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Applications of Scalable Bases in Kernel Approximation

可扩展基在核逼近中的应用

基本信息

批准号：
1716927
负责人：
Thomas Hangelbroek
金额：
$ 13.35万
依托单位：
University of Hawaii
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1716927&HistoricalAwards=false
关键词：
Applications Scalable Bases Kernel Approximation

项目摘要

1716927Hangelbroek This project concerns research in the area of kernel-based meshless approximation methods, and applications to some large-scale scientific computing problems: numerical solution of partial differential equations (e.g., equations governing fluid flow), tomography (e.g., medical and seismic imaging), and scattered data approximation (modeling of irregularly sampled scientific data). The main focus is on generation and use of scalable bases for kernel spaces; this is a new mathematical tool meant to stabilize and accelerate kernel-based algorithms. Graduate students participate in the work of the project. The classical kernel approach is prized for its ability to provide accurate solutions to computational problems with complicated geometry; in this sense, it is a meshless method, which does not require sampling at uniformly placed sites or the careful construction of triangulations, meshes, or other apparatus. However, it can suffer from instability and heavy computational costs when the underlying problems grow in size. It has been shown that in some cases these drawbacks can be mitigated by construction of scalable bases (as developed by the investigator and collaborators), which can be efficiently generated and lead to stabilization of the underlying calculations. The primary applications considered are threefold. First is the development of an adaptive, meshless method for treating elliptic PDEs. The notion of adaptive refinement of meshes is well understood for classical finite elements; this aspect of the project seeks to use the scalable bases local construction (where basis functions decay at a rate determined by the local density of the scattered centers) to develop an adaptive algorithm where the centers are refined, but no remeshing is needed. Second, the investigator employs kernel-based quadrature, accelerated by using the scalable basis as a preconditioner, to treat tomographic problems. The aim here is to develop parameter selection and error analysis for approximate filtered backprojection of tomographic data acquired from a certain class of phantom images. Third, the investigator develops algorithms for scattered data-fitting that reflect local sampling density of the data. Graduate students participate in the work of the project.

小行星1716927 本项目主要研究基于核的无网格逼近方法，并应用于一些大规模科学计算问题：偏微分方程的数值解（例如，控制流体流动的方程），层析成像（例如，医学和地震成像），以及散射数据近似（不规则采样的科学数据的建模）。主要重点是生成和使用内核空间的可扩展基;这是一种新的数学工具，旨在稳定和加速基于内核的算法。研究生参与该项目的工作。经典核方法因其能够为复杂几何的计算问题提供精确的解决方案而受到重视;从这个意义上说，它是一种无网格方法，不需要在均匀放置的站点进行采样，也不需要仔细构建三角形，网格或其他设备。然而，当潜在问题的规模增长时，它可能会受到不稳定性和沉重的计算成本的影响。已经表明，在某些情况下，这些缺点可以通过构建可扩展的基础（如由研究者和合作者开发的）来减轻，这些基础可以有效地生成并导致基础计算的稳定。考虑的主要应用有三个方面。首先是一个自适应的，无网格的方法治疗椭圆偏微分方程的发展。网格自适应细化的概念在经典有限元中得到了很好的理解;该项目的这一方面旨在使用可扩展的基局部构造（其中基函数以分散中心的局部密度确定的速率衰减）来开发自适应算法，其中中心被细化，但不需要重新网格化。其次，调查人员采用基于核的正交，加速使用可扩展的基础作为预处理器，治疗层析成像的问题。这里的目的是开发的参数选择和误差分析的近似过滤反投影的断层数据采集从一定类别的幻影图像。第三，研究者开发了反映数据局部采样密度的离散数据拟合算法。研究生参与该项目的工作。