Localized Kernel Bases: Theory and Applications to Meshless Methods

该奖项支持主要研究人员在基于核的无网格近似方法领域的研究计划，该方法在许多现代大规模科学计算问题中有应用。 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：计算机辅助设计图形、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。 执行这样的任务给传统方法带来了困难，传统方法是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，所谓的“内核方法”，不需要这样的网格，可以处理分散的数据。 除了分析和模拟这些数据外，这些方法还可以数值求解控制大气流动的方程。 该项目将进一步开发内核方法，使其易于使用，速度更快，实现成本更低，并且能够处理来自10万或更多站点的数据。 它将为研究生提供支持，他们将在使用和开发这些方法的理论和应用方面接受培训。该项目的主要目标是开发新的方法和工具，用于攻击分散数据的分析和综合（即，从非均匀分布的地点收集的数据），特别是用于数值求解偏微分方程和非局部扩散问题，例如，在周波学中。 有效地处理这样的问题需要构造局部的、稳定的基、预条件子和其他类似的工具。 为了解决这些问题，PI计划使用他们最近发现的高度本地化的稳定的基于核的基础，开发新的基于核的无网格伽辽金方法。 PI计划进一步开发这些最近发现的基函数的全部潜力，并研究在边界或数据不一致的情况下构建此类基。

This award supports the research program of the Principal Investigators in the area of kernel-based meshless approximation methods, which have applications in a number of modern large-scale scientific computing problems. 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 of a wire fence---that must be carefully tailored to deal with the data sites involved. Newer methods, so-called "kernel methods", do not require such meshes and can handle scattered data. In addition to analyzing and modeling such data, these methods can solve numerically the equations governing, say, atmospheric flow. This project will further the development of 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.The main object of this project is to develop 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 and, in particular, for numerically solving partial differential equations and non-local diffusion problems in peridynamics, for example. Efficiently dealing with such problems requires constructing local, stable bases, preconditioners, and other similar tools. To address such problems, the PIs plan to use their recently discovered highly localized stable kernel-based bases to develop novel kernel-based meshless Galerkin methods. The PIs plan to further develop the full potential of these recently discovered basis functions, and to investigate constructing such bases in situations where boundaries or data that is not quasi uniform occur.