Theory to explain AAALS performance and delineate failure cases

Characterize and explain the performance of the AAA-least squares (AAALS) method for solving two-dimensional Laplace Dirichlet problems on Jordan domains, and delineate the cases in which the AAALS method fails.

Background

The AAALS method approximates Dirichlet boundary data h on a Jordan curve Γ via AAA, then discards poles inside the domain and uses the remaining exterior poles as a basis for a least-squares approximation to the harmonic solution u in the interior. Empirically, the method is fast and accurate across various geometries, including challenging nonconvex domains.

Despite empirical success, the theoretical understanding of why AAALS performs well and where it fails is limited. The authors note that only the beginnings of a theory exist and indicate that advancing the theory will likely rely on properties of Schwarz functions and their singularities near curved boundaries.