Algorithms for the Rational Approximation of Matrix-Valued Functions

Abstract

A selection of algorithms for the rational approximation of matrix-valued functions are discussed, including variants of the interpolatory adaptive Antoulas--Anderson (AAA) method, the rational Krylov fitting (RKFIT) method based on approximate least squares fitting, vector fitting, and a method based on low-rank approximation of a block Loewner matrix. A new method, called the block-AAA algorithm, based on a generalized barycentric formula with matrix-valued weights, is proposed. All algorithms are compared in terms of obtained approximation accuracy and runtime on a set of problems from model order reduction and nonlinear eigenvalue problems, including examples with noisy data. It is found that interpolation-based methods are typically cheaper to run, but they may suffer in the presence of noise for which approximation-based methods perform better.

Data-driven modeling of linear dynamical systems with quadratic output in the AAA framework

“…AAA is an iterative algorithm and builds the partitioning (9) using a greedy search. Assume in step n, AAA has the rational approximant r(s) as in (12) corresponding to the partitioning (9) where the weights {w k } are selected by solving (15). AAA updates (9) via a greedy search by finding ξi ∈ ξ for which the error | r( ξi ) − ĥi | is the largest.…”

The AAAtrig algorithm for rational approximation of periodic functions

“…The error is compared to that of a type (m, m) trigonometric rational function computed by AAAtrig in the left panel of figure 3. AAAtrig clearly provides an excellent approximant with a type (22,22) trigonometric rational function. The FFT method does not reach comparable accuracy until nearly 1,000 terms are used.…”

“…Given its effectiveness, it is surprisingly simple: the original article presented a Matlab implementation in 40 lines of code. Various extensions of AAA have been developed for minimax approximation [29], matrix-valued functions [22], surrogate functions [17] and parametric dynamical systems [31]. In the present work we present an enhancement of AAA for approximating periodic (trigonometric) functions, called 'AAAtrig'.…”

On Loewner data-driven control for infinite-dimensional systems

“…AAA can also be used as a LTI modeling method since it yields a reduced-order rational function that can be interpreted as the transfer function of the surrogate reduced order model (ROM). The AAA algorithm has recently extended for modeling of parametrized dynamics in [9], and for approximation of matrix-valued functions in [11]. Finally, the Vector-Fitting (VF) method is based solely on least squares approximation, and can be also applied for surrogate modeling design.…”

“…Additionally, as for the Loewner method, we will enforce real-valued models. Finally, we restrict the presentation to the SISO case (the MIMO case was addressed in [11]).…”