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

TL;DR

This paper proves the nonsingularity of Kansa collocation matrices with polyharmonic splines and random centers for elliptic equations, advancing understanding of their unisolvence and demonstrating robustness through numerical tests.

Contribution

It establishes the nonsingularity of Kansa matrices with polyharmonic splines and random centers, addressing an open problem in unisolvence for elliptic equations.

Findings

Nonsingularity of Kansa matrices with random centers proven.

Numerical tests show robustness of the method with perturbed centers.

Advances understanding of unisolvence in collocation methods.

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.