Abstract The generation of high-quality staggered unstructured grids is considered, leading to the development of a new optimisation-based strategy designed to construct weighted ‘Regular-Power’ tessellations appropriate for co-volume type numerical discretisation techniques. This new framework aims to extend the conventional Delaunay–Voronoi primal-dual structure; seeking to assemble generalised orthogonal tessellations with enhanced geometric quality. The construction of these grids is motivated by the desire to improve the performance and accuracy of numerical methods based on unstructured co-volume type schemes, including various staggered grid techniques for the simulation of fluid dynamics and hyperbolic transport. In this study, a new hybrid optimisation strategy is proposed; seeking to optimise the geometry, topology and weights associated with general, two-dimensional Regular-Power tessellations using a combination of gradient-ascent and energy-based techniques. The performance of this new method is tested experimentally, with a range of complex, multi-resolution primal-dual grids generated for various coastal and regional ocean modelling applications.
More recently, orthogonal primal-dual pairs have been investigated @cite_10 , with developing the so-called Hodge-Optimized Triangulation (HOT) @cite_31 @cite_44 framework in the context of computer graphics applications. The HOT formulation is a generalised variational approach, in which the minimisation of a coupled primal-dual energy functional is undertaken. Such a formulation is designed to optimise the performance of an associated Discrete Exterior Calculus (DEC) type formulation; aiming to generate generalised primal-dual pairs that are typically centroidal, well-centred and pair-wise orthogonal. In this work, employ a weighted primal-dual structure based on the power diagram and its associated regular triangulation, and construct a coupled, optimisation-based approach designed to minimise the combined HOT functional.
@cite_44 @cite_31 @cite_10
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We introduce Hodge-optimized triangulations (HOT), a family of well-shaped primal-dual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each simplex lies within the simplex, circumcentric duals are often preferred due to the induced orthogonality between primal and dual complexes. We instead promote the use of weighted duals ("power diagrams"). They allow greater flexibility in the location of dual vertices while keeping primal-dual orthogonality, thus providing a valuable extension to the usual choices of dual by only adding one additional scalar per primal vertex. Furthermore, we introduce a family of functionals on pairs of complexes that we derive from bounds on the errors induced by diagonal Hodge stars, commonly used in discrete computations. The minimizers of these functionals, called HOT meshes, are shown to be generalizations of Centroidal Voronoi Tesselations and Optimal Delaunay Triangulations, and to provide increased accuracy and flexibility for a variety of computational purposes. This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the second derivative are combined with the computations for the centering direction. Computations in this approach do not require that primal and dual solutions be feasible. Expressions are given to compute all the higher-order derivatives of the trajectory of interest. The implementation ensures that a suitable potential function is reduced by a constant amount at each iteration.There are several salient features of this approach. An adaptive heuristic for estimating the centering parameter is given. The approach used to compute the step length is also adaptive. A new practical approach to compute the starting point is given. This approach treats primal and dual problems symmetrically.Computational results on a subset of problems available from netlib are given. On mutually tested problems the results show... Motivated by practical numerical issues in a number of modeling and simulation problems, we introduce the notion of a compatible dual complex to a primal triangulation, such that a simplicial mesh and its compatible dual complex (made out of convex cells) form what we call a primal-dual triangulation. Using algebraic and computational geometry results, we show that compatible dual complexes exist only for a particular type of triangulation known as weakly regular. We also demonstrate that the entire space of primal-dual triangulations, which extends the well known (weighted) Delaunay Voronoi duality, has a convenient, geometric parametrization. We finally discuss how this parametrization may play an important role in discrete optimization problems such as optimal mesh generation, as it allows us to easily explore the space of primal-dual structures along with some important subspaces. We propose a new primal-dual algorithmic framework for a prototypical constrained convex optimization template. The algorithmic instances of our framework are universal since they can automatically adapt to the unknown Holder continuity degree and constant within the dual formulation. They are also guaranteed to have optimal convergence rates in the objective residual and the feasibility gap for each Holder smoothness degree. In contrast to existing primal-dual algorithms, our framework avoids the proximity operator of the objective function. We instead leverage computationally cheaper, Fenchel-type operators, which are the main workhorses of the generalized conditional gradient (GCG)-type methods. In contrast to the GCG-type methods, our framework does not require the objective function to be differentiable, and can also process additional general linear inclusion constraints, while guarantees the convergence rate on the primal problem.