New Directions in Mesh-Free Approximation with Localizable Kernels

核函数和径向基函数是有价值的数学工具，通常用于处理大型和具有挑战性的计算问题。它们广泛应用于各种科学和工程分支，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。本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，并通过发展核近似的多尺度、各向异性理论，大大扩展核的近似能力，超越目前的技术水平。结果将有利于那些使用核函数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，同时通过理论和计算方面指导研究生。该项目涉及利用底层核空间中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是处理有限光滑的核，其中存在高度局部化的Riesz基。这样的基础赋予了许多易于理解的分析好处，但它们真正的承诺是改进基于核的算法；研究人员将利用这些局部结构来稳定和加快计算速度，并为分散数据和偏微分方程开发新的无网格数值方法。另一方面涉及高斯函数和其他在空间和频率上都具有极端局域化的无限光滑核。研究者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各种表示系统中的稀疏函数，采用谐波分析和统计技术。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.