Radial Basis Functions

多维空间的数值计算传统上是基于结构网格(如有限差分或谱方法)或非结构网格(如有限元)。即使在后一种情况下，也存在这样的结构，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。与这种网格生成相关联的方面有时会消耗与网格上的后续计算一样多甚至更多的计算机资源。此外，如果不广泛使用区域分解，就几乎不可能在这样的网格上获得高的(谱)计算精度。最近提出了许多无网格方法。径向基函数(RBF)方法在几个方面表现突出，最显著的是它将传统的谱方法推广到完全无网格的环境。此外，实现通常非常简单。例如，20-30行的MatLab通常足以在不规则的3-D区域上求解椭圆型偏微分方程组，以达到频谱精度。伪谱方法和径向基函数法都是在20世纪70年代初由S首先提出的，在求解许多偏微分方程组时，伪谱方法表现出了优越的性能。大约5年前，由本研究人员提出的NSF建议是第一次提出径向基函数作为PS方法中传统基函数的直接替代。最近，在克服高昂的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都取得了重大进展。NSF-DMS最近在本调查员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，现在将继续努力。这些措施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，在极高精度平基函数极限下的新的稳定算法，朝着更快的算法的发展，以及将径向基函数应用于天文/地球物理中最相关的几何中的偏微分方程组，等等。在过去的几十年里，计算方法已经成为如何进行科学和工程的越来越重要的部分，部分原因是计算机硬件的快速发展，但同样也要归功于计算算法的改进。要解决的问题往往需要求解偏微分方程组(PDE)，通常是在几个空间维度的不规则形状区域。径向基函数(RBF)方法是本次赠款的主题，它结合了非常高的精度和无与伦比的几何灵活性，在这方面开辟了全新的机会。若干特别感兴趣的应用领域之一(与NCAR和NOAA的科学家合作)涉及球面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.