Unisolvence of random Kansa collocation by Thin-Plate Splines for the Poisson equation

TL;DR

This paper proves that for the 2D Poisson equation, the collocation matrices using Thin-Plate Splines are almost surely nonsingular when points are randomly chosen on analytic domains, advancing understanding of unisolvence in PDE collocation methods.

Contribution

It establishes the almost sure nonsingularity of collocation matrices with Thin-Plate Splines for the 2D Poisson equation under random point selection, addressing an open problem.

Findings

Collocation matrices are almost surely nonsingular with random points.

Supports the use of Thin-Plate Splines in PDE collocation methods.

Provides theoretical foundation for unisolvence in this context.

Abstract

Existence of sufficient conditions for unisolvence of Kansa unsymmetric collocation for PDEs is still an open problem. In this paper we make a first step in this direction, proving that unsymmetric collocation matrices with Thin-Plate Splines for the 2D Poisson equation are almost surely nonsingular, when the discretization points are chosen randomly on domains with analytic boundary.