New Directions in Scattered Data Analysis via Radial and Related Basis Functions with Applications

插值和近似的误差估计是在任何应用中开发严格算法或数值方法的核心。分散数据问题的一个主要困难是，已知方法收敛的数据生成函数的类比实际中遇到的要小得多。这在通过径向基函数配置数值求解偏微分方程中是有问题的，因为目标函数需要比基函数更平滑，这是双曲问题中的主要限制。这个项目的目标之一是获得误差估计的一个大大扩展类的目标函数。在我们最近的工作插值通过一个限制类的球面基函数的领域，这样的估计。这使人们有希望实现更广泛的目标。另一个重要的目标是开发和实现严格的，计算效率高的算法，数值偏微分方程问题，神经网络，和地球科学的问题，需要分散的数据表面拟合上spheres.The调查的分散数据建模是非常重要的潜在的理解地球上的各种现象。将表面与通过卫星或地面站收集的气象或地球物理数据拟合是这种应用的一个很好的例子。球基函数（SBF）已被用于在这样的问题。径向基函数和周期基函数已广泛应用于各种神经网络，包括用于相控阵雷达测向的架构。 最近，它们被用于求解偏微分方程的无网格数值方法，以及计算机图形学和计算机辅助设计问题。与这些相关的计算是耗时的。最近的进展已使这些困难有所改善，本项目下的工作将继续提高效率。

Error estimates for both interpolation and approximation are central in developing rigorous algorithms or numerical methods in any application. A major difficulty in scattered-data problems has been that the class of data-generating functions for which the methods are known to converge is much smaller than what one encounters in practice. This is problematic in numerically solving partial differential equations via Radial Basis Function collocation, since target functions need to be smoother than the basis functions, which is a major restriction in hyperbolic problems. One of the goals of this project is to obtain error estimates for a greatly expanded class of target functions. In our very recent work on interpolation via a restricted class of Spherical Basis Functions on the sphere, such estimates were obtained. This provides hope that the broader goal is attainable. Another important goal is to develop and implement rigorous, computationally efficient algorithms for numerical partial differential equation problems, neural networks, and problems from the geosciences requiring scattered-data surface fitting on the sphere.The investigation of scattered-data modeling is of great potential importance for the understanding of earth based phenomena of every kind. Fitting surfaces to meteorological or geophysical data collected via satellites or ground stations is a good example of such an application. Spherical basis functions (SBFs) have been used in such problems. Radial and periodic basis functions have been extensively employed in a variety of neural networks, including architectures used for direction-finding via phased-array radar. Very recently, they have been employed in grid-free numerical methods for solving partial differential equations, and to do computer graphics and computer aided design problems. Calculations assocaited with these are time consuming. Recent advances have ameliorated these difficulties and the work under this project will continue to improve efficiency.