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

径向基函数（RBF）是一种已被证明在许多领域具有重要价值的数值技术。例如，三维激光扫描仪将物体（如人脸）转换为“点云”，即对人脸上点的位置进行测量。RBF插值将这些点连接到一个光滑的表面上，这样脸就会成为一个可识别的脸，而不是一团没有连接的标记。rbf已被应用于求解流体流动的偏微分方程，以跟踪飓风和预测天气、港口潮汐流、汽车发动机燃烧流等。不幸的是，rbf也有缺陷。计算不能很好地扩展到大量的自由度，舍入误差可以将预测模型变成无用的随机数生成器，并且在rbf迄今为止取得巨大成功的问题类中有时会出现较差的准确性。这个研究项目的一个目标是在更深层次上理解rbf。为什么它们（大多数时候）如此有效？为什么类似的基于多项式的方法失败了，它们却成功了？rbf和多项式之间的关系是什么？这个项目将探索rbf的基础，以更好地描述它们的应用领域，在可行的地方提高性能，并可能标记一些不适合rbf的应用领域。为了解决它们的缺点，并在更基本的层面上理解rbf， PI将深入研究rbf替代品：它们是多项式与高斯函数的乘积，相当于将Hermite函数的无限区间基扩展到有限区间上的插值和pde求解。PI的早期工作建立了一个严格的收敛误差定理，并通过数值比较显示了在某些应用中Hermite函数优于rbf。传统的单域伪谱方法失败，除非该域是矩形或椭圆，即所谓的张量积域。PI计划扩展这些Hermite伪rbf，以解决使用不规则网格的几何复杂域的多维偏微分方程，这些问题中rbf有时很好，有时会失败。这些复杂的领域包括一个六边形透镜的望远镜，或者一个被海湾环绕、被岛屿穿透的海洋。Hermite函数和rbf易于编程，因此是初步设计、课堂建模以及补充和丰富理论的理想选择。该项目的一个应用目标是推进数值快速原型，也就是说，设计算法，尽管复杂的领域边界，结合代码的简洁性和谱的准确性。

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.