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 whose boundary has an analytic parametrization.

中文翻译：

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泊松方程薄板样条随机堪萨配置的无解性


偏微分方程 Kansa 非对称配置无法解决的充分条件是否存在仍然是一个悬而未决的问题。在本文中，我们朝这个方向迈出了第一步，证明当在边界具有解析参数化的域上随机选择离散化点时，二维泊松方程的薄板样条的不对称配置矩阵几乎肯定是非奇异的。