New Directions in Mesh-Free Approximation with Localizable Kernels

核函数和径向基函数是非常有价值的数学工具，通常用于处理大型和具有挑战性的计算问题。它们广泛应用于科学和工程的各个分支，如计算化学、机器学习、计算机图形学和地球科学，研究人员希望从几何复杂的数据集中近似、建模或提取信息。该项目致力于提高使用核的算法的稳定性和速度，将这些算法应用于一些重要的科学问题，并通过开发多尺度，各向异性的核近似理论，大大扩展核的近似能力，超越当前的最新技术水平。这些结果将有利于那些使用内核作为计算工具的科学领域。在整个项目中，研究者将继续自己对这些问题的研究，同时指导研究生参与理论和计算方面的工作。该项目涉及利用底层核空间内的局部结构来研究非均匀几何和多尺度各向异性核的线性和非线性核近似。这个项目的一个方面对待核有限光滑，存在高度本地化的Riesz基地。这样的基础赋予了一些很好理解的分析的好处，但他们真正的承诺是在改进内核为基础的算法;研究人员将采用这些本地结构，以稳定和加速计算，并开发新的无网格的离散数据和偏微分方程的数值方法。另一个方面涉及高斯和其他无限平滑的内核具有极端的本地化在空间和频率。该研究员将研究和开发非线性近似方法，使用多尺度，各向异性操作来处理函数，这是稀疏的各种表示系统，采用调和分析和统计技术。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Kernels and radial basis functions are valuable mathematical tools that are commonly used to treat large and challenging computational problems. They enjoy widespread use in a variety of branches of science and engineering, like computational chemistry, machine learning, computer graphics, and geosciences, where researchers wish to approximate, model, or extract information from geometrically complicated data sets. This project is devoted to improving the stability and speed of algorithms which use kernels, to apply these algorithms to a number of important scientific problems, and to dramatically extend the approximation power of kernels beyond the current state of the art by developing a multi-scale, anisotropic theory of kernel approximation. Results will benefit those scientific fields that use kernels as a computational tool. Throughout this project, the investigator will continue his own research into these problems, while mentoring graduate students by involving them in theoretical and computational aspects.This project involves exploiting local structures within underlying kernel spaces to study linear and non-linear kernel approximation with non-uniform geometry and multi-scale, anisotropic kernels. One aspect of this project treats kernels of finite smoothness for which there exist highly localized Riesz bases. Such bases confer a number of well-understood analytic benefits, but their true promise is in improving kernel based algorithms; the investigator will employ these local structures to stabilize and speed up computation, and to develop novel mesh-free numerical methods for scattered data and partial differential equations. Another aspect involves Gaussians and other infinitely smooth kernels having extreme localization in both space and frequency. The investigator will study and develop nonlinear approximation methods that use multi-scale, anisotropic operations to treat functions which are sparse in a variety of representation systems, employing techniques from harmonic analysis and statistics.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.