Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

LIPIcs

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art

Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains

The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements