We propose the use of a multishift biconjugate gradient method (BiCG) in combination with a suitable chosen polynomial preconditioning, to efficiently solve the two sets of multiple shifted linear systems arising at each iteration of the iterative rational Krylov algorithm (IRKA) [Gugercin, Antoulas, and Beattie, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 609-638] for H 2optimal model reduction of linear systems. The idea is to construct in advance bases for the two preconditioned Krylov subspaces (one for the matrix and one for its adjoint). By exploiting the shift-invariant property of Krylov subspaces, these bases are then reused inside the model reduction methods for the other shifts. The polynomial preconditioner is chosen to maintain this shift-invariant property. This means that the shifted systems can be solved without additional matrix-vector products. The performance of our proposed implementation is illustrated through numerical experiments.

We discuss a Krylov subspace projection method for model reduction of a special class of quadratic-bilinear descriptor systems. The goal is to extend the two-sided moment-matching method for quadratic-bilinear ODEs to descriptor systems in an efficient and reliable way. Recent results have shown that the direct application of interpolation based model reduction techniques to linear descriptor systems, without any modifications, may lead to poor reduced-order systems. Therefore, for the analysis, we transform the quadratic-bilinear descriptor system into an equivalent quadratic-bilinear ODE system for which the moment-matching is performed. In view of implementation, we provide algorithms that identify the required Krylov subspaces without explicitly computing the projectors used in the analysis. The benefits of our approach are illustrated for the quadratic-bilinear descriptor systems corresponding to semi-discretized Navier-Stokes equations. 1253reduced-order systems obtained from these methods may not be suitable for applications in control or in optimization, where input variation is the main goal.On the other hand, moment-matching methods tend to approximate the input-output behavior of the system well and therefore, unlike trajectory-based methods, these methods are not bound to a specific input. Extending well-known results for linear ODEs, the moment-matching problem has been considered in [5,16] for quadratic-bilinear ODEs for one-sided moments. The latest extension was to two-sided moment-matching for SISO quadratic-bilinear ODEs [7].In this paper, we study a two-sided moment-matching technique for model reduction of Navier-Stokes type quadraticbilinear descriptor systems (1). This class of quadratic-bilinear descriptor systems is different from the one considered in [15] where the vorticity formulation of the Navier-Stokes equations was used. Here, we propose a structured approach for Navier-Stokes type quadratic-bilinear descriptor systems, since the direct implementation of [5,7,16] might lead to an unbounded error in some norm, cf., in particular, [18], where it was shown that the direct extension of moment-matching techniques for linear ODEs to linear DAEs may lead to unbounded H 2 or H ∞ error. An extension of the ideas presented in [18] to a special class of bilinear descriptor systems is presented in [1,8].The first contribution of this paper is to transform the system in (1) into an equivalent quadratic-bilinear ODE system. This is done by introducing projectors similar to those used in [18,21] for linear systems. The second contribution is to reduce the equivalent ODE system by constructing basis matrices for Krylov subspaces without explicitly computing the projectors.The paper is organized as follows: Sect. 2 contains the background theory on the two-sided moment-matching technique for model reduction of quadratic-bilinear ODEs; Sect. 3 presents the transformation of the system (1) to an equivalent ODE system, and shows how two-sided moment-matching can be used to obtain a reduced-order syste...

No abstract