arXiv subjects

Stefano Costa

Publications and source records attributed to Stefano Costa.

4 recordsLinked to original sources

arXiv2026

AAA least squares solution of Helmholtz problems

This paper presents an adaptive numerical framework for solving exterior "sound-soft" scattering problems governed by the Helmholtz equation. By interpreting the Method of Fundamental Solutions through the lens of rational approximation, we introduce an automated strategy for singularity placement based on the analytic continuation of boundary data. The proposed AAALS-Helmholtz algorithm leverages a "continuum" variant of the AAA algorithm to identify the singularities limiting analytic extension, and to ensure an optimal source distribution even for complex, non star-shaped geometries. Furthermore, we establish a formal connection between the Helmholtz and Laplace problems, providing a theoretical justification for the "double poles" technique. The approach offers a robust, meshless alternative to heuristic source placement.

math.NA↗

arXiv2021

AAA-least squares rational approximation and solution of Laplace problems

A two-step method for solving planar Laplace problems via rational approximation is introduced. First complex rational approximations to the boundary data are determined by AAA approximation, either globally or locally near each corner or other singularity. The poles of these approximations outside the problem domain are then collected and used for a global least-squares fit to the solution. Typical problems are solved in a second of laptop time to 8-digit accuracy, all the way up to the corners, and the conjugate harmonic function is also provided. The AAA-least squares combination also offers a new method for avoiding spurious poles in other rational approximation problems, and for greatly speeding them up in cases with many singularities. As a special case, AAA-LS approximation leads to a powerful method for computing the Hilbert transform or Dirichlet-to-Neumann map.

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arXiv2020

Solving Laplace problems with the AAA algorithm

We present a novel application of the recently developed AAA algorithm to the solution of Laplace 2D problems; an application to conformal mapping is also shown as a particular case. These classes of problems have also been addressed by means of dedicated software implementations based on rational function approximation, with unprecedented speed and accuracy. Still we show how the AAA algorithm, conceived to be a flexible and domain independent method, is capable of addressing these situations within a unified framework.

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arXiv2018

An efficient procedure for solving potential field problems: the Conformal Boundary Differences Method

A novel method rooted in the classical Schwarz-Christoffel transformation from the disk is introduced, which allows for fast and accurate solution of potential field problems in possibly inhomogeneous and multiply connected domains: this is for sure its most outstanding feature, circumventing the barriers that have increasingly restricted the scope of conformal mappings in applications since the advent of computers and purely numerical methods. An example problem, derived from a case of practical interest, is analyzed and results are compared with those obtained from FEA.

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