Radial Basis Functions

自然界中的大多数现象都是完全或在一定程度上由偏微分方程描述的。虽然有限差分法在较早的时候已经被用于常微分方程的数值解，但是L.F. Richardson将它们也用于偏微分方程（PDE）现在被认为是计算历史上的一个里程碑事件。在随后的近100年里，在这一领域的数值求解偏微分方程，包括更快的算法，更高的精度，更容易实现，更大的鲁棒性，改进的几何灵活性，更好地扩展大规模并行计算机体系结构等方面，有了巨大的进步，没有一种方法优于在所有这些方面，最好的选择方法的应用领域之间的变化。在需要长时间非常高精度的情况下（例如许多对流主导的方程，例如来自天气，气候或湍流建模），已经发现伪谱（PS）方法表现得特别好，只要不需要局部细化，并且整体几何结构非常简单。径向基函数（Radial Basis Function，简称RBF）最早由Rolland哈代在20世纪70年代初提出（在多元离散节点插值的不同背景下），并由艾德Kansa在1990年进行了求解偏微分方程的测试。许多后续的发展，使用RBF解决偏微分方程来自早期的NSF支持的工作，目前的调查员和他的研究小组在科罗拉多大学。特别是，RBFs被发现减少到PS方法在一定的限制，它也变得很清楚，这个限制是低于最佳在几个方面。一个特别重要的方面是，径向基函数也可以在无网格设置中保持频谱准确性，从而允许一般域形状和易于实现的本地节点细化。目前提出的研究的目的是在RBF领域尚未探索的新机会，无论是在算法方面，如数值稳定性和速度，以及扩展的方法，以新的应用。例如，现在似乎有很好的机会，实现频谱精度也为广泛的自由边界问题，虽然在过去的几十年中，在计算硬件的快速发展是众所周知的，类似的，同样有利的趋势，在数值算法可能不太广泛的赞赏。这两个方面的联合收割机，使数值计算越来越重要的方法，探索广泛的问题，具有巨大的社会影响，如天气和气候建模，海啸预警计算等，目前的工作的主要议题是一个数值方法称为径向基函数（或简称RBFs）。人们发现，在刚才提到的所有领域，它要么非常有前途，要么已经与以前最好的替代品竞争。当用数学术语表述时，关键的挑战是如何最有效地解决所谓的偏微分方程。在这方面，区域预算框架提供了许多新的机会，其他研究小组也越来越多地寻求这些机会。本研究的长期目标是推进RBF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.