Georgia Institute of Technology College of Sciences

I will review a general framework for developing numerical methods working with non-parametrically defined surfaces for various problems. In this talk, I will focus on boundary integral equations. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. Such extension approaches allow us to analyze the well-posedness of the resulting system, develop, systematically and in a unified fashion, numerical schemes for treating a wide range of problems involving differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given. At the end of this talk, I will mention our work in developing multilevel neural network methods for inverting dense and large matrices that arise from boundary integral equations.