Beneath-Beyond Algorithm Research Articles

A convex hull algorithm for a grid minimization of Gibbs energy as initial step in equilibrium calculations in two-phase multicomponent alloys

An efficient method for computing two-phase equilibria in multicomponent alloys is proposed. It relies on a hybrid scheme using as a first step the Quick Hull algorithm with the general dimension Beneath-Beyond algorithm of Barber et al. (1996) [1] adapted for computing the convex hull of the Gibbs energy hypersurface of multicomponent two-phase alloys. First, the salient features of our method are illustrated with calculations of isothermal ferrite–austenite equilibria in Fe–C–Cr, described by the Compound Energy Formalism with two sublattices. Then, computational efficiency is investigated to suggest strategies to achieve the best compromise between efficiency and robustness. Finally, successive equilibrium calculations in a Fe–C–Cr-Mo steel over a large temperature range show the benefit of computing the convex hull before performing the conventional Newton–Raphson search.

An Efficient Solution of Real-Time Fuzzy Regression Analysis to Information Granules Problem

Regression models are well known and widely used as one of the important categories of models in system modeling. In this paper, we extend the concept of fuzzy regression in order to handle real-time implementation of data analysis of information granules. An ultimate objective of this study is to develop a hybrid of a genetically-guided clustering algorithm called genetic algorithm-based Fuzzy C-Means (GAFCM) and a convex hull-based regression approach being regarded as a potential solution to the formation of information granules. It is shown that a setting of Granular Computing helps us reduce the computing time, especially in case of real-time data analysis, as well as an overall computational complexity. We propose an efficient real-time information granules regression analysis based on the convex hull approach in which a Beneath-Beyond algorithm is employed to design sub-convex hulls as well as a main convex hull structure. In the proposed design setting, we emphasize a pivotal role of the convex hull approach or more specifically the Beneath-Beyond algorithm, which becomes crucial in alleviating limitations of linear programming manifesting in system modeling.

Real-time fuzzy regression analysis: A convex hull approach

In this study, we present an enhancement of fuzzy regression analysis with regard to its aspect of real-time processing. Let us recall that fuzzy regression generalizes the concept of classical (numeric) regression in the sense of bringing additional capabilities that allow the model to deal with fuzzy (granular) data. We show that a convex hull method provides a useful vehicle to reduce computing time, which becomes of particular relevance in case of real-time data analysis. Our objective is to develop an efficient real-time fuzzy regression analysis based on the use of convex hull, specifically a Beneath-Beyond algorithm. In this algorithm, the re-construction of convex hull edges depends on incoming vertices while a re-computing procedure can be realized in real-time. We demonstrate the use of the developed enhancement to application to unit performance assessment and air pollution data. An important role of convex hull is contrasted with the limitations of linear programming used in the “ standard” regression.

An Intelligent Data Analysis-Base: Evaluation of Nuclear Power Plants Output Flow

Abstract in order to realize stable electricity generation, nuclear power plant (NPP) generators are evaluated in their performance of generated output power in term of quality and quantity. Therefore, the evaluation is realized on the basis of several influence factor, which have to be analyzed via the exploitation of heterogeneous data sets obtained from scattered locations and different types of sources. In this paper, we stress the pivotal role of extended fuzzy switching regression analysis in handling this type of data, which come from real world of the NPPs industry. The key objective of this study is to implement the enhancement of a convex hull approach in the fuzzy switching regression analysis process which can be viewed as an intelligent data analysis (IDA) approach. This approach is concerned with the effective combination of fuzzy sets theory with the analysis of large amounts of online data. For deploying the multisource data problem, the fuzzy regression analysis based on convex-hull, specifically Beneath-Beyond algorithm. The selected IDA becomes a potential analysis vehicle to successfully reduce the computing time as well as minimize the computational complexity. It is shown that the proposed approach becomes an efficient vehicle for the evaluation of produced output flow by NPPs. The study offers an interesting and practically appealing alterative platform to evaluate the quality and quantity of produced output flow of NPPs.

Average-Case Analysis of the Double Description Method and the Beneath-Beyond Algorithm

This paper deals with the average computational effort for calculating all vertices of a polyhedron described by m inequalities in an n-dimensional space, when we apply the so-called "Double Description Method" (from a dual point of view, i.e. for finding all facets of the convex hull of m given points, this is equivalent to application of the "Beneath-Beyond Algorithm"). Both are incremental algorithms, i.e. they develop the information about the polyhedron stepwise by taking the inequalities/points successively into regard. The average-case analysis is done with respect to the Rotation-Symmetry Model, which is well known from the corresponding analysis of the Simplex Method for linear programming. In this model degenerate problems occur with probability 0. So the (finite) effort to solve those problems has no impact on the expected effort in our model. All the derived results and complexities apply equivalently to both algorithms and to the corresponding primal and dual problems.

A Complete Implementation for Computing General Dimensional Convex Hulls

We present a robust implementation of the Beneath-Beyond algorithm for computing convex hulls in arbitrary dimension. Certain techniques used are of independent interest in the implementation of geometric algorithms. In particular, two important, and often complementary, issues are studied, namely exact arithmetic and degeneracy. We focus on integer arithmetic and propose a general and efficient method for its implementation based on modular arithmetic. We suggest that probabilistic modular arithmetic may be of wide interest, as it combines the advantages of modular arithmetic with the speed of randomization. The use of perturbations as a method to cope with input degeneracy is also illustrated. A computationally efficient scheme is implemented which, moreover, greatly simplifies the task of programming. We concentrate on postprocessing, often perceived as the Achilles' heel of perturbations. Experimental results illustrate the dependence of running time on the various input parameters and attempt a comparison with existing programs. Lastly, we discuss the visualization capabilities of our software and illustrate them for problems in computational algebraic geometry. All code is publicly available.

The quickhull algorithm for convex hulls

The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.