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.

翻译：Loewner框架是一种用于逼近线性和非线性系统的插值方法。本文旨在将该框架扩展至具有任意n个参数的线性参数化系统。主要创新在于建立了适用于任意数量参数的数据驱动实现方法。同样重要的是，我们展示了如何通过避免显式构建规模为$N \times N$的大型n维Loewner矩阵来减轻计算负担。这将计算复杂度从$O(N^3)$降低至约$O(N^{1.4})$，从而有效应对维度灾难，使该方法能够扩展到超大规模数据集。为实现这一目标，我们定义了一种新的广义多元有理函数实现形式。随后引入n维多变量Loewner矩阵，并证明其可通过求解耦合Sylvester方程组进行计算。这些Loewner矩阵的零空间可用于构建多变量重心传递函数。本工作的核心成果在于展示了如何利用一维Loewner矩阵序列计算n维Loewner矩阵的零空间，从而实现计算负担的急剧降低。最后，我们提出两种算法（直接法与迭代法）直接从数据构建能保证（近似）插值特性的多变量（参数化）实现。数值算例验证了该方法的有效性与可扩展性。