Unisolvence of Kansa collocation for elliptic equations by polyharmonic splines with random fictitious centers

TL;DR

The paper addresses the unisolvence (invertibility) of unsymmetric Kansa collocation for elliptic PDEs using polyharmonic splines with random/fictitious centers, focusing on the Poisson equation with Dirichlet boundary conditions. It proves almost-sure nonsingularity of the MQ/IMQ Kansa matrix in any dimension when interior centers $P_i$ are i.i.d. from a density $\sigma\in L^1_+(\Omega)$ and boundary centers $Q_h\in\partial\Omega$, with $\phi(r)=\sqrt{1+(epsilon r)^2}$ (MQ) or $\phi(r)=1/\sqrt{1+(epsilon r)^2}$ (IMQ). The argument combines a determinant representation and a real-analytic/nonzero-analytic function argument in the complex plane, showing branching points at $z=\pm i/\varepsilon$ prevent a zero determinant on a set of positive measure; the result is established by induction on the interior-point count. This provides a theoretical basis for using random/fictitious centers in meshfree Kansa methods and broadens unisolvence results beyond boundary-analytic settings.

Abstract

We make a further step in the unisolvence open problem for unsymmetric Kansa collocation, proving nonsingularity of Kansa matrices with polyharmonic splines and random fictitious centers, for second-order elliptic equations with mixed boundary conditions. We also show some numerical tests, where the fictitious centers are local random perturbations of predetermined collocation points.