Fast Integral Equation Methods for High-Dimensional Diffusion Problems

这个项目的统一主题是发展和分析高维扩散问题的数值方法。特别是，研究将集中在两个方面，即解决边界值问题的热方程在三维热层势。第二个问题是多维数据集的离散高斯变换的计算。 这两个主题都涉及到具有高斯核的积分算子的计算。由于这些运算符是非局部的，朴素算法在数据点的数量上是等比例的。现实的问题可以涉及巨大的数据集，因此，只有当快速的方法，利用某些分析性质的内核，是易于处理的。 本计画将利用高阶Nystrom技巧，结合Chebyshev与指数展开式，以快速求解热传导方程，其能力在许多科学、技术与医学的应用上，都是非常重要的。例如，热传导在地热系统的建模、熔化、焊接以及作为检测乳腺癌的手段的热成像中起着重要作用。通常，任务是从已知的表面温度和通量中识别未知的热源。 这类逆问题的求解需要多次求解正问题，因此速度对数值方法至关重要。 离散高斯变换用于在大数据集中发现关系。这可以是图像、文本或互联网流量中的模式、相关性或积累。具体的应用是，例如，机器识别的面孔，声音和笔迹。 通过让研究生和本科生参与研究，拟议的活动也将有助于未来劳动力教育的卓越和增长。

The unifying theme of this project is the development and analysis ofnumerical methods for high-dimensional diffusion problems. Inparticular, the research will be focused in two areas, namely solvingboundary value problems to the heat equation in three dimensions usingthermal layer potentials. The second issue is the computation of thediscrete Gauss transform for data sets in many dimensions. Bothtopics involve computations with integral operators that have a Gausskernel. Since these operators are non-local, naive algorithms scalequadratically in the number of data points. Realistic problems caninvolve enormous data sets and are therefore tractable only if fastmethods, that exploit certain analytic properties of the kernel, areapplied. This project will employ high-order Nystrom techniques thatare combined with Chebyshev and exponential expansions for the rapidevaluation of integral operators.The ability to solve the heat equation efficiently is fundamental inmany applications of science, technology and medicine. For instance,heat conduction plays an important role in the modeling of geothermalsystems, melting, welding, and in thermography as a means to detectbreast cancer. Often, the task is to identify an unknown heat sourcefrom known surface temperatures and fluxes. The solution of suchinverse problems involves solving the forward problem many times,therefore speed is essential for numerical methods. The discreteGauss transform is used to find relations in large data sets. This canbe patterns, correlations, or accumulations in images, texts, orinternet traffic. Concrete applications are, for instance, machinerecognition of faces, voices and handwriting. By engaging a graduateas well as undergraduate students in research, the proposed activitieswill also contribute to excellence and growth in the education of thefuture workforce.