Research Accomplishments

Antoulas’ accomplishment which should be mentioned rst is his book: Model reduction of large-scale dynamical systems, published by SIAM in 2005. It gives a tutorial and detailed account of the various reduction methods and is quite sought after, as it is still the only textbook in this research area. His major research accomplishments are as follows.

Interpolatory model reduction
About 15 years ago the so-called interpolatory model reduction approach was proposed. This had close connections with what is known in numerical linear algebra as rational Krylov methods. Therein complex points must be chosen where the transfer functions of the original and the reduced systems should match. It can easily be shown that (under mild assumptions) any reduced order system can be obtained from a full-order system by interpolatory projection. Thus the problem of judicious choice of these points becomes a major issue. Stated dierently, one must specify the local behavior of the reduced-order system (i.e. at the interpolation points) in such a way that global properties are satised. Originally, methods of choosing interpolation points involved ad-hoc procedures based on matching the peaks of the frequency response.

(a) Passivity preserving model reduction. His rst major contribution along those lines is an interpolatory reduction method that preserves passivity of the reduced model. He showed, namely, that if the interpolation points are chosen as (some) of the so-called spectral zeros of the original system, passivity (and hence stability) of the reduced system is guaranteed. It is interesting to note that this observation converts the model reduction problem to a generalized eigenvalue problem involving the Hamiltonian of the associated full order system (the original idea belonged to Antoulas, with subsequent contributions from D.C. Sorensen, R. Ionutiu, and J. Rommes).

(b) Optimal H2 model reduction. While working on his book Antoulas derived a simple formula for the H2 norm of a linear system, involving the residues and the poles of the system (see section 5.5.2 of his book). This yielded a straightforward expression for the H2 norm of the error system (dierence between the original and reduced order systems) which in turn, lead to an explicit determination of the rst order necessary optimality conditions. These conditions are interpolatory with interpolation points the mirror images of the reduced-system poles (it turns out that the same optimality conditions were derived earlier by Luenberger using dierent means). Thus the method known as IRKA (Iterative Rational Krylov Algorithm) was put forward. This method is iterative because the poles of the reduced order system are not apriori known. Since then this reduction method has had a wide impact in the model reduction and linear algebra communities (a google search for ‘iterative rational krylov’ gives tens of thousands of hits. Consequently other researchers have picked up this topic (Szyld in the recent SIAM ALA conference presented a preconditioned multi-shift BiCG for H2 model reduction). It should be noted that the contributions to the H2 model reduction problem had up to this point been few and fragmented, of theoretical nature and of no consequence computationally. The IRKA approach made it possible to eciently reduce 5 large-scale systems of order about 10 to the fifth power. Original contributors to this research area are besides Antoulas,S. Gugercin and C.A. Beattie.

Data-driven model reduction
This can also be cast as an interpolatory reduction method. The dierence with the methods mentioned earlier, consists in the fact that instead of a to-be-reduced underlying system, only measurements of its input/output behavior are available (i.e. the frequency response or the S(catering)- parameters of the system are measured). This approach is widely used in many areas including electronic design. Thus given the value of the S-parameters for certain frequencies, the problem consists in constructing low-complexity linear dynamical systems which approximate the given measurements. The contributors in this area are Antoulas and his recent doctoral students.

(a) Multi-input multi-output linear systems. There are several methods available for tackling the problem of constructing a linear system given input/output measurements. The candidate introduced in 1986 in a joint paper with B.D.O. Anderson, a new approach, based on the so called Loewner matrix. The main property of the Loewner matrix is that its rank is equal to the complexity of the underlying system. Two decades later his doctoral student A.J. Mayo introduced the closely related shifted Loewner matrix. This was a breakthrough, as the Loewner, shifted Loewner matrix pencil constructed from input/output measurements, provides a descriptor realization of the underlying system, essentially without computation. As this pencil is singular for large sets of measured data, an SVD-type projection must be added to obtain a low order approximant. This is the natural way to approach this problem. Hence it is both powerful and can be eciently computed, its power lying in the fact that a descriptor realization is obtained from the data in a direct way. Furthermore, because it addresses tangential interpolation, it is the best available method for modeling data with many (several hundred) inputs and outputs.

(b) Recursive construction of models and reduced models from data. Often, either data is not provided all at once or the amount of data provided is high. In such cases, one would like to construct models or reduced models from part of the data, and subsequently update this model as more data becomes available. This leads to recursive construction of models. It turns out that this is a problem rich in structure. For instance, updates are attached to the original models by means of feedback connections. Antoulas studied this topic in dierent contexts several times over the past decades. First came the 1986 recursiveness paper, followed by the 1990 paper on recursiveness in the behavioral framework, and nally the most general version was worked out more recently (in 2010) in the Loewner tangential Interpolation framework; this approach was successfully applied to the construction of reduced order models from large amounts of S-parameter data. All three papers appeared in IEEE Transactions Journals (the rst two in “IEEE-TAC” { Transaction on Automatic Control, and the last one in “IEEE-TCAD” { Transactions on Computer-Aided Design).

(c) Parameter-dependent linear systems. In the use of mathematical models for applications, an important aspect is that such models may depend on parameters (e.g. material properties, shape variations, etc.). The problem then is parametrized model reduction and consists in developing accurate reduced order models which retain the parametric dependence of the original system. By extending the Loewner framework Antoulas and his students developed a novel approach, based on measurement of appropriate responses of the parameter-dependent system. This problem can also be considered as one of approximation of the associated multivatiate transfer function, which is approached here in an innovative way by extending the Loewner matrix. As before, the main property of this extended Loewner matrix is that its rank provides information about the minimal complexity of the underlying system both with respect to the dynamics as well as all the parameters. In the context of multivariate function approximation is should be stressed that the Loewner approach is the only one that allows for an explicit trade-o between accuracy and complexity of the approximate. Results presented in recent conferences have shown that the Loewner approach to parametrized model reduction compares favorably to more established approaches like RB (Reduced Bases) providing in addition, an explicit trade-o between accuracy and complexity.

(d) Nonlinear dynamical systems. The interpolatory model reduction approach has recently been extended to classes of nonlinear systems (see the work of Benner, Breiten and Gu). However, the proposed approaches have drawbacks, (a) Given the dimension of the reduced system, fewer moments are than is possible, are matched. (b) No trade-o between accuracy of t and complexity fo the reduced model is available, in other words, it is not clear how the reduced model complexity should be chosen to achieve a given accuracy. (c) The choice of expansion points is limited to single points with mutliplicity. An attempt to address these issues came by Ionita in his PhD thesis and subsequntly by Antoulas, Gosea and Ionita in their paper submitted to SIAM J. Sci. Computation. They extend the Loewner framework to the case of bilinear systems and quadratic bilinear systems. Consequently, a trade-o is possible between accuracy of t and complexity of the model by means of an SVD of the associated Loewner pencil.