The Loewner framework is an interpolatory framework for the approximation of linear and nonlinear systems. The purpose 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框架是一种用于线性与非线性系统逼近的插值框架。本文旨在将该框架扩展至具有任意数量n个参数的线性参数化系统。本文建立的核心创新之一是为任意参数数量构建基于数据的实现方法。同样重要的是，我们通过避免显式构造规模为$N \times N$的n维Loewner大矩阵，展示了如何减轻计算负担。这将复杂度从$O(N^3)$降低至约$O(N^{1.4})$，从而驯服了维数灾难，并使解决方案可扩展至超大规模数据集。为实现这一目标，我们首先定义了一种新的广义多元有理函数实现方法。随后，我们引入了n维多元Loewner矩阵，并证明其可通过求解一组耦合的Sylvester方程进行计算。这些Loewner矩阵的零空间允许构造多元重心传递函数。本研究的主要成果是展示了如何通过一系列一维Loewner矩阵计算n维Loewner矩阵的零空间，从而大幅降低计算负担。最后，我们提出了两种算法（一种直接算法和一种迭代算法），可直接从数据构造确保（近似）插值的多元（或参数化）实现。数值算例突显了该方法的有效性和可扩展性。