Random Leja points

TL;DR

This paper introduces new random Leja point methods for polynomial interpolation on complex compact sets, offering computationally feasible and modular alternatives with proven convergence properties for holomorphic functions.

Contribution

It proposes novel random Leja point families, including an approximate Metropolis-Hastings version, and analyzes their convergence and implementation advantages over existing methods.

Findings

Random Leja points almost surely lead to convergent interpolation polynomials.

The proposed methods are more modular and easier to implement than deterministic alternatives.

Numerical experiments show competitive accuracy and computational efficiency.

Abstract

Leja points on a compact $K \subset \mathbb{C}$ are known to provide efficient points for interpolation, but their actual implementation can be computationally challenging. So-called pseudo Leja points are a more tractable solution, yet they require a tailored implementation to the compact at hand. We introduce several more flexible random alternatives, starting from a new family we call random Leja points. To make them tractable, we propose an approximate version which relies on the Metropolis-Hastings algorithm with the uniform measure. We also analyse a different family of points inspired by recently introduced randomised admissible meshes, obtained by uniform sampling. When the number of iterations or drawn points is appropriately chosen, we establish that the two resulting families of points provide good points for interpolation. That is, they almost surely lead to convergent interpolating polynomials for holomorphic functions. The two last families of points are readily implemented assuming one knows how to sample uniformly at random in $K$. These makes them more modular than competing deterministic methods. We run numerical experiments to compare the proposed methods in terms of accuracy and computational complexity, for various types of compact sets.