Applications of Scalable Bases in Kernel Approximation

本项目涉及基于核的无网格近似方法领域的研究，并应用于一些大规模的科学计算问题：偏微分方程的数值解（例如，流体流动方程）、层析成像（例如，医学和地震成像）和分散数据近似（不规则采样科学数据的建模）。主要的焦点是为内核空间生成和使用可扩展的基；这是一个新的数学工具，旨在稳定和加速基于核的算法。研究生参与项目工作。经典核方法因其为复杂几何计算问题提供精确解的能力而备受赞誉；从这个意义上讲，它是一种无网格方法，不需要在均匀放置的地点进行采样，也不需要仔细构建三角测量、网格或其他设备。然而，当潜在问题的规模增长时，它可能会遭受不稳定和沉重的计算成本。研究表明，在某些情况下，这些缺点可以通过构建可扩展的基础（由研究者和合作者开发）来减轻，这些基础可以有效地生成并导致底层计算的稳定。考虑的主要应用有三个方面。首先是发展一种自适应的、无网格的方法来处理椭圆偏微分方程。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；该项目的这一方面寻求使用可扩展的基础局部构造（其中基础函数以由分散中心的局部密度决定的速率衰减）来开发一种自适应算法，其中中心被细化，但不需要重新网格化。其次，研究者采用基于核的正交，通过使用可扩展基作为前置条件来加速，来处理层析问题。本文的目的是对从某类幻像中获取的层析成像数据进行近似滤波反投影的参数选择和误差分析。第三，研究者开发了分散数据拟合算法，以反映数据的局部采样密度。研究生参与项目工作。

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.