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：本文提出一种利用径向基函数（RBFs）逼近函数的高效方法，并将其推广至不规则区域上边值问题的求解。该方法基于定义在包围盒内规则网格中心上的RBFs，其中部分中心位于计算域外部。采用过采样配点法对方程进行离散化，仅选取域内配点，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。研究表明，最小二乘问题可分解为规则部分（可利用FFT加速计算）与低秩摄动部分（采用直接求解器单独处理）。摄动秩受域的不规则形状及边界点处边界条件弱施加方式影响。该求解器扩展了先前针对框架及其他过完备集函数逼近问题提出的AZ算法。对于单变量问题，该求解器具有近似最优的对数线性复杂度；对于高维问题虽失去最优性，但仍比直接求解器更快。