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 $\mathcal{H}_2$-optimal 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.