We give a short proof of almost sure invertibility of unsymmetric random Kansa collocation matrices by a class of analytic RBF vanishing at infinity, for the Poisson equation with Dirichlet boundary conditions. Such a class includes popular Positive Definite instances such as Gaussians, Generalized Inverse MultiQuadrics and Matern RBF. The proof works on general domains in any dimension, with any distribution of boundary collocation points and any continuous random distribution of internal collocation points.

翻译：针对带有Dirichlet边界条件的泊松方程，我们给出了一类在无穷远处衰减的解析径向基函数所构成的非对称随机Kansa配点矩阵几乎必然可逆的简洁证明。该类函数包含高斯函数、广义逆多二次函数及Matern径向基函数等常用的正定核函数。该证明适用于任意维度的广义区域，且对边界配点分布无特殊要求，内部配点可采用任意连续随机分布。