Data‐driven approximation methods applied to non‐rational functions

Abstract

In this study, four data-driven interpolation-based methods are compared. The aim is to construct reduced-order models for which the corresponding rational transfer function matches the original non-rational one at selected interpolation points. The primary method of this study is the Loewner framework [2] which addresses this problem in a natural and direct way. The other methods that were studied, vector fitting (VF) [5], and adaptive Antoulas-Anderson (AAA) [4], are instead based on an iterative and adaptive selection procedure. In order to present the adaptive selection as a feature in the Loewner framework and to reduce the time complexity at the same time, we introduced a Loewner CUR method based on the cross approximation algorithm [6,7]. The performance of the above methods is tested by a classical example in approximation theory: the inverse of the Bessel function of the first kind. 1 SISO: Single Input Single Output. Where E, A ∈ C n×n and B, C T ∈ C n×1 , and a scalar D ∈ C. 2 real symmetryH(s) = H(s): we can obtain a real model [2]. Where(·) denotes the complex conjugate and (·) * the complex conjugate transpose.

A bilinear identification‐modeling framework from time domain data

“…Ar, Br, Cr}, linear Loewner framework[2,4]. ;3 H 2r (jω k , jω k ) = Cr(2jω k Er − Ar) −1 O: gen. observability Nr (jω k Er − Ar) −1 Br R: gen.controllability ≈ 2Y II (jω k ) U (jω k ) 2 ;4 Identify the matrix Nr by solving the system: H 2r (jω i , jω i ) = O (i,:) NrR (:,i) ⇒ H 2r (jω i , jω i ) = O (i,:) ⊗ R (:,i) vec(Nr), i = 1, ..., n.;…”

An Efficient, Memory-Saving Approach for the Loewner Framework

“…Low-rank approximations to and may be computed by adaptive cross approximation [ 4 ], particularly suited for hierarchical matrices, the CUR decomposition [ 11 ] as in [ 22 , 38 ], or related schemes. These approaches select a certain number of columns and rows of the original matrices in a greedy fashion based on various heuristics, and a core matrix is utilised to compute a low-rank approximation.…”

“…Moreover, the memory constraints coming from the allocation of and are still present. Alternatively, one can focus on accelerating solely the step of the SVD calculation by employing Krylov methods (see, e.g., [ 3 , 19 , 25 , 41 ] to name a few), by using the randomized SVD [ 31 ] to compute the dominant singular triplets instead of the full SVD or other types of inexact SVD-type decompositions (adaptive cross approximation [ 4 ], particularly suited for hierarchical matrices, or a CUR decomposition [ 11 ] as in [ 22 , 38 ]).…”

Model order reduction approach to the one-dimensional collisionless closure problem

“…This can be done using the so-called Loewner framework 33 . The method has found wide applications, not limited to model order reduction [34][35][36][37][38] . First, let us generalise the problem.…”