Fast, matrix-free matrix-vector product with the Löewner matrix

The Löwner framework is one of the most successful data-driven model order reduction techniques. Given k right interpolation data and h left interpolation data, the standard layout of this approach is composed of two stages. First, the kh x kh Löwner matrix L and shifted Löwner matrix S are constructed. Then, an SVD of L-ζS, ζ belonging to one of the data sets, provides the projection matrices used to compute the sought reduced model. These two steps become numerically challenging for large k and h in terms of both computational time and storage demand. We show how the structure of L and S can be exploited to reduce the cost of performing (L-ζS)x while avoiding the explicit allocation of L and S.