Received: 07 August 2018 Accepted: 15 August 2018 DOI: 10.1002/pamm.201800368 Data-driven approximation methods applied to non-rational functions. Dimitrios S. Karachalios 1, * , Ion Victor Gosea 1,** , and Athanasios C. Antoulas 1,2,*** 1 Max Planck Institute for Dynamics of Complex Technical Systems, Data-Driven System Reduction and Identification (DRI) group, Sandtorstraße 1 Magdeburg 39106. 2 Rice University and Baylor College of Medicine, Houston 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. c  2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Computational linear algebra deals with one of the most challenging problems encountered in recent years, that of the big data analysis. In combination with control theory, the problem mainly focuses on reducing and approximating input-output complex systems with low order models [1]. This note focuses on the interpolation-based reduction in the Loewner framework, which can construct low order models from data. The inverse of the Bessel function of the first kind is selected as a test example in order to present some of the challenges arising in approximation theory. In particular, the Bessel function is non-rational and is defined over the complex plane, so we have to deal with an approximation problem in a two-dimensional domain. Moreover, it is known that such non-rational functions are feasible with linear systems of infinite dimension. The Bessel function of the first kind and of m ∈ N order is defined as the following contour integral eq.[1]. J m (s)= 1 2πi  e ( s 2 )(t- 1 t ) t -m-1 dt. (1) The m =0 case will be examined in this study. Our goal is to approximate the non-rational function H(s)= 1 J0(s) ,s ∈ C in the rectangle domain Ω = [0, 10] × [−1, 1] ⊂ C by a rational function such as the G(s)= C(sE − A) -1 B + D, s ∈ Ω. The following notation (E, A, B, C, D) indicates a linear time invariant system in a descriptor-form representation [2]. 2 Methods The Loewner Framework Given a set of pairs of complex numbers with the conjugates, {(s k , φ k ): k =1, ..., N =2n} with s k ∈ C, φ k ∈ C, we devided the data in two disjoint sets, the left data as: (µ i ,ν i ), i =1, ..., n and the right data as: (λ j ,w j ), j =1, ..., n. We considered only the SISO 1 case with real symmetry 2 . The objective is to find H r (s) ∈ C such that: H r (µ i ) ≈ ν i and H r (λ j ) ≈ w j with r ∈ N and r ≪ n. The associated Loewner and shifted Loewner matrices L and L s are defined as: L (ij) = ν i − w j µ i − λ j , L s(ij) = µ i ν i − w j λ j µ i − λ j , i, j =1, ..., n. (2) Loewner SVD (Loew-SVD) In the usual case of redundant data, we used the rank revealing SVD factorization of the Loewner matrices as described in [2]. For the singular Loewner pencil (L, L s ), applying SVD we chose the first r left and right singular vectors, as projection matrices X ∈ C n×r , Σ ∈ R r×r , Y ∈ C n×r . The uncompressed singular model is: {E = −L, A = −L s , B = V, C = W}. An approximate interpolant is constructed as described in the following theorem: Theorem 2.1 The quadruple { ˆ E = −X * LY, ˆ A = −X * L s Y, ˆ B = X * V, ˆ C = WY}, is then the realization of an approximate data interpolant of order r. The approximant can be explicitly computed as: H r (s)= ˆ C(s ˆ E − ˆ A) -1 ˆ B with s ∈ C. Loewner CUR (Loew-CUR) Another approach to getting a low rank approximant H r is to replace the SVD with the CUR decomposition so as to reduce the time complexity. The following algorithm allow us to obtain a matrix decomposition which in general preserves the sparsity and the physical meaning of the initial data [6, 7]. In CUR decomposition, C stands for the columns, R for the rows and U is a co-prime matrix with a low rank. ∗ Corresponding author: karachalios@mpi-magdeburg.mpg.de,**gosea@mpi‐magdeburg.mpg.de,***aca@rice.edu 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 symmetry ¯ H(s)= H(¯ s): we can obtain a real model [2]. Where ¯ (·) denotes the complex conjugate and (·) ∗ the complex conjugate transpose. PAMM · Proc. Appl. Math. Mech. 2018;18:e201800368. www.gamm-proceedings.com c  2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 of 2 https://doi.org/10.1002/pamm.201800368