Paul Leopardi, Alvise Sommariva, Marco Vianello
Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for compressed Least Squares fitting.
Giacomo Gigante, Paul Leopardi
The algorithm devised by Feige and Schechtman for partitioning higher dimensional spheres into regions of equal measure and small diameter is combined with David and Christ's construction of dyadic cubes to yield a partition algorithm suitable to any connected Ahlfors regular metric measure space of finite measure.
Paul Leopardi, Ari Stern
This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge-Dirac operator, which is a square root of the abstract Hodge-Laplace operator considered by Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354]. Dirac-type operators are central to the field of Clifford analysis, where recently there has been considerable interest in their discretization. We prove a priori stability and convergence estimates, and show that several of the results in finite element exterior calculus can be recovered as corollaries of these new estimates.