Abstract:The traditional view in numerical conformal mapping is that once the boundary correspondence function has been found, the map and its inverse can be evaluated by contour integrals. We propose that it is much simpler, and 10-1000 times faster, to represent the maps by rational functions computed by the AAA algorithm. To justify this claim, first we prove a theorem establishing root-exponential convergence of rational approximations near corners in a conformal map, generalizing a result of D. J. Newman in 1964. … Show more
“… Fig. 3 The conformal map of a circular pentagon onto the unit disk has been computed and then approximated numerically by a rational function of degree 70 [ 14 , 60 ] by the AAA algorithm. The poles cluster exponentially at the corners, where the map is singular …”
Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning
confidence: 99%
“…Figure 3 turns to our first problem of scientific computing. Following methods presented in [ 14 , 60 ], a region E in the complex plane bounded by three line segments and two circular arcs has been conformally mapped onto the unit disk, and the map has then been approximated to about eight digits of accuracy by AAA approximation, which finds a rational function with . This process is entirely adaptive, based on no a priori information about corners or singularities, yet it clusters the poles near the corners just as in Figs.…”
Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning
confidence: 99%
“…Newman’s theorem has been a great stimulus to further research in rational approximation theory [ 11 – 13 , 23 , 36 , 39 , 41 , 62 ]. It has not, however, had much impact on scientific computing until very recently with the discovery that it can be the basis of root-exponentially converging numerical methods for the solution of partial differential equations (PDEs) in domains with corner singularities [ 6 , 14 – 16 , 59 ]. The aim of this paper is to contribute to building the bridge between approximation theory and numerical computation.…”
“… Fig. 3 The conformal map of a circular pentagon onto the unit disk has been computed and then approximated numerically by a rational function of degree 70 [ 14 , 60 ] by the AAA algorithm. The poles cluster exponentially at the corners, where the map is singular …”
Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning
confidence: 99%
“…Figure 3 turns to our first problem of scientific computing. Following methods presented in [ 14 , 60 ], a region E in the complex plane bounded by three line segments and two circular arcs has been conformally mapped onto the unit disk, and the map has then been approximated to about eight digits of accuracy by AAA approximation, which finds a rational function with . This process is entirely adaptive, based on no a priori information about corners or singularities, yet it clusters the poles near the corners just as in Figs.…”
Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning
confidence: 99%
“…Newman’s theorem has been a great stimulus to further research in rational approximation theory [ 11 – 13 , 23 , 36 , 39 , 41 , 62 ]. It has not, however, had much impact on scientific computing until very recently with the discovery that it can be the basis of root-exponentially converging numerical methods for the solution of partial differential equations (PDEs) in domains with corner singularities [ 6 , 14 – 16 , 59 ]. The aim of this paper is to contribute to building the bridge between approximation theory and numerical computation.…”
“…However, virtually any exponential clustering is in fact sufficient, provided it scales with n −1/2 as n → ∞, and Section 2 is devoted to presenting theorems to establish this claim. Quite apart from their application to Laplace problems, we believe these results represent a significant addition to the approximation theory literature, as well as shedding light on the clustered poles observed experimentally in [15] and [23]. Section 3 describes our algorithm, which depends on placing sample points on the boundary with exponential clustering to match that of the poles outside.…”
A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a result of D. J. Newman in 1964 in approximation theory, we first prove that such approximations can achieve root-exponential convergence for a wide range of problems, all the way up to the corner singularities. We then develop a numerical method to compute approximations via linear least-squares fitting on the boundary. Typical problems are solved in < 1s on a desktop to 8-digit accuracy, with the accuracy guaranteed in the interior by the maximum principle. The computed solution is represented globally by a single formula, which can be evaluated in a few microseconds at each point.
“…In [12], the Adaptive Antoulas-Anderson (AAA) greedy algorithm for computing a barycentric rational approximant has been presented. This recent method leads to impressively well-conditioned bases, which can be used in various fields, such as in computing conformal maps, or in rational minimax approximations (see [13,14]). Note that a similar approach has been considered in [15] for kernel-based interpolation.…”
In this work, we extend the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater-Hormann and to AAA approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in De Marchi et al. (2020). Numerical tests show that it yields an accurate approximation of discontinuous functions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
