Fast Integral Equation Methods for High-Dimensional Diffusion Problems

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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.