Preconditioned Multishift BiCG for $\mathcal{H}_2$-Optimal Model Reduction

Abstract

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.

Nested Krylov methods for shifted linear systems

“…[10,20]. Since even the one-time application of a shift-and-invert preconditioner can be computationally costly, polynomial preconditioners have been developed for shifted problems in [1]. They have the advantage of preserving the shift-invariance (2) and being computational feasible at the same time.…”

“…The presented work is a new approach to the iterative solution of (1). We use nested Krylov methods that use an inner Krylov method as a preconditioner for an outer Krylov iteration, cf.…”

“…Therefore, only the seed shift τ has to be chosen and in order to invert P(σ) we only need to decompose P. However, the one-time decomposition of P can numerically be very costly and the suitable choice of the seed shift τ is difficult for a large range of shifts σ 1 , .., σ Nσ , as has been pointed out by [20]. In [1], a polynomial preconditioner is suggested that cheaply approximates P −1 .…”

“…Since the shift-and-invert matrix has to be solved by a direct method, this approach can be computationally very costly. This can be overcome by either a multi-grid approach [5,21] or by an approximation of the shift-and-invert preconditioner using a polynomial approximation as shown in [1]. Most recently, many shift-and-invert preconditioners have been applied in a flexible Krylov method in order to capture a wider range of shifts, cf.…”

A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems

“…The first linear system is called the seed system, others are the add systems. Such problem occurs in many scientific and engineering applications, such as structural dynamics [1,2], quantum chromodynamics [3], web search ranking [4], control theory [5,6] and so on. Therefore, there is a strong need for establishing efficient solutions of Eq.…”