Preconditioning 2D Integer Data for Fast Convex Hull Computations

In order to accelerate computing the convex hull on a set of n points, a heuristic procedure is often applied to reduce the number of points to a set of s points, s ≤ n, which also contains the same hull. We present an algorithm to precondition 2D data with integer coordinates bounded by a box of size p × q before building a 2D convex hull, with three distinct advantages. First, we prove that under the condition min(p, q) ≤ n the algorithm executes in time within O(n); second, no explicit sorting of data is required; and third, the reduced set of s points forms a simple polygonal chain and thus can be directly pipelined into an O(n) time convex hull algorithm. This paper empirically evaluates and quantifies the speed up gained by preconditioning a set of points by a method based on the proposed algorithm before using common convex hull algorithms to build the final hull. A speedup factor of at least four is consistently found from experiments on various datasets when the condition min(p, q) ≤ n holds; the smaller the ratio min(p, q)/n is in the dataset, the greater the speedup factor achieved.

Convex optimization framework for intermediate deadline assignment in soft and hard real-time distributed systems

It is generally challenging to determine end-to-end delays of applications for maximizing the aggregate system utility subject to timing constraints. Many practical approaches suggest the use of intermediate deadline of tasks in order to control and upper-bound their end-to-end delays. This paper proposes a unified framework for different time-sensitive, global optimization problems, and solves them in a distributed manner using Lagrangian duality. The framework uses global viewpoints to assign intermediate deadlines, taking resource contention among tasks into consideration. For soft real-time tasks, the proposed framework effectively addresses the deadline assignment problem while maximizing the aggregate quality of service. For hard real-time tasks, we show that existing heuristic solutions to the deadline assignment problem can be incorporated into the proposed framework, enriching their mathematical interpretation.

On Some Infinite Convex Invariants

The present study on some infinite convex invariants. The origin of convexity can be traced back to the period of Archimedes and Euclid. At the turn of the nineteenth centaury , convexicity became an independent branch of mathematics with its own problems, methods and theories. The convexity can be sorted out into two kinds, the first type deals with generalization of particular problems such as separation of convex sets[EL], extremality[FA], [DAV] or continuous selection Michael[M1] and the second type involved with a multi- purpose system of axioms. The theory of convex invariants has grown out of the classical results of Helly, Radon and Caratheodory in Euclidean spaces. Levi gave the first general definition of the invariants Helly number and Radon number. The notation of a convex structure was introduced by Jamison[JA4] and that of generating degree was introduced by Van de Vel[VAD8]. We also prove that for a non-coarse convex structure, rank is less than or equal to the generating degree, and also generalize Tverberg’s theorem using infinite partition numbers. Compare the transfinite topological and transfinite convex dimensions

Convex extendable trees

The concept of convex extendability is introduced to answer the problem of finding the smallest distance convex simple graph containing a given tree. A problem of similar type with respect to minimal path convexity is also discussed.

Convex linear combination processes for compositions

Aitchison and Bacon-Shone (1999) considered convex linear combinations of compositions. In other words, they investigated compositions of compositions, where the mixing composition follows a logistic Normal distribution (or a perturbation process) and the compositions being mixed follow a logistic Normal distribution. In this paper, I investigate the extension to situations where the mixing composition varies with a number of dimensions. Examples would be where the mixing proportions vary with time or distance or a combination of the two. Practical situations include a river where the mixing proportions vary along the river, or across a lake and possibly with a time trend. This is illustrated with a dataset similar to that used in the Aitchison and Bacon-Shone paper, which looked at how pollution in a loch depended on the pollution in the three rivers that feed the loch. Here, I explicitly model the variation in the linear combination across the loch, assuming that the mean of the logistic Normal distribution depends on the river flows and relative distance from the source origins

Refinement criteria for global illumination using convex funcions

In several computer graphics areas, a refinement criterion is often needed to decide whether to go on or to stop sampling a signal. When the sampled values are homogeneous enough, we assume that they represent the signal fairly well and we do not need further refinement, otherwise more samples are required, possibly with adaptive subdivision of the domain. For this purpose, a criterion which is very sensitive to variability is necessary. In this paper, we present a family of discrimination measures, the f-divergences, meeting this requirement. These convex functions have been well studied and successfully applied to image processing and several areas of engineering. Two applications to global illumination are shown: oracles for hierarchical radiosity and criteria for adaptive refinement in ray-tracing. We obtain significantly better results than with classic criteria, showing that f-divergences are worth further investigation in computer graphics. Also a discrimination measure based on entropy of the samples for refinement in ray-tracing is introduced. The recursive decomposition of entropy provides us with a natural method to deal with the adaptive subdivision of the sampling region

A collocation method for high frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

A new blind equalization algorithm based on convex combination

The convex combination is a mathematic approach to keep the advantages of its component algorithms for better performance. In this paper, we employ convex combination in the blind equalization to achieve better blind equalization. By combining the blind constant modulus algorithm (CMA) and decision directed algorithm, the combinative blind equalization (CBE) algorithm can retain the advantages from both. Furthermore, the convergence speed of the CBE algorithm is faster than both of its component equalizers. Simulation results are also given to verify the proposed algorithm.