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4:00 pm Wednesday, October 3, 2012 Geometry-Analysis: Discretizing Compact Manifolds with Minimum Energy: Quasi-uniformity and Complexityby Ed Saff (Vanderbilt) in HB 227- Abstract: The problem of finding configurations of points that are optimally-distributed on a set appears in a number of guises including best-packing problems, coding theory, geometrical modeling, statistical sampling, radial basis approximation and golf-ball design (i.e., where to put the dimples). This talk will focus on classical and recent results concerning geometrical properties of N-point configurations {x_i}_{i=1}^N on a compact metric set A (with metric m) that minimize a weighted Riesz s-energy functional of the form \Sum_{i\neq j} w(x_i, x_j)/m(x_i, x_j)^s for a given `weight' function w on A X A and a parameter s > 0. Specically, if A supports an (Ahlfors) a-regular measure \mu, we prove that whenever s > a , any sequence of weighted minimal Riesz s-energy N-point congurations on A (for`nice' weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as N grows large. Furthermore, if A is an a-rectifiable compact subset of Euclidean space with positive and finite - a dimensional Hausdorff measure, one may choose the weight w to generate a quasi-uniform sequence of congurations that also has (as N -> oo) a prescribed positive continuous limit distribution with respect to a-dimensional Hausdor measure. By utilizing varying weights we also show how to substantially reduce the complexity of energy computations for near optimal configurations. This is joint work with S. Borodachov, D. Hardin and T. Whitehouse.
Submitted by hardt@rice.edu |