Radial Basis Functions

在多个空间维度上的数值计算传统上是基于结构化网格（如有限差分或谱方法）或非结构化网格（如有限元）。即使在后一种情况下，也存在一种结构，即需要计算出哪些节点子集应该连接到局部三角形、四面体等。与这些网格代相关的方面有时消耗的计算机资源与网格上的后续计算一样多，甚至更多。此外，如果不广泛使用域分解，在这种网格上实现高（谱）计算精度几乎是不可能的。近年来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光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域上求解椭圆偏微分方程，达到光谱精度。伪光谱（PS）方法和rbf方法都是在20世纪70年代初提出的。PS方法在求解许多偏微分方程方面的优越性能早就得到了证明。本研究人员在大约5年前提出的NSF建议是rbf第一次在PS方法中作为传统基函数的直接替代提出的。最近，在克服高计算机成本和早先被认为严重限制RBF方法的数值病态方面取得了重大进展。最近NSF-DMS支持的工作由目前的研究者在这一领域开辟了许多进一步的机会，现在将继续追求。其中包括以完全稳定的方式将光谱精度与局部节点聚类相结合，在高精度平面基函数极限下的一种新的稳定算法，向更快算法的发展，以及在与天文学/地球物理学最相关的几何中将rbf应用于偏微分方程，等等。在过去的几十年里，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。通常，要解决的任务，当用数学术语表述时，需要解偏微分方程（PDEs），通常在几个空间维度的不规则形状区域中。本赠款的主题是径向基函数（RBF）方法，它在这方面开辟了全新的机会，结合了非常高的准确性和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，这是有效的气候建模和太阳动力学研究的关键问题。

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.