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Barycentric Interpolation Based on Equilibrium Potential

Kelong Zhao, shuhuang Xiang

TL;DR

BIEP extends barycentric interpolation to general bounded regions in the complex plane by tying node and pole placement to the logarithmic equilibrium potential. Nodes are computed from the equilibrium density via Symm's integral equation, enabling polynomial interpolation, while poles are introduced through a companion region $F$ for rational interpolation, yielding convergence rates tied to the potential and Robin constant $V_E$. The approach demonstrates near-optimal convergence for analytic functions and substantial improvements for near-singular problems, with practical applications including solving 2-D Laplace boundary-value problems. The work provides actionable algorithms, parameter guidelines, and open-source code, broadening the applicability of high-order interpolation beyond intervals and circles.

Abstract

We present a novel barycentric interpolation algorithm designed for analytic functions $f\in\mathcal{A}(E)$ defined on the complex plane. The algorithm, which encompasses both polynomial and rational interpolation, is tailored to handle singularities near $E$. Our method is applicable to regions $E$ bounded by piecewise smooth Jordan curves, and it imposes no connectivity restrictions on the region. The key feature of our approach lies in efficiently computing discrete points via the numerical solution of Symm's integral equation, enabling the construction of polynomial or rational barycentric interpolants. Furthermore, our method provides relevant parameters for the equilibrium potential, such as Robin's constant, which can be used to estimate convergence rates. Numerical experiments demonstrate the convergence rate achieved by our method in comparison to the theoretical convergence rate.

Barycentric Interpolation Based on Equilibrium Potential

TL;DR

BIEP extends barycentric interpolation to general bounded regions in the complex plane by tying node and pole placement to the logarithmic equilibrium potential. Nodes are computed from the equilibrium density via Symm's integral equation, enabling polynomial interpolation, while poles are introduced through a companion region for rational interpolation, yielding convergence rates tied to the potential and Robin constant . The approach demonstrates near-optimal convergence for analytic functions and substantial improvements for near-singular problems, with practical applications including solving 2-D Laplace boundary-value problems. The work provides actionable algorithms, parameter guidelines, and open-source code, broadening the applicability of high-order interpolation beyond intervals and circles.

Abstract

We present a novel barycentric interpolation algorithm designed for analytic functions defined on the complex plane. The algorithm, which encompasses both polynomial and rational interpolation, is tailored to handle singularities near . Our method is applicable to regions bounded by piecewise smooth Jordan curves, and it imposes no connectivity restrictions on the region. The key feature of our approach lies in efficiently computing discrete points via the numerical solution of Symm's integral equation, enabling the construction of polynomial or rational barycentric interpolants. Furthermore, our method provides relevant parameters for the equilibrium potential, such as Robin's constant, which can be used to estimate convergence rates. Numerical experiments demonstrate the convergence rate achieved by our method in comparison to the theoretical convergence rate.
Paper Structure (12 sections, 1 theorem, 48 equations, 12 figures, 3 algorithms)

This paper contains 12 sections, 1 theorem, 48 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Polynomial interpolation of the equilibrium potential
  3. Rational interpolation of the equilibrium potential
  4. Algorithms of the interpolation
  5. Polynomial interpolation based on equilibrium potential
  6. Rational interpolation based on equilibrium potential
  7. Experimental results
  8. Polynomial interpolants
  9. Rational interpolants for near-singularities functions
  10. Further discussions on the rational interpolation
  11. Application: Solving the Laplace equation
  12. Conclusion

Key Result

Theorem 2.2

\newlabelthm:den0 If $\partial E$ is a bounded piecewise simply smooth boundary and a family of point sets $\{x_i^{(n)}\}_{i=0}^n\subseteq \partial E$ obeys a positive density function $w$ of a unit measure $\mu$ on $\partial E$, then it holds and $\mu_n\stackrel{*}{\rightarrow}\mu$.

Figures (12)

  • Figure 1: The contours of the potential $U_{\mu_n}$ ($n=200$) in the 'lollipop' (left) and 'ice cream cone' (right) domains. These points (blue) resemble identically charged particles that repel each other and exhibit tip aggregation.
  • Figure 1: Schematic diagram for $\partial E$ is a curve segment.
  • Figure 1: Left: Interpolation nodes (blue) and the discrete potential $U_{\mu_n}$ generated by the 300 nodes. Right: The convergence rates compared with the theoretical convergence rates $\rho^n$, $\rho=\exp(-0.4180)$ (blue dashed), $\rho=\exp(-0.2248)$ (yellow-brown dashed) and $\rho=\exp(-0.1115)$ (violet dashed), respectively.
  • Figure 1: Potential contours of the nodes (blue) and poles (red) for $n=100$. The values in the figure are the potential differences $(c_1+c_2)$ corresponding to different $F$.
  • Figure 1: Left: Uniform norm error for approximating several harmonic functions using the real part of polynomial interpolation, for different $n$. Right: Pointwise error for $u(z)=\log|z-1|$ when $n=300$. The violet "+" marks the location of the singularity.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Definition 2.1
  • Theorem 2.2