  When balanced truncation is used for model order reduction, one has to solve
a pair of Lyapunov equations for two Gramians and uses them to construct a
reduced-order model. Although advances in solving such equations have been
made, it is still the most expensive step of this reduction method. Parametric
model order reduction aims to determine reduced-order models for
parameter-dependent systems. Popular techniques for parametric model order
reduction rely on interpolation. Nevertheless, the interpolation of Gramians is
rarely mentioned, most probably due to the fact that Gramians are symmetric
positive semidefinite matrices, a property that should be preserved by the
interpolation method. In this contribution, we propose and compare two
approaches for Gramian interpolation. In the first approach, the interpolated
Gramian is computed as a linear combination of the data Gramians with positive
coefficients. Even though positive semidefiniteness is guaranteed in this
method, the rank of the interpolated Gramian can be significantly larger than
that of the data Gramians. The second approach aims to tackle this issue by
performing the interpolation on the manifold of fixed-rank positive
semidefinite matrices. The results of the interpolation step are then used to
construct parametric reduced-order models, which are compared numerically on
two benchmark problems.
