Localized Kernel Bases with Application to Meshless Methods

在计算机辅助设计图形学、数据挖掘、医学成像、学习网络、地球科学和许多其他领域，涉及分析和合成从空间或表面上的分散地点获得的数据的问题出现在不同的领域。该奖项将支持开发新的方法和工具，以利用核心方法攻击零散数据的分析和综合--即从非均匀分布的站点收集的数据。在这个奖项的支持下，研究人员最近发现了来自流形上特殊核的高度局部化的碱基，这一突破将在提出的问题和他们调查这些问题的方法中发挥关键作用。为由核构成的逼近空间寻找一个良好的、稳定的基的问题与以低计算代价确定良好的局部化基密切相关。用“标准”核基处理核内插或最小二乘逼近的主要困难之一是配置矩阵是满的，而且常常是病态的。研究人员和合作者最近的工作表明，与某些核相关的拉格朗日型插值函数关于它们的中心是指数局部化的，从数值实验来看，似乎在计算上是廉价的。该奖项将支持探索这些新发现的基本功能的全部潜力。分析和建模来自分散、不规则放置的站点的数据的需求经常出现在不同的领域：计算机辅助设计图形学、数据挖掘、医学成像、学习网络和地球科学，以及许多其他领域。例如，天气预报或气候模拟是基于由卫星、地面站或海上站上的传感器在分散的地点收集的地球物理数据。执行这类任务对传统方法来说是困难的，这些方法基于在统一放置的地点收集数据，或者需要构建“网格”(如铁丝网)，必须仔细定制以处理所涉及的数据地点。较新的方法，即所谓的核方法，不需要这样的网格，可以处理分散的数据。该奖项将进一步开发这些内核方法，使它们易于使用、速度更快、实现成本更低，并能够处理来自10万个或更多站点的数据。它将为研究生提供支持，他们将接受使用和开发这些方法的理论和应用方面的培训。

Problems involving the analysis and synthesis of data taken from scattered sites in space or on surfaces arise in diverse fields -- computer-aided design graphics, data mining, medical imaging, learning networks, geoscience, and many other areas. This award will support the development of new methods and tools for attacking the analysis and synthesis of scattered data -- i.e. data collected from non-uniformly distributed sites -- by means of kernel methods. The investigators supported by this awards recently discovered highly localized bases derived from special kernels on manifolds, and this breakthrough will play a key role both in the problems proposed and in their approach to investigating them. The problem of finding a good, stable basis for an approximation space made from kernels is closely connected with determining well-localized bases at low computational cost. One of the major difficulties in dealing with kernel interpolation or least squares approximation with the ``standard'' kernel basis is that collocation matrices are full and frequently ill conditioned. Recent work by the investigators and collaborators showed that Lagrange-type interpolating functions associated with certain kernels are exponentially localized about their centers and, from numerical experiments, appear to be computationally inexpensive. This award will support exploring the full potential of these newly discovered basis functions.The need for analyzing and modeling data taken from scattered, irregularly placed sites arises frequently in diverse fields: computer-aided design graphics, data mining, medical imaging, learning networks, and geoscience, in addition to many other areas. For example, weather prediction or climate modeling is based on geophysical data collected at scattered sites, by sensors on satellites, ground stations, or stations at sea. Carrying out such tasks presents difficulties for traditional methods, which are based on collecting data at uniformly placed sites or which require constructing ``meshes'' (think wire fence) that must be carefully tailored to deal with the data sites involved. Newer methods, the so-called kernel methods, do not require such meshes and can handle scattered data. This award will further the development of these kernel methods, making them easy to use, faster, less expensive to implement, and capable of handling data from a hundred thousand or more sites. It will provide support for graduate students, who will be trained in both the theoretical and the applied aspects of using and developing these methods.