Applications and analysis of discrete energy and polarization

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This award supports the research program of the Principal Investigators concerned primarily with mathematical and computational aspects of "discrete energy problems." These are problems with classical roots -- the simplest example is that of determining the configuration of a finite number of point charges on a conducting shell interacting via mutual electrostatic repulsion -- yet are relevant to several modern large-scale scientific computing applications. Representing surfaces and volumes by a large but finite number of points is useful for data sampling methods and for simulating complex geophysical processes such as climate changes and heat diffusion in Earth's mantle. Minimal energy methods for particles interacting via pairwise repulsive interactions will be used to generate such approximating point configurations. Graduate students and post-doctoral fellows will play an active role in the research program, which will involve collaboration with scientists at the National Center for Atmospheric Research.The investigative team is (i) developing an asymptotic analysis of discrete energy problems with an external field for nonintegrable Riesz kernels and is applying the theory to develop computationally efficient algorithms for generating large-scale node sets modeling a specified nonuniform density; (ii) analyzing energy asymptotics for periodic problems and their connection to optimal lattice (crystalline) structures; (iii) investigating the geometrical properties of optimal and near optimal polarization configurations on manifolds, especially for large numbers of points.