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，在许多领域都具有很高的价值。例如，三维激光扫描仪将物体（例如人脸）转换为“点云”，即测量点在脸上的位置。 RBF插值将点连接到光滑的表面，因此脸部表现为可识别的脸部，而不是无连接的标记云。 RBF已应用于求解流体流动的部分微分方程，以追踪飓风和预测天气，港口中的潮流，汽车发动机中的燃烧流量等等。不幸的是，RBF也有缺陷。计算尺度较差到大量的自由度，圆形错误可以将预测模型转化为无用的随机数发生器，并且在RBF迄今已取得巨大成功的问题类别中，有时会出现较差的准确性。该研究项目的目标之一是在更深层次的水平上了解RBF。他们为什么在很多时候工作得这么好？当类似的基于多项式的方法失败时，为什么它们会胜利？ RBF和多项式之间的关系是什么？该项目将探索RBF的基础，以更好地描述其应用领域，在可行的地方提高性能，并可能将某些应用程序域标记为不适合RBF。为了应付自己的缺点并在更基本的层面上理解RBF，PI将深入研究RBF-SUBSTITUTS：这些是与高斯的多项式的产物，相当于将Hermite的无限间隔基础函数扩展以在限制间隔内进行插入和PDE分解。 PI的早期工作建立了严格的合并和错误定理以及数值比较，显示了在某些应用中，Hermite功能对RBF的优越性。除非域是矩形或椭圆形，否则常规的单域伪谱方法会失败。 PI计划使用不规则的网格扩展这些HERMITE伪RBF，以在几何复杂的域中求解多维PDE，而RBF有时是好的，有时会发生故障的问题。如此复杂的领域包括带有六角形镜头的望远镜或带有海湾并用岛屿刺穿的海洋。 Hermite功能和RBF易于编程，因此非常适合初步设计，教室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制作，即设计算法，尽管域边界复杂，但将代码的简洁性与光谱精度相结合。