Gaussian-Localized Polynomial Approximation: A Well-Conditioned Spectral Method for Solving Partial Differential Equations in Complicated Domains

径向基函数(RBF)是一种在许多领域都被证明具有重要价值的数值技术。例如，三维激光扫描仪将人脸等对象转换为“点云”，即测量脸部各点的位置。径向基函数插值将这些点连接到一个平滑的曲面中，这样人脸就会显示为可识别的人脸，而不是一大堆未连接的标记。径向基函数法已被应用于求解流体流动的偏微分方程组，以跟踪飓风、预报天气、港口潮汐流动、汽车发动机燃烧流动等。不幸的是，RBF也有缺陷。计算不适合于大量自由度，舍入误差可能会将预测模型变成无用的随机数生成器，并且在RBF迄今取得巨大成功的问题类中有时存在较低的准确性。本研究项目的目的之一是在更深层次上理解RBFS。为什么它们工作得这么好(大部分时间)？当类似的基于多项式的方法失败时，他们为什么会获胜？径向基函数和多项式之间有什么关系？这个项目将探索RBFS的基础，以便更好地描述它们的应用领域，在可行的情况下提高性能，并可能将一些应用领域标记为不适合RBFS。为了克服它们的缺点并在更基本的层面上理解RBF，PI将深入研究RBF-代词：这些代词是多项式与高斯的乘积，相当于将Hermite函数的无限区间基扩展到有限区间上的插补和偏微分方程组求解。PI的早期工作建立了严格的收敛和误差定理，并进行了数值比较，表明了Hermite函数在某些应用中比RBF函数的优越性。传统的单域伪谱方法失效，除非域是矩形或椭圆形，即所谓的张量积域。PI计划扩展这些Hermite伪RBF，以使用不规则网格来求解几何复杂区域中的多维PDE，其中RBF有时是好的，有时是失败的。这种复杂的区域包括一个带有六角形透镜的望远镜，或者一个被海湾环绕并被岛屿穿透的海洋。Hermite函数和RBF易于编程，因此非常适合初步设计、课堂建模以及补充和丰富理论。该项目的一个应用目标是推进数字快速原型制作，即设计出尽管区域边界复杂，但代码简洁性与光谱准确性相结合的算法。

Radial basis functions (RBF) are a numerical technology that has proven to be of great value in many fields. For example, three-dimensional laser scanners convert objects, such as a human face, into a "point cloud," that is, measurements of the position of points on the face. RBF interpolation connects the dots into a smooth surface so that face appears as a recognizable face instead of a cloud of unconnected markers. RBFs have been applied to solve the partial differential equations of fluid flow so as to track hurricanes and predict weather, tidal flows in harbors, combusting flows in an automobile engine, and so on. Unfortunately, RBFs also have flaws. Calculations scale poorly to a large number of degrees of freedom, round-off errors can turn a forecasting model into a useless random number generator, and poor accuracy is sometimes present in problem classes where RBFs have hitherto been a great success. One goal of this research project is to understand RBFs at a deeper level. Why do they work so well (much of the time)? Why do they triumph when similar polynomial-based methods fail? What is the relationship between RBFs and polynomials? This project will explore the foundations of RBFs to better delineate their domain of application, improve performance where feasible, and potentially mark some application domains as unsuitable for RBFs. To cope with their shortcomings and to also understand RBFs at a more fundamental level, the PI will intensively study RBF-substitutes: these are products of polynomials with Gaussians, equivalent to extending the infinite interval basis of Hermite functions to interpolation and PDE-solving on a finite interval. Earlier work of the PI established a rigorous convergence-and-error theorem and also numerical comparisons showing the superiority of Hermite functions to RBFs in some applications. Conventional single-domain pseudospectral methods fail unless the domain is a rectangle or ellipse, a so-called tensor product domain. The PI plans to extend these Hermite pseudo-RBFs to solve multidimensional PDEs in geometrically-complicated domains using irregular grids, problems where RBFs are sometimes good and sometimes failures. Such complicated domains include a telescope with a hexagonal lens or an ocean ringed with bays and pierced with islands. Hermite functions and RBFs are easy to program and therefore ideal for preliminary design, classroom modeling, and complementing and enriching theory. An applied goal of the project is to advance numerical rapid prototyping, that is, to devise algorithms that, despite complicated domain boundaries, combine brevity of code with spectral accuracy.