  The Loewner framework is an interpolatory approach designed for approximating
linear and nonlinear systems. The goal here is to extend this framework to
linear parametric systems with an arbitrary number n of parameters. One main
innovation established here is the construction of data-based realizations for
any number of parameters. Equally importantly, we show how to alleviate the
computational burden, by avoiding the explicit construction of large-scale
n-dimensional Loewner matrices of size $N \times N$. This reduces the
complexity from $O(N^3)$ to about $O(N^{1.4})$, thus taming the curse of
dimensionality and making the solution scalable to very large data sets. To
achieve this, a new generalized multivariate rational function realization is
defined. Then, we introduce the n-dimensional multivariate Loewner matrices and
show that they can be computed by solving a coupled set of Sylvester equations.
The null space of these Loewner matrices then allows the construction of the
multivariate barycentric transfer function. The principal result of this work
is to show how the null space of the n-dimensional Loewner matrix can be
computed using a sequence of 1-dimensional Loewner matrices, leading to a
drastic computational burden reduction. Finally, we suggest two algorithms (one
direct and one iterative) to construct, directly from data, multivariate (or
parametric) realizations ensuring (approximate) interpolation. Numerical
examples highlight the effectiveness and scalability of the method.
