Maximum principles for vectorial approximate minimisers on non-convex functionals

We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group. Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler–Lagrange system of PDE.

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

Rapid preconditioning of data for accelerating convex hull algorithms

Given a dataset of two-dimensional points in the plane with integer coordinates, the method proposed reduces a set of n points down to a set of s points s ≤ n, such that the convex hull on the set of s points is the same as the convex hull of the original set of n points. The method is O(n). It helps any convex hull algorithm run faster. The empirical analysis of a practical case shows a percentage reduction in points of over 98%, that is reflected as a faster computation with a speedup factor of at least 4.

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.

Orthogonal packing of identical rectangles within isotropic convex regions

A mixed integer continuous nonlinear model and a solution method for the problem of orthogonally packing identical rectangles within an arbitrary convex region are introduced in the present work. The convex region is assumed to be made of an isotropic material in such a way that arbitrary rotations of the items, preserving the orthogonality constraint, are allowed. The solution method is based on a combination of branch and bound and active-set strategies for bound-constrained minimization of smooth functions. Numerical results show the reliability of the presented approach. (C) 2010 Elsevier Ltd. All rights reserved.

Algorithms for Maximum Independent Set in Convex Bipartite Graphs

A bipartite graph G = (V, W, E) is convex if there exists an ordering of the vertices of W such that, for each v. V, the neighbors of v are consecutive in W. We describe both a sequential and a BSP/CGM algorithm to find a maximum independent set in a convex bipartite graph. The sequential algorithm improves over the running time of the previously known algorithm and the BSP/CGM algorithm is a parallel version of the sequential one. The complexity of the algorithms does not depend on |W|.

Convex Hulls in Image Processing : A Scoping Review

The demands of image processing related systems are robustness, high recognition rates, capability to handle incomplete digital information, and magnanimous flexibility in capturing shape of an object in an image. It is exactly here that, the role of convex hulls comes to play. The objective of this paper is twofold. First, we summarize the state of the art in computational convex hull development for researchers interested in using convex hull image processing to build their intuition, or generate nontrivial models. Secondly, we present several applications involving convex hulls in image processing related tasks. By this, we have striven to show researchers the rich and varied set of applications they can contribute to. This paper also makes a humble effort to enthuse prospective researchers in this area. We hope that the resulting awareness will result in new advances for specific image recognition applications.

Convex combinations of long memory estimates from different sampling rates

Convex combinations of long memory estimates using the same data observed at different sampling rates can decrease the standard deviation of the estimates, at the cost of inducing a slight bias. The convex combination of such estimates requires a preliminary correction for the bias observed at lower sampling rates, reported by Souza and Smith (2002). Through Monte Carlo simulations, we investigate the bias and the standard deviation of the combined estimates, as well as the root mean squared error (RMSE), which takes both into account. While comparing the results of standard methods and their combined versions, the latter achieve lower RMSE, for the two semi-parametric estimators under study (by about 30% on average for ARFIMA(0,d,0) series).

Integral representation with convex capacities that are squeeze of (additive) probability measures

In this paper I will investigate the conditions under which a convex capacity (or a non-additive probability which exhibts uncertainty aversion) can be represented as a squeeze of a(n) (additive) probability measure associate to an uncertainty aversion function. Then I will present two alternatives forrnulations of the Choquet integral (and I will extend these forrnulations to the Choquet expected utility) in a parametric approach that will enable me to do comparative static exercises over the uncertainty aversion function in an easy way.

Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs

We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. We prove the almost sure convergence of these decomposition methods when the relatively complete recourse assumption holds. We also prove the almost sure convergence of these algorithms when applied to risk-averse multistage stochastic linear programs that do not satisfy the relatively complete recourse assumption. The analysis is first done assuming the underlying stochastic process is interstage independent and discrete, with a finite set of possible realizations at each stage. We then indicate two ways of extending the methods and convergence analysis to the case when the process is interstage dependent.

Efficacy of linear and convex transducers for ultrasound-guided transvaginal follicle aspiration

A new device was developed to hold linear transducers for transvaginal follicle aspiration. Efficacy of follicle aspiration was compared using a linear 6 MHz and a convex 5 MHz transducer. Fifty-five cows were submitted to follicle aspiration at random days of the estrous cycle. Aspirations were conducted with linear (n = 28) and convex (n = 38) transducers with 18 G needles at a negative pressure corresponding to 13 ml H2O/min. A greater number of follicles were aspirated using convex than to linear probe (12.4 versus 7.8, respectively, P < 0.05). Mean number of oocytes and recovery rates were similar for convex (5.4 and 48.6%) and linear (4.6 and 59.3%) transducers. Limited space between the linear transducer and needle guide restricted access to some portions of the ovary, reducing the number of follicles aspirated using a linear transducer. The newly developed adaptor allowed greater stability, holding the ovaries firmly against the linear transducer. This diminished mobility permitted a similar number of oocytes to be recovered with both transducers. In conclusion, this new adaptor provided a low cost alternative for routine follicle aspiration and oocyte recovery in cattle. (C) 2002 Elsevier B.V. All rights reserved.

Projections Onto Convex Sets through Particle Swarm Optimization and its application for remote sensing image restoration

Image restoration attempts to enhance images corrupted by noise and blurring effects. Iterative approaches can better control the restoration algorithm in order to find a compromise of restoring high details in smoothed regions without increasing the noise. Techniques based on Projections Onto Convex Sets (POCS) have been extensively used in the context of image restoration by projecting the solution onto hyperspaces until some convergence criteria be reached. It is expected that an enhanced image can be obtained at the final of an unknown number of projections. The number of convex sets and its combinations allow designing several image restoration algorithms based on POCS. Here, we address two convex sets: Row-Action Projections (RAP) and Limited Amplitude (LA). Although RAP and LA have already been used in image restoration domain, the former has a relaxation parameter (A) that strongly depends on the characteristics of the image that will be restored, i.e., wrong values of A can lead to poorly restoration results. In this paper, we proposed a hybrid Particle Swarm Optimization (PS0)-POCS image restoration algorithm, in which the A value is obtained by PSO to be further used to restore images by POCS approach. Results showed that the proposed PSO-based restoration algorithm outperformed the widely used Wiener and Richardson-Lucy image restoration algorithms. (C) 2010 Elsevier B.V. All rights reserved.

Do static sources respond to massive scalar particles from the Hawking radiation as uniformly accelerated ones do in the inertial vacuum?

We examine the recently found equivalence for the response of a static scalar source interacting with a massless Klein-Gordon field when the source is (i) static in Schwarzschild spacetime, in the Unruh vacuum associated with the Hawking radiation, and (ii) uniformly accelerated in Minkowski spacetime, in the inertial vacuum, provided that the source's proper acceleration is the same in both cases. It is shown that this equivalence is broken when the massless Klein-Gordon field is replaced by a massive one.