径向基函数与拟蒙特卡洛方法研究
A study on radial basis function and quasi-Monte Carlo methods
AI总结：
本研究通过数值积分视角揭示径向基函数与拟蒙特卡洛方法的内在联系，提出基于积分方程核函数构造高效径向基函数的方法，并给出构造、误差界及节点布置的重要结果。
摘要：
径向基函数（RBF）和拟蒙特卡洛（QMC）方法是处理具有复杂且移动边界几何的高维问题的两种非常有前景的方案，因为它们不依赖于维度且本质上无网格。这两种策略看似不相关，且迄今是独立发展的。前者主要用于求解偏微分方程（PDE）、神经网络、几何生成、散乱数据处理，并具有插值理论的数学论证[1]；而后者通常用于在蒙特卡洛方法（MCM）背景下评估高维积分[2]。本通讯的目的是尝试在数值积分的基础上建立它们的内在联系。积分方程的核函数被发现是构造高效径向基函数的关键。文中还给出了关于径向基函数构造、误差界和节点布置的一些重要结果。需要强调的是，这里的径向基函数是基于积分分析而非复杂的插值和原生空间分析建立的。
英文摘要：
The radial basis function (RBF) and quasi Monte Carlo (QMC) methods are two very promising schemes to handle high-dimension problems with complex and moving boundary geometry due to the fact that they are independent of dimensionality and inherently meshless. The two strategies are seemingly irrelevant and are so far developed independently. The former is largely used to solve partial differential equations (PDE), neural network, geometry generation, scattered data processing with mathematical justifications of interpolation theory [1], while the latter is often employed to evaluate high-dimension integration with the Monte Carlo method (MCM) background [2]. The purpose of this communication is to try to establish their intrinsic relationship on the grounds of numerical integral. The kernel function of integral equation is found the key to construct efficient RBFs. Some significant results on RBF construction, error bound and node placement are also presented. It is stressed that the RBF is here established on integral analysis rather than on the sophisticated interpolation and native space analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
