Discrete Energy and Polarization Problems on Manifolds with Applications

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The configuration of points on a surface at which computations are carried out affects how well one can approximate functions of the surface or the shape of the surface itself. Configurations that optimize some notion of energy associated with the arrangement of points can lead to better approximations. This research project explores efficient implementation of such optimal configurations, to provide improved tools for data sampling methods and for simulating complex geophysical processes such as climate changes and heat diffusion in Earth's mantle.Optimization problems for energy are of significance in discretizing various surfaces and volumes. For example, point configurations generated by a suitably weighted Riesz energy can approximate a prescribed density on the object. The investigative team is (i) implementing numerically the theory developed by the principal investigators by exploring efficient algorithms for the computation of near optimal energy configurations on manifolds, and (ii) developing the asymptotic theory of periodic versions of optimal energy problems and especially their connection to lattice (crystalline) structures. Undergraduate and graduate students as well as post-doctoral fellows play an active role in the research program.