Optimal power diagrams via function approximation. (September 2018) - Xiao, Yanyang; Chen, Zhonggui; Cao, Juan; Zhang, Yongjie Jessica; Wang, Cheng
Abstract: In this paper, we present a novel method for generating cell complexes with anisotropy conforming to the Hessian of an arbitrary given function. This is done by variationally optimizing the discontinuous piecewise linear approximation of the given functions over power diagrams. The resulting cell complexes corresponding to the approximations are referred to as Optimal Power Diagram (OPD). A hybrid optimization technique, coupling a modified Monte Carlo method with a local search strategy, is tailored for effectively solving the specific optimization task. In contrast to the Optimal Voronoi Tessellation (OVT) method (Budninskiy et al., 2016), our OPD method does not restrict the target functions to be convex, providing more diverse classes of tessellations of the domain. Furthermore, our OPD method generally yields smaller approximation errors than the OVT method, which uses underlaid approximants. We conduct several experiments to demonstrate the efficacy of our optimization algorithm in finding good local minima and generating high-quality anisotropic polytopal meshes. Graphical abstract: Highlights: A novel optimal power diagram (OPD) method is proposed for generating high-quality cell complexes. The anisotropy of the resulting power cells conforms to the Hessian of an arbitrary given function. A modified Monte Carlo method with a local search strategy is tailored for effective optimization.