Unisolvence of unsymmetric random Kansa collocation by Gaussians and other analytic RBF vanishing at infinity

TL;DR

This paper proves that certain analytic radial basis function collocation matrices are almost surely invertible for solving Poisson equations with Dirichlet boundary conditions, regardless of domain or point distribution.

Contribution

It provides a short proof of invertibility for unsymmetric random Kansa collocation matrices using analytic RBFs that vanish at infinity, applicable in general settings.

Findings

Invertibility holds almost surely for a broad class of analytic RBFs.

The proof applies to arbitrary domains and distributions of collocation points.

Includes popular RBFs like Gaussians, Inverse MultiQuadrics, and Matern functions.

Abstract

We give a short proof of almost sure invertibility of unsymmetric random Kansa collocation matrices by a class of analytic RBF vanishing at infinity, for the Poisson equation with Dirichlet boundary conditions. Such a class includes popular Positive Definite instances such as Gaussians, Generalized Inverse MultiQuadrics and Matern RBF. The proof works on general domains in any dimension, with any distribution of boundary collocation points and any continuous random distribution of internal collocation points.