Join us for this research seminar, part of the Numerical analysis and scientific computing seminar series.
The goal of model order reduction is to construct low-dimensional systems with the same structure and similar response characteristics as the original large-scale complex system. When- ever the model description is unknown or not available, the modeling and reduction steps have to be directly inferred from measured or computed data.
Rational approximation seeks to find a rational function that can approximate a given function by minimizing the distance between the two functions in some prescribed norm. In recent years, many methods have been proposed (for some of these, see 2-4) with corresponding high accuracy algorithms implemented in toolboxes such as Chebfun or RKToolbox.
The Loewner framework (LF) is a data-driven model reduction method that constructs high-fidelity models from measurements. The transfer function of the Loewner model is a rational function that interpolates the sampled frequency response. For an overview, see the book chapter in 1. For a recent note on approximating non-rational functions, see 5.
The LF relies on compressing the given data set and extracting meaningful quantities, instead of solving optimization problems through an iteration. It is a direct method that uses projection matrices chosen as the singular vector matrices of the corresponding Loewner matrix. The rational function computed through the LF is written in barycentric representation. We compare the accuracy of our method to that of others such as AAA in 3 (also a data-driven method) and minimax in 4 for approximating various data sets. We use as test cases non- rational functions typically encountered in approximation theory, such as the signum function as well as transfer functions corresponding to physical phenomena modeled as dynamical systems.
[1] A. C. Antoulas, S. Lefteriu and A. C. Ionita, A tutorial introduction to the Loewner framework for model reduction, In Model Reduction and Approximation, Chapter 8, pp. 335–376, 2017.
[2] M. Berljafa and S. Gu ?ttel, The RKFIT algorithm for nonlinear rational approximation, SIAM J. Sci. Comput., 39(5):A2049–A2071, 2017.
[3] Y. Nakatsukasa, O. Sete and L. N. Trefethen, The AAA algorithm for rational approximation, SIAM J. Sci. Comput., 40(3), A1494-A1522, 2018.
[4] S. I. Filip, Y. Nakatsukasa, L. N. Trefethen and B. Beckermann, Rational minimax approximation via adaptive barycentric representations, SIAM J. Sci. Comput., 40(4), A2427–A2455, 2018.
[5] D.S. Karachalios, I. V. Gosea, and A. C. Antoulas, Data-driven approximation methods applied to non- rational functions, Proceedings in Applied Mathematics and Mechanics 18 (1), 1–2, 2018.
