Possible Number Of Iterations Research Articles

A simple method for rejection sampling efficiency improvement on SIMT architectures

We derive a probability distribution for the possible number of iterations required for a SIMT (single instruction multiple thread) program using rejection sampling to finish creating a sample across all threads. This distribution is found to match a recently proposed distribution from Chakraborty and Gupta (in: Communications in statistics: theory and methods, 2015) that was shown as a good approximation of certain datasets. This work demonstrates an exact application of this distribution. The distribution can be used to evaluate the relative merit of some sampling methods on the GPU without resort to numerical tests. The distribution reduces to the expected geometric distribution in the single thread per warp limit. A simplified formula to approximate the expected number of iterations required to obtain rejection iteration samples is provided. With this new result, a simple, efficient layout for assigning sampling tasks to threads on a GPU is found as a function of the rejection probability without recourse to more complicated rejection sampling methods.

THE ALGORITHM FOR DETERMINING THE STARTING POINT IN THE SIMULATION BY THE METHOD OF POSSIBLE DIRECTIONS

Context. The problem of determining the starting point when of G. Zoutendijk’s method of possible directions is considered, namely the case where such a point can be arbitrary or even unknown. Presented in the article allows to determine the further direction of movement in such a way that with a minimum number of study points to obtain the most accurate result. The solution of the above problem relates to the description of complex surfaces that can be represented by a gully function. The definition of the starting point is used in the practice of modeling the development of environmental risks in technogenic pollution of intersections of territories. Objective. The solution of the problem of choosing the starting point when using the method of possible directions for describing complex surfaces ravine function. Method. The article uses the method of G. Zoutendijk for solving tasks Chebyshev’s approximation. Results. An algorithm for determining the starting point for a problem with linear constraints is introduced. An approach for solving complex tasks of representation of gully functions with the minimum possible number of iterations is presented. The results of mathematical modeling are tested in practice to solve the problem of description of the contaminated area by hydrogen’s radioisotope – tritium. Conclusions. An approach to determining the starting point in the calculations using the method of possible directions of G. Zoutendijk that allows you to select a point to build from it the motion vector with the specified step and to determine the direction of the second point to draw the next step. This takes into account the limitations, in particular – the values at each new point must be expressed in a non-negative number. For the given normalization, there are a number of features that should be taken into account in the calculation and construction of algorithms for solving applied problems according to the proposed approach. For example, some iterations can lead to more calculations and transformations on each iteration, but the number of iterations is smaller than other types of iterations. This depends on the size of the task and the number of constraints in each case. The presented approach can be used in the development of methods, models and algorithms for the description of rough terrain to solve the problems of visualization of the process of pollution by dangerous substances.

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

Tradeoffs Between Convergence Speed and Reconstruction Accuracy in Inverse Problems

Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the $\ell _1$ -minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm.

Infinite impulse response systems modeling by artificial intelligent optimization methods

Artificial Intelligent Optimization (AIO) algorithms learn from the past searches via using a group of individuals or agents. These Artificial Intelligence-based optimizing techniques are able to solve complex optimization problems with complicated constraints. They find the optimal in the low possible number of iterations, where optimal means the best from all possibilities selected from a special point of view. This paper presents a research on employing AIO methods with aim to Infinite Impulse Response (IIR) system modeling for design and optimization of IIR digital filters. The proposed methods cover a variety of AIO methods; algorithm based on evolution strategy (genetic algorithm) and heuristic algorithms (particle swarm optimization, population-based; gravitational search algorithm, and inclined planes system optimization, both population-based and based on Newton’s laws). In this paper, the IIR system modeling is solved as a constrained single-objective optimization problem in the Mean Squared Error (MSE) fitness function and is evaluated for two different benchmark IIR plants with high and low orders. To evaluate performance, efficiency and efficacy of the methods, two important criteria are used: “Indicator of Success (IoS)” and “Degree of Reliability (DoR)”. In addition, the effect of decreasing population size (search agents) is analyzed on the performance and efficiency of the algorithms. Simulation results clarify the success of the research in terms of the MSE, IoS and DoR.

Diseño óptimo del pico de lanzamiento de puentes lanzados de hormigón pretensado

The design of the launching nose of an incrementally launched bridge determines its constructive process and, therefore, also its dimensions. The optimization of the launching nose can raise as a first step to improve the design of a launched bridge. The conventional design process of a launching nose is based on trial and error method to reduce bending moment of prestressed concrete deck at the foremost support during launch. In this way, there is no guarantee that the obtained solution is the best among all the possible solutions since they all depend on the experience and intuition of a designer, and they are also restricted by a limited number of possible iterations. Given that launched bridges constitute an important constructive typology, all the available capacities of design innovation should be incorporated, among which it can be found numerical optimization. This research work proposes an objective and rigorous formulation to optimize a launching nose of launched bridge under real constraints that a bridge designer can encounter in practice. Comparing the results obtained by conventional process and that by optimization techniques, it can be verified that some of the assumptions, considered in classical design methods of a launching nose, are not based on any theoretical foundation. This fact demonstrates the utility of numerical optimization to improve a design.

Improved Optimization Formulations for Launching Nose of Incrementally Launched Prestressed Concrete Bridges

The conventional design process of a launching nose of an incrementally launched bridge is based on trial-and-error methods to reduce bending moment of prestressed concrete deck at the foremost support during launch. With this method, there is no guarantee that the obtained solution is the best among all the possible solutions, since they all depend on the experience and intuition of a designer, and they are also restricted by a limited number of possible iterations. Given that launched bridges constitute an important constructive typology, all the available capacities of design innovation should be incorporated, including numerical optimization. This research work proposes an objective and rigorous formulation to optimize the launching nose of a launched bridge under real constraints that a bridge designer can encounter in practice. Comparing the results obtained by conventional process and those obtained by optimization techniques allows us to verify that some of the assumptions considered in classical design methods of a launching nose are not based on any theoretical foundation. This fact demonstrates the utility of numerical optimization to improve a design.

An Adaptive Weighted Bisection Method For Finding Roots Of Non-Linear Equations

In this paper, an algorithm for finding the roots of non-linear equations is developed by introducing a weight in the formula of the Bisection Method (BM). Initially, we use a fixed weight to solve an equation in the least possible number of iterations. In a second stage, we develop a method termed as the Adaptive Weighted Bisection Method (AWBM) in order to update the weight at each iteration. The adaptation is achieved by minimizing the function values of the iterates with respect to the weight. Our numerical experiments show that, the AWBM, based on minimizing a function value with respect to the weight, achieves quadratic convergence. However, the method differs from the classic second order Newton-Raphson by guaranteeing convergence through bracketing.

Reduced complexity in-phase/quadrature-phase M-QAM turbo equalization using iterative channel estimation

A reduced complexity trellis-based turbo equalizer known as the in-phase (I)/quadrature-phase (Q) turbo equalizer (TEQ-IQ) invoking iterative channel impulse response (CIR) estimation is proposed. The underlying principle of TEQ-IQ is based on equalizing the I and Q component of the transmitted signal independently. This requires the equalization of a reduced set of separate I and Q signal components in comparison to all of the possible I/Q phasor combinations considered by the conventional trellis-based equalizer. It was observed that the TEQ-IQ operating in conjunction with iterative CIR estimation was capable of achieving the same performance as the full-complexity conventional turbo equalizer (TEQ-CT) benefiting from perfect CIR information for both 4- and 16-quadrature amplitude modulation (QAM) transmissions, while attaining a complexity reduction factor of 1.1 and 12.2, respectively. For 64-QAM, the TEQ-CT receiver was too complex to be investigated by simulation. However, by assuming that only two turbo equalization iterations were required, which is the lowest possible number of iterations, the complexity of the TEQ-IQ was estimated to be a factor of 51.5 lower than that of the TEQ-CT. Furthermore, at BER = 10/sup -3/ the performance of the TEQ-IQ 64-QAM receiver using iterative CIR estimation was only 1.5 dB away from the associated decoding performance curve of the nondispersive Gaussian channel.

Predicting humidity effect on adsorption capacity of activated carbon for water-immiscible organic vapors

This paper presents a simple numerical solution to the method of Manes (Proceedings of the Engineering Foundation Conference Bavaria, West Germany, 1984: 335) developed from the Polanyi adsorption potential theory to predict the effect of water humidity on the adsorption capacity of activated carbon for a water-immiscible organic vapor. The input parameters for the solution are the pure vapor adsorption isotherms quantified using the Qi, Hay and Rood (QHR) equation in Qi et al. (J. Environ. Eng., 124(11);1998:1130) for water, and the Dubinin–Radushkevich equation in Dubinin (Progress in Surface and Membrane Science, vol. 9, 1975, Academic Press) for the organic chemical. The solution employs a simple algorithm of decrement search and requires the least possible number of iterations in calculation. Predicted results compare favorably with the adsorption data for benzene in the relative humidity range 0–90%. This numerical approach is straightforward and better suited than the original graphical technique for use in dynamic simulation of activated carbon adsorbers.

Fractal-based image coding with fast decoder convergence

In this paper, a new class of fractal image coders is described. As in previously published fractal coding algorithms, the encoder models each image to be coded as an attractor of a simple affine mapping, and finds the parameters in such a mapping by a blockwise image analysis. The image code consists of the parametric description of the mapping, which is selected to provide the best affine fit to the image in the l 2 sense. The decoder uses the parametric description it receives to synthesize the attractor of the mapping through a simple iterative procedure. Although the resulting coders are strongly related to previously published fractal coders, they are modified in a way that gives exact decoder convergence towards the attractor in the lowest possible number of iterations — typically three or less. In contrast to what has been the case with previous fractal coding algorithms, this number of iterations is image independent. The resulting decoder can be implemented in a computationally very efficient pyramid structure. Also, a coder offering non-iterative decoding is included as a special case. The paper describes the modifications of previously known fractal coders that are necessary to obtain the fast convergence. It is shown that the image quality remains unimpaired by these modifications. The computational complexity of the decoding algorithm in the new coder class is analyzed, and an efficient pyramid structure for the decoder is outlined. Finally, a coding example is given.

Experience with a Matrix Norm Estimator

Fortran 77 codes for estimating the 1-norm of a real or complex matrix were presented by Higham in [ACM Trans. Math. Software, 14 (1988), pp. 381–396]. The codes have found use in various applications and have been adopted by two program libraries. Further observations about the norm estimation algorithm and experience in using it are reported here. In particular, an example is given where the algorithm requires nearly the maximum possible number of iterations.

A class of linear programming problems requiring a large number of iterations

Abstract : A coordinate-free description of the simplex algorithm (for nondegenerate linear programming problems) is supplied, and is used to show that the number of iterations can be larger than was previously known. For 0 m n, there is constructed a nondegenerate linear programming problem whose bounded (n - m)-dimensional feasible region is defined by means of m linear equality constraints in n nonnegative variables, and in which, after starting from the worst choice of an initial feasible vertex, m(n - m - l) + 1 simplex iterations are required in order to reach the optimal vertex. It is conjectured that this is the maximum possible number of iterations (for arbitrary 0 m n), but the conjecture is proved only for n m + 4. (Author)

Letter to the Editor—Examples Relating to the Simplex Method

An example is given showing that the most common variant of the simplex algorithm does not satisfy a condition conjectured by Saaty in an analysis of the maximum possible number of iterations. Situations are exhibited in which the simplex method encounters all vertices of the constraint polyhedron.