The unifying theme of this project is the development and analysis of numerical methods for high-dimensional diffusion problems. In particular, the research will be focused in two areas, namely solving boundary value problems to the heat equation in three dimensions using thermal layer potentials. The second issue is the computation of the discrete Gauss transform for data sets in many dimensions. Both topics involve computations with integral operators that have a Gauss kernel. Since these operators are non-local, naive algorithms scale quadratically in the number of data points. Realistic problems can involve enormous data sets and are therefore tractable only if fast methods, that exploit certain analytic properties of the kernel, are applied. This project will employ high-order Nystrom techniques that are combined with Chebyshev and exponential expansions for the rapid evaluation of integral operators.
The ability to solve the heat equation efficiently is fundamental in many applications of science, technology and medicine. For instance, heat conduction plays an important role in the modeling of geothermal systems, melting, welding, and in thermography as a means to detect breast cancer. Often, the task is to identify an unknown heat source from known surface temperatures and fluxes. The solution of such inverse problems involves solving the forward problem many times, therefore speed is essential for numerical methods. The discrete Gauss transform is used to find relations in large data sets. This can be patterns, correlations, or accumulations in images, texts, or internet traffic. Concrete applications are, for instance, machine recognition of faces, voices and handwriting. By engaging a graduate as well as undergraduate students in research, the proposed activities will also contribute to excellence and growth in the education of the future workforce.