2014/09/09 by David W. A. Bourne, Bourne, David, Steven Roper +3 · 1 citation
Materials Science · #Block Copolymer Self-Assembly #FOS: Mathematics #Machine Learning in Materials Science #Numerical Analysis (math.NA) #Polymer crystallization and properties
paper · pdf · doi:10.48550/arxiv.1409.2786
openalex publication_date 2014/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we develop a numerical method for solving a class of\noptimization problems known as optimal location or quantization problems. The\ntarget energy can be written either in terms of atomic measures and the\nWasserstein distance or in terms of weighted points and power diagrams\n(generalized Voronoi diagrams). The latter formulation is more suitable for\ncomputation. We show that critical points of the energy are centroidal power\ndiagrams, which are generalizations of centroidal Voronoi tessellations, and\nthat they can be approximated by a generalization of Lloyd's algorithm (Lloyd's\nalgorithm is a common method for finding centroidal Voronoi tessellations). We\nprove that the algorithm is energy decreasing and prove a convergence theorem.\nNumerical experiments suggest that the algorithm converges linearly. We\nillustrate the algorithm in two and three dimensions using simple models of\noptimal location and crystallization. In particular, we test a conjecture about\nthe optimality of the BCC lattice for a simplified model of block copolymers.\n