Preconditioned multishift BiCG for H2-optimal model reduction

Modern methods for H2-optimal model order reduction include the Iterative Rational Krylov Algorithm (IRKA, [Gugerkin, Antoulas, and Beattie, 2008]) and Parameterized Model Reduction (PMOR, [Baur, Beattie, Benner, and Gugercin, 2011]). In every IRKA or PMOR iteration, two sequences of shifted linear systems need to be solved, one for the shifted matrices and one for their adjoint, using the same sequence of shifts. In this paper we propose a computationally efficient way of solving both sequence of problems together using the bi-conjugate gradient method (BiCG). The idea is to construct in advance bases for the two Krylov subspaces (one for the matrix and one for its adjoint), suitably preconditioned. These bases are then reused inside the parametrized model reduction methods for the other shifts, without the need for additional matrix-vector products. The performance of our proposed implementation ideas is illustrated through numerical experiments.