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

高斯局部多项式逼近：求解复杂域中偏微分方程的良好条件谱方法

• 批准号: 1521158
• 负责人: John Boyd
• 金额: $ 15.91万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1521158&HistoricalAwards=false
• 关键词: Gaussian; Localized; Polynomial; Approximation; Well

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.

径向基函数（RBF）是一种数值技术，已被证明在许多领域具有巨大价值。例如，三维激光扫描仪将人脸等物体转换为“点云”，即面部各点位置的测量结果。 RBF 插值将这些点连接成一个光滑的表面，以便面部显示为可识别的面部，而不是一堆未连接的标记。 RBF 已被应用于求解流体流动的偏微分方程，以跟踪飓风并预测天气、港口的潮汐流、汽车发动机的燃烧流等。不幸的是，RBF 也有缺陷。计算很难扩展到大量自由度，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 迄今为止取得巨大成功的问题类别中有时会出现精度较差的情况。该研究项目的目标之一是更深入地了解 RBF。为什么它们（大部分时间）工作得这么好？当类似的基于多项式的方法失败时，为什么它们会取得胜利？ RBF 和多项式之间有什么关系？该项目将探索 RBF 的基础，以更好地描述其应用领域，在可行的情况下提高性能，并可能将某些应用领域标记为不适合 RBF。为了克服它们的缺点并在更基础的层面上理解 RBF，PI 将深入研究 RBF 替代品：这些是多项式与高斯函数的乘积，相当于将 Hermite 函数的无限区间基础扩展到插值和 PDE 求解有限区间。 PI 的早期工作建立了严格的收敛与误差定理，并且通过数值比较显示了 Hermite 函数在某些应用中相对于 RBF 的优越性。除非域是矩形或椭圆形（即所谓的张量积域），否则传统的单域伪谱方法就会失败。 PI 计划扩展这些 Hermite 伪 RBF，以使用不规则网格解决几何复杂域中的多维偏微分方程，这些问题中 RBF 有时有效，有时失败。此类复杂的领域包括带有六角形透镜的望远镜或海湾环绕、岛屿穿插的海洋。 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值快速原型设计，即设计算法，尽管域边界复杂，但仍将代码的简洁性与光谱精度结合起来。