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

Alvise Sommariva, Marco Vianello

TL;DR

The paper tackles the persistent issue of unisolvence (invertibility) for unsymmetric Kansa collocation matrices in Poisson problems with Dirichlet boundary conditions on general domains. It introduces a broad class of analytic radial basis functions vanishing at infinity, including Gaussians, Generalized Inverse Multiquadrics, and Matérn RBFs, with scaled forms $\phi_\varepsilon$, and analyzes random interior collocation points together with fixed boundary points. The central contribution is a rigorous proof that the Kansa matrix is almost surely nonsingular for any $m\ge 1$ and $n\ge 0$, achieved via a determinant function $F(P)=\det(K(P))$ that is analytic and not identically zero, together with a limiting invertible matrix as points go to infinity. This yields unisolvence guarantees for these RBFs on general domains and informs practical choices of the shape parameter to balance conditioning and accuracy.

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.

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

TL;DR

The paper tackles the persistent issue of unisolvence (invertibility) for unsymmetric Kansa collocation matrices in Poisson problems with Dirichlet boundary conditions on general domains. It introduces a broad class of analytic radial basis functions vanishing at infinity, including Gaussians, Generalized Inverse Multiquadrics, and Matérn RBFs, with scaled forms , and analyzes random interior collocation points together with fixed boundary points. The central contribution is a rigorous proof that the Kansa matrix is almost surely nonsingular for any and , achieved via a determinant function that is analytic and not identically zero, together with a limiting invertible matrix as points go to infinity. This yields unisolvence guarantees for these RBFs on general domains and informs practical choices of the shape parameter to balance conditioning and accuracy.

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.
Paper Structure (2 sections, 2 theorems, 16 equations)

This paper contains 2 sections, 2 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Unisolvence of random Kansa collocation

Key Result

Theorem 1

Let $\phi:[0,+\infty)\to \mathbb{R}$ be a radial function such that: Moreover, $K_n$ be the Kansa collocation matrix by $\phi_\varepsilon(r)=\phi(\varepsilon r)$ defined above for equation (poisson) in $\Omega\subset \mathbb{R}^ d$, $d\geq 2$, where $\{Q_h\}$ is any fixed set of $m$ distinct points on $\partial\Omega$, and $\{P_i\}$ is a sequence of i.i.d. (independe

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 1