems is of fundamental importance in studying a wide range of complex physical phenomena. However, the ever-increasing need for accuracy leads to extremely large-scale dynamical systems whose simulations impose huge computational demands. Model reduction offers one remedy to this problem by producing simpler reduced models that are both easier to analyze and faster to simulate while accurately replicating the original behavior. Interpolatory model reduction methods have emerged as effective candidates for very large-scale problems due to their ability to produce high-fidelity (optimal in some cases) reduced models for linear and bilinear dynamical systems with modest computational cost. In this paper, we will briefly review the interpolation framework for model reduction and describe a well studied flow control problem that requires model reduction of a large scale system of differential algebraic equations. We show that interpolatory model reduction produces a feedback control strategy that matches the structure of much more expensive control design methodologies.

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arXiv2014

Quadrature-Based Vector Fitting: Implications For H2 System Approximation

Vector Fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements. The original method, which we refer to as VF, is based on a least-squares fit to the measurements by a rational function, using an iterative reallocation of the poles of the approximant. We show that one can improve the performance of VF significantly, by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an H2 error measure. Our modified approach, designated here as QuadVF, helps recover the original transfer function with better global fidelity (as measured with respect to the H2 norm), than the localized least squares approximation implicit in VF. We extend the new framework also to incorporate derivative information, leading to rational approximants that minimize system error with respect to a discrete Sobolev norm. We consider the convergence behavior of both VF and QuadVF as well, and evaluate potential numerical ill-conditioning of the underlying least-squares problems. We investigate briefly VF in the case of noisy measurements and propose a new formulation for the resulting approximation problem. Several numerical examples are provided to support the theoretical discussion.

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arXiv2013

Multipoint Volterra Series Interpolation and H2 Optimal Model Reduction of Bilinear Systems

In this paper, we focus on model reduction of large-scale bilinear systems. The main contributions are threefold. First, we introduce a new framework for interpolatory model reduction of bilinear systems. In contrast to the existing methods where interpolation is forced on some of the leading subsystem transfer functions, the new framework shows how to enforce multipoint interpolation of the underlying Volterra series. Then, we show that the first-order conditions for optimal H2 model reduction of bilinear systems require multivariate Hermite interpolation in terms of the new Volterra series interpolation framework; and thus we extend the interpolation-based first-order necessary conditions for H2 optimality of LTI systems to the bilinear case. Finally, we show that multipoint interpolation on the truncated Volterra series representation of a bilinear system leads to an asymptotically optimal approach to H2 optimal model reduction, leading to an efficient model reduction algorithm. Several numerical examples illustrate the effectiveness of the proposed approach.

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arXiv2013

Nonlinear Parametric Inversion using Interpolatory Model Reduction

Nonlinear parametric inverse problems appear in several prominent applications; one such application is Diffuse Optical Tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of large-scale discretized, parametrized, partial differential equations (PDEs) in the forward model. In this paper, we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in DOT by drastically reducing the computational cost of solving the forward problems. The key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis; a similar observation holds for Jacobian evaluations as well. This motivates the use of system-theoretic model order reduction methods. We discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model. Within the DOT setting, these surrogate models can approximate both the cost functional and the associated Jacobian with very little loss of accuracy while significantly reducing the cost of the overall inversion process. Four numerical examples illustrate the efficiency of the proposed approach. Although we focus on DOT in this paper, we believe that our approach is applicable much more generally.

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arXiv2013

Model Reduction of Descriptor Systems by Interpolatory Projection Methods

In this paper, we investigate interpolatory projection framework for model reduction of descriptor systems. With a simple numerical example, we first illustrate that employing subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded H2 or H-infinity errors due to the mismatch of the polynomial parts of the full and reduced-order transfer functions. We then develop modified interpolatory subspace conditions based on the deflating subspaces that guarantee a bounded error. For the special cases of index-1 and index-2 descriptor systems, we also show how to avoid computing these deflating subspaces explicitly while still enforcing interpolation. The question of how to choose interpolation points optimally naturally arises as in the standard state space setting. We answer this question in the framework of the H2-norm by extending the Iterative Rational Krylov Algorithm (IRKA) to descriptor systems. Several numerical examples are used to illustrate the theoretical discussion.

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arXiv2012

Interpolatory Weighted-H2 Model Reduction

This paper introduces an interpolation framework for the weighted-H2 model reduction problem. We obtain a new representation of the weighted-H2 norm of SISO systems that provides new interpolatory first order necessary conditions for an optimal reduced-order model. The H2 norm representation also provides an error expression that motivates a new weighted-H2 model reduction algorithm. Several numerical examples illustrate the effectiveness of the proposed approach.

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arXiv2012

Interpolatory H-infinity Model Reduction

We introduce an interpolation framework for H-infinity model reduction founded on ideas originating in optimal-H2 interpolatory model reduction, realization theory, and complex Chebyshev approximation. By employing a Loewner "data-driven" framework within each optimization cycle, large-scale H-infinity norm calculations can be completely avoided. Thus, we are able to formulate a method that remains effective in large-scale settings with the main cost dominated by sparse linear solves. Several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better H-infinity performance than those produced by balanced truncation; these models often are as good as (and occasionally better than) those models produced by optimal Hankel norm approximation. In all cases, these reduced models are produced at far lower cost than is possible either with balanced truncation or optimal Hankel norm approximation.

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arXiv2012

On the ADI method for the Sylvester Equation and the optimal-$\mathcal{H}_2$ points

The ADI iteration is closely related to the rational Krylov projection methods for constructing low rank approximations to the solution of Sylvester equation. In this paper we show that the ADI and rational Krylov approximations are in fact equivalent when a special choice of shifts are employed in both methods. We will call these shifts pseudo H2-optimal shifts. These shifts are also optimal in the sense that for the Lyapunov equation, they yield a residual which is orthogonal to the rational Krylov projection subspace. Via several examples, we show that the pseudo H2-optimal shifts consistently yield nearly optimal low rank approximations to the solutions of the Lyapunov equations.

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arXiv2011

Recycling BiCG with an Application to Model Reduction

Science and engineering problems frequently require solving a sequence of dual linear systems. Besides having to store only few Lanczos vectors, using the BiConjugate Gradient method (BiCG) to solve dual linear systems has advantages for specific applications. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms -- for example, in quantum Monte Carlo (QMC) methods for electronic structure calculations -- leads to a quadratic error bound. Since our focus is on sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other bi-Lanczos based methods, such as CGS, BiCGSTAB, QMR, and TFQMR. We develop an augmented bi-Lanczos algorithm and a modified two-term recurrence to include recycling in the iteration. The recycle spaces are approximate left and right invariant subspaces corresponding to the eigenvalues closest to the origin. These recycle spaces are found by solving a small generalized eigenvalue problem alongside the dual linear systems being solved in the sequence. We test our algorithm in two application areas. First, we solve a discretized partial differential equation (PDE) of convection-diffusion type. Such a problem provides well-known test cases that are easy to test and analyze further. Second, we use recycling BiCG in the Iterative Rational Krylov Algorithm (IRKA) for interpolatory model reduction. IRKA requires solving sequences of slowly changing dual linear systems. We show up to 70% savings in iterations, and also demonstrate that for a model reduction problem BiCG takes (about) 50% more time than recycling BiCG.

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arXiv2011

Structure-preserving tangential interpolation for model reduction of port-Hamiltonian Systems

Port-Hamiltonian systems result from port-based network modeling of physical systems and are an important example of passive state-space systems. In this paper, we develop the framework for model reduction of large-scale multi-input/multi-output port-Hamiltonian systems via tangential rational interpolation. The resulting reduced-order model not only is a rational tangential interpolant but also retains the port-Hamiltonian structure; hence is passive. This reduction methodology is described in both energy and co-energy system coordinates. We also introduce an $\mathcal{H}_2$-inspired algorithm for effectively choosing the interpolation points and tangential directions. The algorithm leads a reduced port-Hamiltonian model that satisfies a subset of $\mathcal{H}_2$-optimality conditions. We present several numerical examples that illustrate the effectiveness of the proposed method showing that it outperforms other existing techniques in both quality and numerical efficiency.

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arXiv2011

Convergence of the Iterative Rational Krylov Algorithm

The Iterative Rational Krylov Algorithm (IRKA) of [8] is an interpolatory model reduction approach to the optimal $\mathcal{H}_2$ approximation problem. Even though the method has been illustrated to show rapid convergence in various examples, a proof of convergence has not been provided yet. In this note, we show that in the case of state-space symmetric systems, IRKA is a locally convergent fixed point iteration to a local minimum of the underlying $\mathcal{H}_2$ approximation problem.

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arXiv2011

Inexact Solves in Interpolatory Model Reduction

We investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem. We give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods. We show that when a Petrov-Galerkin framework is employed for the inexact solves, the associated reduced order model is an exact interpolatory model for a nearby full-order system; thus demonstrating backward stability. We also give evidence that for $\h2$-optimal interpolation points, interpolatory model reduction is robust with respect to perturbations due to inexact solves. Finally, we demonstrate the effecitveness of direct use of inexact solves in optimal ${\mathcal H}_2$ approximation. The result is an effective model reduction strategy that is applicable in realistically large-scale settings.

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