We describe an efficient method for the approximation of functions using radial basis functions (RBFs), and extend this to a solver for boundary value problems on irregular domains. The method is based on RBFs with centers on a regular grid defined on a bounding box, with some of the centers outside the computational domain. The equation is discretized using collocation with oversampling, with collocation points inside the domain only, resulting in a rectangular linear system to be solved in a least squares sense. The goal of this paper is the efficient solution of that rectangular system. We show that the least squares problem splits into a regular part, which can be expedited with the FFT, and a low rank perturbation, which is treated separately with a direct solver. The rank of the perturbation is influenced by the irregular shape of the domain and by the weak enforcement of boundary conditions at points along the boundary. The solver extends the AZ algorithm which was previously proposed for function approximation involving frames and other overcomplete sets. The solver has near optimal log-linear complexity for univariate problems, and loses optimality for higher-dimensional problems but remains faster than a direct solver.

翻译：本文描述了一种利用径向基函数（RBF）进行函数逼近的高效方法，并将其推广为求解不规则区域边值问题的求解器。该方法基于在包围盒上规则网格定义的RBF中心点，其中部分中心点位于计算域外部。方程采用过采样配点法离散化，仅将配点布置在域内，从而形成需按最小二乘意义求解的矩形线性系统。本文旨在高效求解该矩形系统。我们证明最小二乘问题可分解为规则部分（可通过快速傅里叶变换加速）与低秩扰动部分（采用直接求解器单独处理）。扰动的秩受区域不规则形状及边界上弱施加边界条件的影响。该求解器拓展了先前针对框架及其他过完备集合函数逼近提出的AZ算法。对于单变量问题，该求解器具有近最优的对数线性复杂度；对于高维问题虽失去最优性，但仍快于直接求解器。