Applications of Scalable Bases in Kernel Approximation

1716927 Hangelbroek这个项目涉及基于核的无网格近似方法领域的研究，以及在一些大型科学计算问题中的应用：偏微分方程组的数值解(例如，控制流体流动的方程)、层析成像(例如，医学和地震成像)和散乱数据近似(对不规则采样的科学数据进行建模)。主要关注于内核空间可伸缩基的生成和使用；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。经典的核方法因其能够为复杂几何的计算问题提供准确的解决方案而备受赞誉；从这个意义上说，它是一种无网格法，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剖分、网格或其他设备。然而，当潜在问题的规模增大时，它可能会受到不稳定和高昂的计算成本的影响。已经证明，在某些情况下，这些缺点可以通过构建可伸缩的基(如研究人员和合作者开发的)来缓解，这种基可以有效地生成并导致基本计算的稳定。考虑的主要应用有三个方面。首先是发展一种自适应、无网格的方法来处理椭圆型偏微分方程组。网格自适应细化的概念对于经典有限元是很好理解的；该项目的这一方面寻求使用可伸缩的基局部构造(其中基函数以由分散中心的局部密度确定的速率衰减)来开发一种自适应算法，其中中心被细化，但不需要重新划分网格。其次，调查者使用基于核的求积，通过使用可伸缩基作为预条件算子来加速，以处理层析问题。这里的目的是为从某一类体模图像获取的层析数据的近似滤波反投影进行参数选择和误差分析。第三，研究人员开发了反映数据局部采样密度的散乱数据拟合算法。研究生参与该项目的工作。

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.