Explicit Iterative Algorithm Research Articles

Objective priors from maximum entropy in data classification

Lack of knowledge of the prior distribution in classification problems that operate on small data sets may make the application of Bayes’ rule questionable. Uniform or arbitrary priors may provide classification answers that, even in simple examples, may end up contradicting our common sense about the problem. Entropic priors (EPs), via application of the maximum entropy (ME) principle, seem to provide good objective answers in practical cases leading to more conservative Bayesian inferences. EP are derived and applied to classification tasks when only the likelihood functions are available. In this paper, when inference is based only on one sample, we review the use of the EP also in comparison to priors that are obtained from maximization of the mutual information between observations and classes. This last criterion coincides with the maximization of the KL divergence between posteriors and priors that for large sample sets leads to the well-known reference (or Bernardo’s) priors. Our comparison on single samples considers both approaches in prospective and clarifies differences and potentials. A combinatorial justification for EP, inspired by Wallis’ combinatorial argument for entropy definition, is also included.The application of the EP to sequences (multiple samples) that may be affected by excessive domination of the class with the maximum entropy is also considered with a solution that guarantees posterior consistency. An explicit iterative algorithm is proposed for EP determination solely from knowledge of the likelihood functions. Simulations that compare EP with uniform priors on short sequences are also included.

TAGE iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations

In the present paper, we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation. Difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm. The schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method. Computational results are presented comparing the two methods in terms of accuracy and execution times. The results indicate the advantage of using parallel implementation of the new method.

An Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces

In this paper, we introduce a new explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive maps on Hilbert spaces. An application and a numerical result are given for illustration.

Iterative Algorithms for Finding Common Solutions to Variational Inclusion Equilibrium and Fixed Point Problems

AbstractThe main purpose of this paper is to introduce an explicit iterative algorithm to study the existence problem and the approximation problem of solution to the quadratic minimization problem. Under suitable conditions, some strong convergence theorems for a family of nonexpansive mappings are proved. The results presented in the paper improve and extend the corresponding results announced by some authors.

Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space X whose norm is uniformly Gâteaux differentiable and T : C → C be a continuous pseudo-contraction with a nonempty fixed point set F ( T ) . For arbitrary given element u ∈ C and for t ∈ ( 0 , 1 ) , let { y t } be the unique continuous path such that y t = ( 1 − t ) T y t + t u . Assume that y t → p ∈ F ( T ) as t → 0 . Let { α n } , { β n } and { γ n } be three real sequences in (0, 1) satisfying the following conditions: (i) α n + β n + γ n = 1 ; (ii) lim n → ∞ α n = lim n → ∞ β n = 0 ; (iii) lim n → ∞ β n 1 − γ n = 0 ; or (iii)′ ∑ n = 0 ∞ α n 1 − γ n = ∞ . Let { ϵ n } be a summable sequence of positive numbers. For arbitrary initial datum x 0 = x 0 0 ∈ C and a fixed n ≥ 0 , construct elements { x n m } as follows: x n m + 1 = α n u + β n x n + γ n T x n m , m = 0 , 1 , 2 , … . Suppose that there exists a least positive integer N ( n ) satisfying the following condition: ‖ T x n N ( n ) + 1 − T x n N ( n ) ‖ ≤ γ n − 1 ( 1 − γ n ) ϵ n . Define iteratively a sequence { x n } in an explicit manner as follows: x n + 1 = x n + 1 0 = x n N ( n ) + 1 = α n u + β n x n + γ n T x n N ( n ) , n ≥ 0 . Then { x n } converges strongly to a fixed point of T . For all the continuous pseudo-contractive mappings for which is possible to construct the sequence x n , this result improves and extends a recent result of Yao et al. [Yonghong Yao, Yeong-Cheng Liou, Rudong Chen, Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., 67 (2007) 3311–3317].

Loosening of elastic inclusions

A numerical method is presented for simulating the occurrence of localized slip and separation along the interfaces of multiple, randomly distributed, circular elastic inclusions in an infinite elastic plane. The method is an extension of a direct boundary integral approach previously described elsewhere for solving problems involving perfectly bonded circular inclusions. Here, we allow displacement discontinuities to develop along the inclusion/matrix interfaces in accordance with a linear Mohr–Coulomb yield condition combined with a tensile strength cut-off. The displacements, tractions, and displacement discontinuities on the inclusion boundaries are all represented by truncated Fourier series, and an explicit iterative algorithm is adopted to determine zones of slip and separation under the prevailing loading conditions. Several examples are given to demonstrate the accuracy and generality of the approach.

Optimal Voltage Design for Static Shape Control of Smart Structures with Nonlinear Piezoelectric Actuators

In this paper, static shape control of structures with nonlinear piezoelectric actuators is investigated. Firstly, an adhesive element is developed to model smart plates with nonlinear piezoelectric actuator patches. The adhesive element combines a pair of collocated four-node quadrilateral elements for the upper and lower adherents and a pseudo-adhesive layer element with constant shear and peel strains along its thickness. Secondly, after establishing the equilibrium equations with nonlinear actuating forces, an explicit iterative algorithm for finding the optimal control voltage distribution on the nonlinear piezoelectric actuators is formulated, and the converging speed of this iterative algorithm is also estimated. Finally, a simulation example is given to illustrate that the present algorithm is effective in finding the optimal control voltage distribution for shape control of nonlinearly actuated structures. Significant effect of the nonlinear piezoelectric constitutive relationship on the optimal control voltage is also observed.

A HIGH-EFFICIENCY PARALLEL SOLUTION OF THE RADIATIVE TRANSFER EQUATION

Solution of the radiative transfer equation for nontrivial geometry and optical properties is a massive numerical problem, and can benefit from parallel processing to reduce the time to solution. Spatial parallelization is presented, applied to the DOTS algorithm with finite-volume directional discretization, and an explicit pseudo-time-stepping iterative solution with multigrid acceleration. Two- and three-dimensional benchmark problems were investigated with up to 16 processors on a shared-memory vector machine. In contrast to previous studies, high parallel efficiency was demonstrated up to the highest number of processors. The reasons for the good performance can be traced to the explicit iterative algorithm, which permits easy parallelization without affecting the rate of convergence; and to the use of high-bandwidth shared memory, which significantly reduces communication overhead.

MODELING OF DIELECTRIC FLUID SOLIDIFICATION WITH CHARGED PARTICLES IN ELECTRIC FIELDS AND REDUCED GRAVITY

A mathematical model and an explicit finite-difference iterative integration algorithm for two-dimensional laminar steady flow and solidification of an incompressible, viscous, electrically conducting but neutrally charged melt containing electrically charged panicles and exposed to an externally applied electrostatic field were developed. The system of governing electrohydrodynamic equations was derived from a combination of Maxwell's equations and the Navier-Stokes equations, including thermally induced buoyancy, latent heat release, and Joule heating, while accounting for the mushy region. Physical properties were treated as arbitrarily temperature-dependent. Numerical results demonstrate the existence of strong electrothermoconvective motion in the melt and quantify its influence on the amount of accrued solid, deposition pattern of the electrically charged particles inside the accrued solid, and the melt/solid interface shape.