A refined nonlinear least-squares method for the rational approximation problem

TL;DR

This work tackles the tendency of the adaptive AAA algorithm to produce high-degree rational approximants to meet a prescribed tolerance $\tau$. It proposes NL-AAA, a nonlinear least-squares refinement that, in combination with a greedy interpolation strategy, aims to meet the tolerance with the smallest feasible degree $k$ while guaranteeing monotone error decay. The authors derive Wirtinger gradients for the true nonlinear error and its Levy, SK, and WF refinements, showing that Whitfield’s iteration aligns with the true $L^2$ objective at convergence, whereas Levy and SK may not. Numerical experiments on synthetic function-approximation tasks and model-order-reduction benchmarks (including ISS and a vibrating plate) demonstrate that NL-AAA achieves smaller degrees with more reliable, monotone convergence and improved accuracy over classical AAA, highlighting its practical impact for data-driven modeling and reduced-order modeling. The work thus provides both theoretical insights and practical algorithms for robust, low-order rational approximation in scientific computing.

Abstract

The adaptive Antoulas-Anderson (AAA) algorithm for rational approximation is a widely used method for the efficient construction of highly accurate rational approximations to given data. While AAA can often produce rational approximations accurate to any prescribed tolerance, these approximations may have degrees larger than what is actually required to meet the given tolerance. In this work, we consider the adaptive construction of interpolating rational approximations while aiming for the smallest feasible degree to satisfy a given error tolerance. To this end, we introduce refinement approaches to the linear least-squares step of the classical AAA algorithm that aim to minimize the true nonlinear least-squares error with respect to the given data. Furthermore, we theoretically analyze the derived approaches in terms of the corresponding gradients from the resulting minimization problems and use these insights to propose a new greedy framework that ensures monotonic error convergence. Numerical examples from function approximation and model order reduction verify the effectiveness of the proposed algorithm to construct accurate rational approximations of small degrees.

Figures (5)

• Figure 1: Convergence behavior of the classical AAA algorithm and the newly proposed NL-AAA method on two test functions: In both cases, NL-AAA decreases the error monotonically. While for the absolute value, both methods show a very similar performance, for the ReLU function, the proposed method obtains smaller approximation errors significantly faster than the classical AAA approach.
• Figure 2: Denominator functions of degree $k = 14$ of the AAA approximations to $\lvert x \rvert$ and $\textrm{relu}(x)$ on the real interval $[-1, 1]$: While the variation of the approximating denominator for $\lvert x \rvert$ is constrained to only two order of magnitude, the denominator varies nearly $17$ orders of magnitude in the approximation of $\textrm{relu}(x)$.
• Figure 3: Convergence and approximation results of AAA and NL-AAA for the functions $\lvert \sin(3\pi x) \rvert$ and $\textrm{triWave}(x)$: For both example functions, the classical AAA has a very unsteady error convergence while the proposed NL-AAA provides a monotonic decay. For the $\textrm{triWave}$ function, the final approximation by AAA shows visible deviations from the given data while NL-AAA provides an indistinguishable approximation.
• Figure 4: Convergence and approximation results of AAA and NL-AAA for the transfer function of the international space station model: The proposed NL-AAA provides a monotonic convergence in the $\ell_{2}$ error but also convinces in the $\ell_{\infty}$ error. For larger model orders, the error of the classical AAA is visibly larger than for NL-AAA in both error metrics.
• Figure 5: Convergence and approximation results of AAA and NL-AAA for the transfer function of the vibrating plate: The proposed NL-AAA method provides a smooth and reliable decaying error behavior in both considered error metrics, while the classical AAA has rapid changes in its approximation quality with an overall larger error. For model order $50$, the frequency response shows that AAA introduces localized inaccuracies while NL-AAA fits the data without visible differences.

Theorems & Definitions (5)

• Remark 1
• Lemma 1
• proof
• Theorem 1
• proof