Theory and Applications of Localized Kernel Bases to Meshfree Methods

从分散的、不规则放置的站点获取的数据分析和建模的需求在不同的领域中频繁出现：大气科学、人工智能、计算机辅助设计图形、数据挖掘、医学成像、学习网络和许多其他领域。 例如，天气预测或气候建模是基于在分散的地点收集的地球物理数据，通过卫星上的传感器，地面站或海上站。 执行这些任务给传统方法带来了困难，传统方法基于在统一放置的站点收集数据，或者需要构建“网格”（想想铁丝网），必须仔细定制以处理所涉及的数据站点。 较新的方法，即所谓的核方法，是无网格的，可以处理分散的数据。 研究人员进一步发展了核方法的理论，在此基础上，可以开发出更容易使用，更快，实现成本更低，并且能够处理来自10万个或更多站点的数据的算法。分散数据问题对任何方法都提出了挑战，包括基于一个（条件）正定函数的传统核类型算法。 在无网格方法、机器学习、神经网络和其他情况下，自然会出现分散的数据。 理想情况下，处理这些数据需要本地稳定的基础，预处理器和其他类似的工具。 在最近的工作领域和旋转的空间中，没有边界的存在，调查人员已经开发了新的基地有关的某些类别的内核。 对于边界固有的问题，其基础需要进一步发展。 这个项目的一个关键部分涉及一个新的想法，外推数据稍微超出一个紧凑的域的边界，以提高该方法的近似能力。 调查的另一个关键领域涉及数据的局部近似阶数，这比典型的准统一假设要普遍得多。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

The need for analyzing and modeling data taken from scattered, irregularly placed sites arises frequently in diverse fields: atmospheric science, artificial intelligence, computer-aided design graphics, data mining, medical imaging, learning networks, and 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, are meshfree and can handle scattered data. The investigators further develop the theory of kernel methods, on the basis of which algorithms can be developed that are easier to use, faster, less expensive to implement, and capable of handling data from a hundred thousand or more sites.Scattered data problems present a challenge for any method, including the traditional kernel-type algorithms based on translates of one (conditionally) positive definite function. Scattered data occur naturally in meshfree methods, machine learning, neural nets, and other situations. Dealing with such data, ideally, requires local, stable bases, preconditioners, and other similar tools. In recent work on the sphere and rotations in space where no boundary is present, the investigators have developed novel bases related to certain classes of kernels. For problems where boundaries are inherent, their bases need further development. One key part of this project involves a novel idea of extrapolating data slightly beyond the boundary of a compact domain to enhance the approximation power of the method. Another key area of investigation involves local approximation orders for data that is far more general than the typical quasi-uniform assumptions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.