Parametric model order reduction with a small \(\mathcal {H}_{2}\)-error using radial basis functions

Given optimal interpolation points sigma sub(1),..., sigma sub( r ), the \(\mathcal {H}_{2}\)-optimal reduced order model of order r can be obtained for a linear time-invariant system of order n>>r by simple projection (whereas it is not a trivial task to find those interpolation points). Our approach to linear time-invariant systems depending on parameters \(p\in \mathbb {R}d}\) is to approximate their parametric dependence as a so-called metamodel, which in turn allows us to set up the corresponding parametrized reduced order models. The construction of the metamodel we suggest involves the coefficients of the characteristic polynomial and radial basis function interpolation, and thus allows for an accurate and efficient approximation of sigma sub(1)(p),..., sigma sub( r )(p). As the computation of the projection still includes large system solves, this metamodel is not sufficient to construct a fast and truly parametric reduced system. Setting up a medium-size model without extra cost, we present a possible answer to this. We illustrate the proposed method with several numerical examples.