Core Concepts
Kansa collocation with Thin-Plate Splines ensures unisolvence for the Poisson equation with randomly chosen discretization points.
Abstract
Abstract:
- Unisolvence conditions for Kansa unsymmetric collocation for PDEs are unresolved.
- Thin-Plate Splines with random discretization points on analytic domains ensure nonsingular collocation matrices.
Introduction:
- Kansa collocation is widely used but lacks a theoretical foundation for unisolvence.
- Greedy approaches address singularity issues in collocation matrices.
Unisolvence of Random Kansa Collocation:
- Thin-Plate Splines without polynomial addition guarantee nonsingular collocation matrices.
- TPS are scale-invariant and avoid scaling issues with RBF.
Data Extraction:
- "Kansa unsymmetric collocation, originally proposed in the mid ’80s [11], has become over the years a popular meshless method for the numerical solution of boundary value problems for PDEs."
- "TPS without polynomial addition can guarantee unisolvence in the interpolation framework."
Stats
"Kansa unsymmetric collocation, originally proposed in the mid ’80s [11], has become over the years a popular meshless method for the numerical solution of boundary value problems for PDEs."
"TPS without polynomial addition can guarantee unisolvence in the interpolation framework."
Quotes
"Since the numerical experiments by Hon and Schaback show that Kansa’s method cannot be well-posed for arbitrary center locations, it is now an open question to find sufficient conditions on the center locations that guarantee invertibility of the Kansa matrix."