\({\mathcal{H}}_{2}\)-Optimal Model Reduction of Linear Quadratic-Output Systems by Multivariate Rational Interpolation

Abstract. This paper addresses the [Formula: see text]-optimal approximation of linear dynamical systems with quadratic-output functions, also known as linear quadratic-output systems. Our major contributions are threefold. First, we derive interpolatory first-order optimality conditions for the linear quadratic-output [Formula: see text] minimization problem. These conditions correspond to the mixed-multipoint tangential interpolation of the full-order linear- and quadratic-output transfer functions, and generalize the Meier–Luenberger optimality framework for the [Formula: see text]-optimal model reduction of linear time-invariant systems. Second, given the optimal interpolation data, we show how to enforce the interpolatory optimality conditions explicitly by Petrov–Galerkin projection of the full-order model. Third, to find the optimal interpolation data, we build on this projection framework and propose a generalization of the iterative rational Krylov algorithm for the [Formula: see text]-optimal model reduction of linear quadratic-output systems, called [Formula: see text]. Upon convergence, [Formula: see text] produces reduced linear quadratic-output systems that satisfy the interpolatory optimality conditions. The method only requires solving shifted linear systems and matrix-vector products, thus making it suitable for large-scale problems. Numerical examples are included to illustrate the effectiveness of the proposed method. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://doi.org/10.5281/zenodo.18829841 . [Formula: see text]