Kernel approximation with scalable bases

在不同的科学领域中，涉及到从空间中分散的地点、从表面或从更复杂的结构中获取的数据的近似和表示问题。内核方法因其处理此类非结构化数据的能力而受到重视。它们也是无网格方法的重要例子，这些计算工具在不需要网格或三角剖分等伴随结构的情况下成功地发挥作用。主要研究人员计划研究可伸缩局部化基(一种处理核的强大新方法)的分析和计算性质，并使用这些基来解决基于核的逼近中的一些基本挑战。研究人员希望产生对处理非结构化数据和大规模数据集的科学家和工程师有用的算法。在整个项目中，研究人员将通过让研究生参与这项研究的理论和应用方面来指导他们。与其他计算工具(如小波、样条和分段多项式有限元)不同，核方法不显式地涉及缩放操作：随着数据变得更加密集，核不需要扩张。这允许核从复杂的几何配置中产生近似，其中单个比例不明显。然而，它经常导致涉及条件较差的大型线性系统的问题，其解既慢又不稳定。已经证明，在许多情况下，潜在的函数空间具有稳定的、可以快速构造的高度局部化的基。有令人信服的证据表明，许多计算问题的近线性处理是可能的，并且可能存在快速算法来处理涉及内核的许多基本运算。提出的研究试图在新的环境下(即，对于新的核和新的流形)产生可伸缩的基，理解和克服由边界效应和高度不均匀的数据排列引起的基本问题，并最终利用这些基实现并行的、快速的核逼近算法。

Problems involving the approximation and representation of data taken from scattered sites in space, from surfaces, or from more complicated structures arise in diverse scientific fields. Kernel methods are valued for their ability to treat such unstructured data. They are also important examples of meshless methods, computational tools that function successfully without the need of accompanying structures like grids or triangulations. The principal investigator plans to study analytic and computational properties of scalable localized bases (a powerful new methodology for working with kernels) and to use these bases to address some fundamental challenges in kernel based approximation. The investigator expects to generate algorithms that will be of use to scientists and engineers who work with unstructured data and large-scale datasets. Throughout the project, the investigator will mentor graduate students by involving them in both the theoretical and the applied aspects of this research.Unlike other computational tools (like wavelets, splines and piecewise polynomial finite elements), kernel methods do not explicitly involve a scaling operation: as data becomes more dense, the kernel is not required to be dilated. This allows kernels to produce approximates from complex geometrical configurations where a single scale is not apparent. However, it often leads to problems involving large, poorly conditioned linear systems, the solution of which is both slow and unstable. It has been demonstrated that in many cases the underlying function spaces possess stable, highly localized bases that can be constructed rapidly. There is compelling evidence that near linear processing of many computational problems is possible, and fast algorithms may exist to treat many of the basic operations involving kernels. The proposed research seeks to generate scalable bases in new settings (i.e., for new kernels and on new manifolds), to understand and overcome fundamental problems stemming from boundary effects and highly non-uniform arrangements of data, and ultimately to implement parallelized, fast algorithms for kernel approximation with these bases.