A numerical evaluation of some collinear scaling algorithms for unconstrained

Collinear scaling algorithms for unconstrained minimization were first proposed by Davidon (1977,80) so that they may incorporate more information about the problem than is possible with quasi–Newton algorithms. Sorensen (1980,82), and Ariyawansa (1983,90) have derived collinear scaling algorithms as natural extensions of quasi–Newton algorithms. In this paper we describe the results of a comprehensive numerical evaluation of four members in the classes of collinear scaling algorithms derived by Sorensen (1980,82) and Ariyawansa (1983,90), relative to the quasi–Newton algorithms they extend.     

On some topological properties of numerical algorithms

A result quantity in a numerical algorithm is considered as a function of the input data, roundoff and truncation errors. In order to investigate this functional relationship using the methods of mathematical analysis a structural model of the numerical algorithm calledR-automaton is introduced. It is shown that the functional dependence defined by anR-automaton is a continuous rational function in a neighborhood of any data point except in a point set, the Lebesgue measure of which is zero. An effective general-purpose algorithm is presented to compute the derivative of any result quantity with respect to the individual roundoff and truncation errors. Some ways of generalizing theR-automation model without losing the results achieved are finally suggested.     

A numerical method for some nonlinear differential equation models in biology

In this paper, the authors present a full discretization of nonlinear generalisations of the Fischer and Burgers equations with the zero flux on the boundary. Efficiency of the method is derived via a numerical comparison between their numerical solution and the exact solution.     

A comparative study of crossover in differential evolution

In order to understand the role of crossover in differential evolution, theoretical analysis and comparative study of crossover in differential evolution are presented in this paper. Two new crossover methods, namely consecutive binomial crossover and non-consecutive exponential crossover, are designed. The probability distribution and expectation of crossover length for binomial and exponential crossover used in this paper are derived. Various differential evolution algorithms with different crossover methods including mutation-only differential evolution are comprehensively compared at system level instead of parameter level. Based on the theoretical analysis and simulation results, the effect of crossover on the reliability and efficiency of differential evolution algorithms is discussed. Some insights are revealed.     

A semi-analytical numerical method for solving evolution and elliptic partial differential equations

A new method for analyzing initial–boundary value problems for linear and integrable nonlinear partial differential equations (PDEs) has been introduced by one of the authors.     

A brief review of some application driven fast algorithms for elliptic partial differential equations

Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.     

Distributed mixed variant differential evolution algorithms for unconstrained global optimization

This paper proposes a novel distributed differential evolution algorithm called Distributed Mixed Variant Differential Evolution (dmvDE). To alleviate the time consuming trial-and-error selection of appropriate Differential Evolution (DE) variant to solve a given optimization problem, dmvDE proposes to mix effective DE variants with diverse characteristics in a distributed framework. The novelty of dmvDEs lies in mixing different DE variants in an island based distributed framework. The 19 dmvDE algorithms, discussed in this paper, constitute various proportions and combinations of four DE variants (DE/rand/1/bin, DE/rand/2/bin, DE/best/2/bin and DE/rand-to-best/1/bin) as subpopulations with each variant evolving independently but also exchanging information amongst others to co-operatively enhance the efficacy of the distributed DE as a whole. The dmvDE algorithms have been run on a set of test problems and compared to the distributed versions of the constituent DE variants. Simulation results show that dmvDEs display a consistent overall improvement in performance than that of distributed DEs. The best of dmvDE algorithms has also been benchmarked against five distributed differential evolution algorithms. Simulation results reiterate the superior performance of the mixing of the DE variants in a distributed frame work. The best of dmvDE algorithms outperforms, on average, all five algorithms considered.     

New algorithms for the evaluation of complex bifurcation points in ordinary differential equations. a comparative numerical study

Four iterative algorithms (two of them new) for the evaluation of complex (Hopf) bifurcation points in ordinary differential equations are compared. A comparison of effectiveness of the proposed algorithms is made for two examples taken from chemical reaction engineering. The applicability of the algorithms for the solution of various types of problems is also discussed.     

A strategy for parallel implementation of finite difference numerical schemes associated to evolution differential equations

Within the framework of parallel numerical treatment, there are many more results on stationary (or time-frozen) PDE's than on their evolution (or time-dependent) counterparts. This is justified as computer simulation of time flow – an intrinsically sequential phenomenum – leads naturally to a sequential algorithm, thus inhibiting any search for concurrent ones. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the convergence of solutions of some evolution differential equations

Two conjectures concerning the asymptotic behaviour of solutions of two types of evolution differential equations are presented.     

Mathematical analysis of memristor through fractal-fractional differential operators: A numerical study

The newly generalized energy storage component, namely, memristor, which is a fundamental circuit element so called universal charge-controlled mem-element, is proposed for controlling the analysis and coexisting attractors. The governing differential equations of memristor are highly nonlinear for mathematical relationships. The mathematical model of memristor is established in terms of newly defined fractal-fractional differential operators so called Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractal-fractional differential operator. A novel numerical approach is developed for the governing differential equations of memristor on the basis of Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractal-fractional differential operator. We discussed chaotic behavior of memristor under three criteria such as (i) varying fractal order, we fixed fractional order; (ii) varying fractional order, we fixed fractal order; and (ii) varying fractal and fractional orders simultaneously. Our investigated graphical illustrations and simulated results via MATLAB for the chaotic behaviors of memristor suggest that newly presented Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractal-fractional differential operators generate significant results as compared with classical approach.     

On spectral radii of matrices governing some numerical estimating algorithms with incomplete data

Many estimating procedures are carried out with incomplete data by means of different types of EM algorithms. They allow us to obtain maximum likelihood parameter estimates in classical inference and also estimates based on the posterior mode in Bayesian inference. This paper analyzes in detail the spectral radii of the Jacobian matrices algorithm as a possible way to evaluate convergence rates. The eigenvalues of such matrices are explicitly obtained in some cases and, in all of them, a geometric convergence rate is, at least, guaranteed near the optimum. Finally, a comparison between the leading eigenvalues of EM and direct and approximate EM-Bayes algorithms may suggest the efficiency of each case.     

On spectral radii of matrices governing some numerical estimating algorithms with incomplete data

Summary Many estimating procedures are carried out with incomplete data by means of different types of EM algorithms. They allow us to obtain maximum likelihood parameter estimates in classical inference and also estimates based on the posterior mode in Bayesian inference. This paper analyzes in detail the spectral radii of the Jacobian matrices algorithm as a possible way to evaluate convergence rates. The eigenvalues of such matrices are explicitly obtained in some cases and, in all of them, a geometric convergence rate is, at least, guaranteed near the optimum. Finally, a comparison between the leading eigenvalues of EM and direct and approximate EM-Bayes algorithms may suggest the efficiency of each case.     

A numerical study of augmented penalty function algorithms for terminally constrained optimal control problems

Three augmented penalty function algorithms are tested and compared with an ordinary penalty function algorithm for two demonstration optimal control problems. Although the augmented penalty function is quite helpful in solving control problems with terminal state constraints, the convergence can be improved significantly by providing systematic increases in the penalty constant.     

Global convergence of some differential equation algorithms for solving equations involving positive variables

This paper describes some sufficient conditions for global convergence in five differential equation algorithms for solving systems of non-linear algebraic equations involving positive variables. The algorithms are continuous time versions of a modified steepest descent method, Newton's method, a modified Newton's method and two algorithms using the transpose of the Jacobian in place of the inverse of the Jacobian in Newton's method. It is shown that under a set of mildly restrictive conditions the Jacobian transpose algorithm has qualitatively the same convergence as Newton's method.     

An ensemble of discrete differential evolution algorithms for solving the generalized traveling salesman problem

In this paper, an ensemble of discrete differential evolution algorithms with parallel populations is presented. In a single populated discrete differential evolution (DDE) algorithm, the destruction and construction (DC) procedure is employed to generate the mutant population whereas the trial population is obtained through a crossover operator. The performance of the DDE algorithm is substantially affected by the parameters of DC procedure as well as the choice of crossover operator. In order to enable the DDE algorithm to make use of different parameter values and crossover operators simultaneously, we propose an ensemble of DDE (eDDE) algorithms where each parameter set and crossover operator is assigned to one of the parallel populations. Each parallel parent population does not only compete with offspring population generated by its own population but also the offspring populations generated by all other parallel populations which use different parameter settings and crossover operators. As an application area, the well-known generalized traveling salesman problem (GTSP) is chosen, where the set of nodes is divided into clusters so that the objective is to find a tour with minimum cost passing through exactly one node from each cluster. The experimental results show that none of the single populated variants was effective in solving all the GTSP instances whereas the eDDE performed substantially better than the single populated variants on a set of problem instances. Furthermore, through the experimental analysis of results, the performance of the eDDE algorithm is also compared against the best performing algorithms from the literature. Ultimately, all of the best known averaged solutions for larger instances are further improved by the eDDE algorithm.     

A note on some algorithms for matroids

We use results on the optimal basis of a matroid to modify two algorithms, one which finds the circuits of a matroid from the bases, and one which finds the bases from the cocircuits.     

A new differential mutation base generator for differential evolution

A new differential mutation base strategy for differential evolution (DE), namely best of random, is proposed. The best individual among several randomly chosen individuals is chosen as the differential mutation base while the other worse individuals are donors for vector differences. Hence both good diversity and fast evolution speed can be obtained in DE using the new differential mutation base. A comprehensive comparative study is carried out over a set of benchmark functions. Numerical results show that a better balance of reliability and efficiency can be obtained in differential evolution implementing the new generator of differential mutation base, especially in functions with high dimension.     

A numerical investigation of rank-two ellipsoid algorithms for nonlinear programming

We study the performance of some rank-two ellipsoid algorithms when used to solve nonlinear programming problems. Experiments are reported which show that the rank-two algorithms studied are slightly less efficient than the usual rank-one (center-cut) algorithm. Some results are also presented concerning the growth of ellipsoid asphericity in rank-one and rank-two algorithms.