DeepAI
Log In Sign Up

AAA-least squares rational approximation and solution of Laplace problems

07/04/2021
by   Stefano Costa, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

05/08/2019

Solving Laplace problems with corner singularities via rational functions

A new method is introduced for solving Laplace problems on 2D regions wi...
01/26/2020

Solving Laplace problems with the AAA algorithm

We present a novel application of the recently developed AAA algorithm t...
11/09/2019

Numerical conformal mapping with rational functions

New algorithms are presented for numerical conformal mapping based on ra...
08/16/2019

An algorithm for real and complex rational minimax approximation

Rational minimax approximation of real functions on real intervals is an...
11/21/2020

Zolotarev's fifth and sixth problems

In an influential 1877 paper, Zolotarev asked and answered four question...
01/14/2023

Nonlinear approximation of functions based on non-negative least squares solver

In computational practice, most attention is paid to rational approximat...
09/22/2015

Modifying iterated Laplace approximations

In this paper, several modifications are introduced to the functional ap...