Upcoming Event: Oden Institute Seminar

The Loewner Framework and the Kolmogorov Superposition Theorem

Thanos Antoulas, Rice University

Abstract

The Loewner framework is an interpolatory approach for the approximation of linear and nonlinear systems. The purpose here is to extend this framework to linear systems depending on several parameters.To achieve this, we introduce the n-dimensional multivariate Loewner matrices and show that they can be computed by solving a set of coupled Sylvester equations. The null space of these Loewner matrices allows the construction of multivariate rational functions in barycentric form.

Surprisingly, the null space of n-dimensional Loewner matrices can be computed using a sequence of one-dimensional Loewner matrices. As a consequence, a decoupling of the variables is achieved, which leads to a drastic reduction of the computational burden. This decoupling also leads to a result similar to the Kolmogorov Superposition Theorem, for rational functions. 

Concluding, we suggest two algorithms for constructing, directly from data, multivariate (parametric) realizations ensuring (approximate) interpolation. Numerical examples highlight the effectiveness and scalability of the method.

Reference: A.C. Antoulas, I.V. Gosea, C. Poussot-Vassal, On the Loewner framework, the Kolmogorov Superposition Theorem, and the curse of dimensionality, SIAM Review, Volume 67, Pages 737-770 (2025).

Biography

Thanos Antoulas is a professor of Electrical and Computer Engineering at Rice University. He is also an adjunct professor at the Baylor College of Medicine, Houston. Between 2016 and 2022 he was Max-Planck Fellow at the MPI in Magdeburg. His research interests are in data-driven modeling and model reduction. His contributions include the development of the Loewner framework for linear and non-linear systems. He is the author of the book "Model reduction of large-scale systems", published by SIAM, 20 years ago, and considered a standard textbook in this area.