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Center for Nonlinear Studies |
Constructing point sets in Euclidean regions or surfaces is a task with many applications. Of interest to us are the use of point sets for, among other things, numerical integration, grid generation for solving PDEs, meshless methods for PDEs, ensemble averages of solutions of stochastic PDEs, surrogate optimization, response surface building, reduced-order modeling, hypercube point sampling, and data compression, clustering, and segmentation. In this talk, we first focus on grid generation. We list several methods for determining triangulations of Euclidean regions and several means for assessing the quality of uniform grids. We also touch on 3D, adaptively determined non-uniform grids, and anisotropic grid generation as well as grids on surfaces, including spheres. At the end we briefly consider two related topics in hypercube points sampling, namely how one can generate improved Latin hypercube samples and how the discrepancy of some high-discrepancy point sets can be lowered through a simple Latinization process. Underlying the discussion is our use of centroidal Voronoi tessellation point sets; these are point sets that are simultaneously generators of Voronoi tessellation and centers of mass of the corresponding Voronoi cells.
Host: Todd Ringler