Radial Basis Functions

自然界中的大多数现象要么完全用偏微分方程描述，要么在某种程度上用偏微分方程式描述。虽然有限差分法早先被用于常微分方程组的数值解，但1910年由L.F.理查森提出的将有限差分法也用于偏微分方程的提议现在被认为是计算史上的一个里程碑式的事件。在接下来的近100年里，在这个数值求解偏微分方程组的领域中，在许多方面都取得了巨大的进步，包括更快的算法、更高的精度、更容易的实现、更强的稳健性、更好的几何灵活性、更好的大规模并行计算机体系结构的伸缩性等。没有一种方法在所有这些方面都是优秀的，并且最佳的方法选择因应用领域而异。在需要很长时间内具有很高精度的情况下(例如许多以对流为主的方程，例如由天气、气候或湍流模拟引起的)，伪谱(PS)方法已被发现执行得特别好，只要不需要局部细化，并且总体几何形状非常简单。径向基函数(RBF)最早是由Rolland Hardy(在不同背景下的多元离散节点内插)在20世纪70年代初由S提出的，并于1990年由Ed Kansa用于求解偏微分方程组。使用径向基函数求解偏微分方程组的许多后续发展都来自于本研究人员和他在科罗拉多大学的研究小组早期由NSF支持的工作。特别是，人们发现RBF在一定的限制下简化为PS方法，而且很明显，这个限制在几个方面不是最优的。一个特别重要的方面是，RBF可以在无网格设置下保持频谱精度，允许一般的域形状和易于实现的局部节点细化。目前拟议的研究旨在寻找径向基函数领域中尚未探索的新机会，包括在算法方面，如数值稳定性和速度，以及将该方法扩展到新的应用。例如，现在看来，对于广泛的自由边界问题，也有很好的机会实现频谱精度。虽然过去几十年计算硬件的快速进步是众所周知的，但在数值算法方面类似和同样有利的趋势可能不那么广泛地被认识到。这两个方面结合在一起，使得数值计算成为探索具有重大社会影响的广泛问题的越来越重要的方法，如天气和气候建模、海啸预警计算等。本工作的主要主题是一种称为径向基函数(简称RBF)的数值方法。人们发现，在刚才提到的所有领域，它要么非常有希望，要么已经与以前最好的替代方案相媲美。当用数学术语表述时，关键挑战是如何最有效地解决被称为偏微分方程组的问题。RBF在这里提供了许多新的机会，其他研究小组也越来越多地追求这些机会。本研究的长期目标是推动成果预算编制方法，使其更容易适用于上述任务。

Most phenomena in nature are either entirely or to some significant extent described by partial differential equations. Although finite differences had been used earlier for the numerical solution of ordinary differential equations, the proposal in 1910 by L.F. Richardson to use them also for partial differential equations (PDEs) is now recognized as a landmark event in the history of computing. In the nearly 100 years that has followed, there has been vast progress on many fronts in this area of numerically solving PDEs, including faster algorithms, higher accuracies, easier implementations, greater robustness, improved geometric flexibility, better scaling for massively parallel computer architectures, etc. No single method excels in all these respects, and the best choice of method varies between application areas. In cases requiring very high accuracy over long times (such as many convection-dominated equations, for example arising from weather, climate, or turbulence modeling), pseudospectral (PS) methods have been found to perform especially well, as long as local refinement is not needed, and the overall geometry is quite simple. Radial basis functions (RBFs) were first proposed by Rolland Hardy (in the different context of multivariate scattered node interpolation) in the early 1970's, and were tested for solving PDEs by Ed Kansa in 1990. Much subsequent development on using RBFs for solving PDEs have come from earlier NSF-supported works by the present investigator and his research group at University of Colorado. In particular, RBFs were found to reduce to PS methods in a certain limit, and it has also become clear that this limit is less than optimal in several respects. A particularly important aspect is that RBFs can maintain spectral accuracy also in meshfree settings, allowing both general domain shapes and easy-to-implement local node refinements. The presently proposed research aims towards still unexplored new opportunities in the RBF area, both with regard to algorithmic aspects, such as numerical stability and speed, as well as extending the method to new applications. For example, there now appears to be excellent chances of achieving spectral accuracy also for a wide range of free boundary problems.While the rapid advances in computational hardware during the last decades are well known, the similar and equally favorable trend in terms of numerical algorithms might be less widely appreciated. Both aspects combine to make numerical computations an increasingly important approach for exploring a wide range of issues with great societal impact, such as weather and climate modeling, tsunami early warning calculations, etc. The main topic of the present work is a numerical methodology known as Radial Basis Functions (or RBFs for short). It has been found to be either very promising or already competitive with the best previous alternatives in all of the areas just mentioned. When formulated in mathematical terms, the key challenge becomes how to most effectively solve something known as partial differential equations. RBFs offer here numerous new opportunities, which are increasingly pursued also by other research groups. The long term goal of the present research to advance the RBF methodology to the point that it becomes still more readily applicable for tasks such as those outlined above.