Radial Basis Functions

径向基函数

• 批准号: 0611681
• 负责人: Bengt Fornberg
• 金额: $ 26.68万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0611681&HistoricalAwards=false
• 关键词: Radial; Basis; Functions

Numerical computations in multiple space dimensions have traditionally been based on structured grids (e.g. finite differences or spectral methods) or on unstructured grids (e.g. finite elements). Even in the latter case, there is structure in the sense that one needs to work out which subsets of nodes should be connected into local triangles, tetrahedra, etc. Aspects associated with such grid generations can at times consume as much or even more computer resources than do the subsequent computations on the grid. Furthermore, it has been all but impossible to achieve high (spectral) computational accuracy on such grids without resorting to extensive use of domain decompositions. Numerous mesh-free methods have been proposed recently. The Radial Basis Function (RBF) approach stands out in several respects, most notably in that it generalizes traditional spectral methods to entirely mesh-free settings. Furthermore, implementations are usually remarkably simple. For example, 20-30 lines of Matlab typically suffices for solving an elliptic PDE on an irregular 3-D domain, to spectral accuracy. Both pseudospectral (PS) methods and RBFs were first proposed in the early 1970's. The superior performance of PS methods for solving many PDEs was demonstrated early on. A NSF proposal by the present investigator about 5 years ago was the first time RBFs were presented in terms of being a direct replacement for the traditional basis functions in PS methods. Recently, significant progress has been made both on overcoming the high computer cost and the numerical ill-conditioning that earlier were thought to severely limit the RBF approach. Recent NSF-DMS supported work by the present investigator has opened up numerous further opportunities in this area, which will now be pursued. These include combining spectral accuracy with local node clustering in a fully stable way, a new stable algorithm in the extremely high accuracy flat basis function limit, developments towards faster algorithms, and the application of RBFs to PDEs in the geometries that are most relevant in astro/geophysics, etc.In the last several decades, computational methods have become an increasingly essential part of how science and engineering are conducted, partly because of a rapid evolution of computer hardware, but equally much thanks to improvements in computational algorithms. Very often, the task to be solved, when formulated in mathematical terms, require the solution of partial differential equations (PDEs), often in irregularly shaped regions in several space dimensions. The Radial Basis Function (RBF) methodology, which is the subject of the present grant, opens entirely new opportunities in this regard, combining very high accuracy with unsurpassed geometric flexibility. One of several application areas of particular interest (pursued in collaboration with scientists at NCAR and NOAA) concerns geophysical and astrophysical modeling in spherical geometries, which are critical issues for effective climate modeling and for studies of solar dynamics.

多维空间的数值计算传统上是基于结构化网格（例如有限差分或谱方法）或非结构化网格（例如有限元）。即使在后一种情况下，也有结构的意义上说，一个需要工作出哪些节点的子集应该连接到本地三角形，四面体等方面与这种网格生成有时可以消耗尽可能多的甚至更多的计算机资源比做网格上的后续计算。此外，它一直是所有，但不可能实现高（光谱）计算精度，这样的网格，而不诉诸广泛使用的域分解。最近提出了许多无网格方法。径向基函数（RBF）方法在几个方面脱颖而出，最值得注意的是，它将传统的谱方法推广到完全无网格的设置。此外，实现通常非常简单。例如，20-30行Matlab通常足以在不规则的3-D域上求解椭圆PDE，达到谱精度。伪谱（PS）方法和径向基函数都是在20世纪70年代初首次提出的。PS方法在求解许多偏微分方程时的上级性能在早期就得到了证明。本研究者在大约5年前提出的NSF建议是第一次提出将RBF作为PS方法中传统基函数的直接替代。最近，已经取得了显着的进展，克服了高的计算机成本和数值病态，早期被认为是严重限制了RBF方法。本调查员最近在国家安全基金-移民管理局的支持下开展的工作在这一领域开辟了许多进一步的机会，现在将继续开展这些工作。这些包括以完全稳定的方式将谱精度与局部节点聚类相结合，在极高精度平坦基函数极限下的新稳定算法，朝着更快算法的发展，以及在与天文/地球物理学最相关的几何中将径向基函数应用于偏微分方程等。在过去的几十年中，计算方法已经成为科学和工程如何进行的越来越重要的一部分，部分原因是计算机硬件的快速发展，但同样要归功于计算算法的改进。很多时候，要解决的任务，当制定在数学方面，需要解决偏微分方程（PDE），往往在不规则形状的区域在几个空间维度。径向基函数（RBF）方法，这是本赠款的主题，在这方面开辟了全新的机会，结合了非常高的精度与无与伦比的几何灵活性。特别感兴趣的几个应用领域之一（与NCAR和NOAA的科学家合作进行）涉及球形几何的地球物理和天体物理建模，这是有效气候建模和太阳动力学研究的关键问题。