径向基函数逼近中的若干问题研究

The fundamental problem of data science which has many applications in compressed sensing, meshless method for partial differential equations, machine learning and neural network is to rebuild the information from the sample data. From the numerical approximation viewpoint, to simulate the function itself or its functional (such as derivative) based on the sampling data (function value or functional value with random distribution). In recent years, the data come from mainstream research area possesses a characteristic of multi-dimensional, and scattered and random. In this project, an effective way to deal with such problems-reproducing kernel methods, especially radial basis function method discussed. The study will address the following three aspects: (1) such problems usually involves large-scale computational problems, we intend to construct a new base as the pre-conditions to improve computational efficiency; (2) to discuss the quasi-interpolation method as a tool to approximate the functional value numerically; thus (3)the theory could be applied to differential equations numerical solution, machine learning and compressed sensing and other aspects. The study of this project will enrich and improve the theoretical system of radial basis function interpolation methods and the quasi-interpolation method, and moreover, it will promote the development of other applied field of study.

在压缩感知、无网格微分方程数值解、机器学习和神经网络等众多重要应用领域提出的数据科学问题实质是从采样数据出发对信息进行重建。从数值逼近的角度来说就是:给定函数的一些带有随机分布的函数值(或泛函值)采样数据，对该函数本身及其泛函（譬如导函数、甚至高阶导函数）进行数值模拟。近年来的主流研究热点向数据具有多维、散乱和随机的特点聚集，本项目拟对处理此类问题的有效方法-再生核方法特别是径向基函数方法进行讨论。研究内容涉及以下三个方面：（1）此类问题通常涉及大规模计算问题，我们拟构造新的适用于计算的基底作为预条件处理，以提高计算效率；(2)讨论拟插值方法作为工具对泛函进行数值逼近的性质；从而(3)将这些理论应用于微分方程数值解、机器学习和压缩感知等方面。本项目的研究在理论上将丰富和完善径向基函数插值方法和拟插值方法的理论体系，并将为其他应用研究领域提供新的计算方法、注入新的活力。

本项目对径向基函数逼近的方法及其应用进行了研究，为无网格微分方程数值解、机器学习等应用领域提供了理论支撑和高效算法。研究内容包括：1. 基于核的概率密度函数估计问题，我们采用统计距离，即Wasserstein的距离来衡量两个概率密度函数的差异，得到了收敛阶；2. 径向基函数multiquadric拟插值方法在微分方程数值解中的应用，我们把multiquadric拟插值方法从一维推广到多维，进一步将其应用于多维空间域的微分方程问题中；3. 基于径向基函数法的分数阶微分方程数值解法，我们在行方法的基础上用径向基函数方法求分数阶Cattaneo方程和 Klein-Gordon方程的数值解，并给出了这个方法的无条件稳定性和收敛性证明。