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

在任何应用中，对插值和近似的误差估计都是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别比实际遇到的要小得多。这在通过径向基函数搭配数值求解偏微分方程时是有问题的，因为目标函数需要比基函数更平滑，这是双曲型问题的一个主要限制。这个项目的目标之一是获得一个极大扩展的目标函数类的误差估计。在我们最近通过球面上的一类受限的球面基函数进行插值的工作中，我们得到了这样的估计。这为实现更广泛的目标提供了希望。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，用于数值偏微分方程问题、神经网络和地球科学问题，这些问题需要在球体上进行分散数据表面拟合。对离散数据建模的研究对于理解各种地球现象具有重要的潜在意义。通过卫星或地面站收集的气象或地球物理数据拟合曲面是这种应用的一个很好的例子。球面基函数（SBFs）已被用于这类问题。径向和周期基函数已广泛应用于各种神经网络中，包括用于通过相控阵雷达测向的体系结构。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。与这些相关的计算非常耗时。最近的进展改善了这些困难，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.