Broyden, Fletcher, Goldfarb And Shanno Research Articles

Predicting Optimized Dissolution of Selected African Copperbelt Copper-cobalt-bearing Ores by Means of Neural Network Prediction and Response Surface Methodology Modeling

While the uncertainty brought about by a varying feed mineralogy was taken into consideration, the paper investigated the modeling and prediction of the leaching behavior of complex copper-cobalt bearing ores, using an artificial neural network (ANN) with a backforward algorithm. The process optimization is further conducted using the response surface methodology (RSM) employing the Box-Behnken design (BBD). Seven (7) parameters were considered in a multiple linear regression according to the L12 screening plan (27) of Plackett–Burman. From the seven parameters, four including solid percentage (15, 27.5, 40%), time (45, 90, 135 min), particle size passing (53, 75, 105 µm), and Fe2+ ion concentration (2, 4, 6 g/L) are modeled with L27(34) BBD. With a composite desirability of 0.94, leaching yields of 93.46% Cu and 89.43% Co were obtained. The neural network algorithm used is the BFGS (Broyden, Fletcher, Goldfarb and Shanno) algorithm multilayer perceptron with the hyperbolic tangent activation function for the hidden layer and a linear activation function for the neural output. The Multilayer perceptron {4–7-1} structure was chosen as a suitable arrangement for Cu leaching. Comparing the predicted values and those obtained experimentally resulted with a correlation coefficient of 0.9552 for the data trained in the artificial neural network and 0.8742 for the data obtained with the response surface methodology. The synergy of these 2 techniques shows that the prediction can be achieved by means of the ANN giving the values of the root mean square errors (RMSE) of 0.0115, 0.00624, 0.0229, respectively, for the training, testing and validation sets for copper recovery while the correlational study between variables could be done through the RSM. The above includes only the 95% confidence interval while the remaining 5% would be uncertain. The above results and conclusion are accompanied by the relative uncertainty as the ore mineralogy varies. The combination of the synergistic use of ANN and RSM with the sensitivity analysis has approached the process to the physics of the Multi-criteria decision-making.Graphical

Comparison of optimization algorithms for selecting the fractional frequency in Fourier form unit root tests

ABSTRACT We compare the performance of unit root tests which include flexible Fourier trends in their testing processes. The algorithms considered are those of Broyden, Fletcher, Goldfarb and Shanno (BFGS), Berndt, Hall, Hall and Hausman (BHHH), Simplex, Genetic and grid search (GS). The simulation results indicate that derivative-free methods, such as Genetic and Simplex, have advantages over hill-climbing methods, such as BFGS and BHHH in providing accurate fractional frequencies for fractional frequency flexible Fourier form (FFFFF) unit root test. When the parameters are estimated under the alternative hypothesis of the FFFFF type of unit root test, the grid search and derivative-free methods provide unbiased and efficient estimations. We also provide the asymptotic distribution of the FFFFF unit root test. We extend the FFFFF unit root test to a panel version in order to increase the power of the test. Finally, the empirical analyses of Covid19 unit root test show that derivative-free methods, are better than other methods. However, for big data and accurate estimation of the frequency parameters, the Simplex methodology using the bootstrap process is preferred.

The Comparison of Power and Optimization Algorithms on Unit Root Testing with Smooth Transition

The aim of this study is to search for a better optimization algorithm in applying unit root tests that inherit nonlinear models in the testing process. The algorithms analyzed include Broyden, Fletcher, Goldfarb and Shanno (BFGS), Gauss---Jordan, Simplex, Genetic, sequential quadratic programming and extensive grid-search. The simulation results indicate that the derivative free methods, such as Genetic and Simplex, have advantages over hill climbing methods, such as BFGS and Gauss---Jordan, in obtaining accurate critical values for the Leybourne et al. (J Time Ser Anal 19:83---97, 1998) (LNV) and Sollis (J Time Ser Anal 25:409---417, 2004) unit root tests. Besides, we extend our analysis by including exponential smooth transition type of trend function in to unit root testing which is not used in the previous literature. The same result also holds true for our newly proposed unit root test with exponential smooth transition function type of trend model. Furthermore, we realize that there is a gap in the unit root studies that the newly proposed tests are not analyzed between each other's data generating process (DGP). Hence, we investigate the power comparison of different nonlinear unit root test under various DGP including nonlinear unit root tests and find interesting results such as LNV type unit root test can manage to capture state dependent nonlinearity when the transition speed is high. Finally, we have used the Australian real interest rate parity hypothesis to empirically verify the results that we have obtained in the simulation studies.

Minimization of Gibbs energy with phase stability test for determining vapor–liquid equilibrium of hydrogen-middle distillate systems at hydrotreating conditions

The effect of applying the minimization of the Gibbs energy with phase stability tests on the prediction of the amount of vaporized middle distillate in presence of hydrogen at hydrotreating (HDT) conditions (70.9bars and 250–420°C) is studied. The Peng–Robinson (PR) thermodynamic model is used to fit the H2–pseudocomponents interaction coefficients with experimental data and solving vapor–liquid equilibrium (VLE) by minimizing the Gibbs energy with stability tests. The steepest descent and BFGS (Broyden, Fletcher, Goldfarb, and Shanno) quasi Newton minimization methods are used to solve the VLE with stability test. The optimization of interaction parameters is done by the nonlinear systems Simplex method with a computational tool. The methodology was validated with information reported in the literature of two binary systems: (1) trans-2-hexen-1-ol (C6H12O)/CO2 at 30°C and 65.3–150.0bars, and (2) H2S/CH4 at −83°C and 40.53bars. The developed methodology is useful, allows for calculating the VLE through the catalytic bed and the variation in vapor and liquid flows, compositions and thermo physical properties.

Prediction of capillary gas chromatographic retention times of fatty acid methyl esters in human blood using MLR, PLS and back-propagation artificial neural networks

Quantitative structure–retention relationship (QSRR) models correlating the retention times of fatty acid methyl esters in high resolution capillary gas chromatography and their structures were developed based on non-linear and linear modeling methods. Genetic algorithm (GA) was used for the selection of the variables that resulted in the best-fitted models. Gravitational index (G2), number of cis double bond (N cDB) and number of trans double bond (N tDB) were selected among a large number of descriptors. The selected descriptors were considered as inputs for artificial neural networks (ANNs) with three different weights update functions including Levenberg–Marquardt backpropagation network (LM-ANN), BFGS (Broyden, Fletcher, Goldfarb, and Shanno) quasi-Newton backpropagation (BFG-ANN) and conjugate gradient backpropagation with Polak–Ribiére updates (CGP-ANN). Computational result indicates that the LM-ANN method has better predictive power than the other methods. The model was also tested successfully for external validation criteria. Standard error for the training set using LM-ANN was SE = 0.932 with correlation coefficient R = 0.996. For the prediction and validation sets, standard error was SE = 0.645 and SE = 0.445 and correlation coefficient was R = 0.999 and R = 0.999, respectively. The accuracy of 3–2–1 LM-ANN model was illustrated using leave multiple out-cross validations (LMO-CVs) and Y-randomization.

Training Algorithm Performance for Image Classification by Neural Networks

Adaptive training is critical for image classification by artificial neural networks (ANNS). While the machine learning community has been enthusiastic in developing various training algorithms, little research has been conducted to evaluate the performance of these algorithms in image classification by neural networks. We introduce and evaluate nine commonly-used training algorithms in terms of their performance in land-cover classification from remotely sensed data by the multi-layer-perceptron (MLP) neural networks. ALP has been considered as the most popular neural network architecture. The training algorithms we consider are Steepest Gradient Descent, Gradient Descent with Momentum, Resilient Propagation, Fletcher-Reeves, Polak-Ribiere, Powell-Beale, Scaled Conjugate Gradient, BFGS (Broyden, Fletcher, Goldfarb, and Shanno), and Levenberg-Marquardt. We use each algorithm to train the MLP networks multiple times using identical training samples, and then apply each of the resultant network models to derive land-cover information from a Landsat Enhanced Thematic Mapper Plus (ETM+) image. The training algorithms are further evaluated according to their training efficiency, capability of convergence, classification accuracy, and stability of the classification accuracy. It is found that the performance of these algorithms varies substantially and selecting an appropriate algorithm can lead to a fast and efficient training and an increase in land-cover classification accuracy by artificial neural networks.

Flow curve prediction of Al–Mg alloys under warm forming conditions at various strain rates by ANN

This paper describes an approach based on artificial neural network (ANN) to identify the material flow curves of strain hardened 5083-H111 and 5754-O Al–Mg alloys at the temperature ranges from room temperature (RT) to 300 °C and a strain rate of 0.0016–0.16 s −1. The tensile tests were performed to determine the material responses at various temperatures and strain rates. An ANN model was developed to predict the flow curves of the materials in terms of experimental data. The input parameters of the model are strain rate, temperature, and strain while tensile flow stress is the output. A three layer feed-forward network was trained with BFGS (Broyden, Fletcher, Goldfarb, and Shanno) algorithm. The amount of the neurons in the hidden layer was determined by determining of the root mean square error (RMSE) values for each material. Results reveal that the predicted values in the ANN model are in very good agreement with the experimental data. The ANN model, described in this paper, is an efficient quantitative tool to evaluate and predict the deformation behavior of 5083-H111 and 5754-O Al–Mg alloys for tensile test at prescribed deformation conditions.

Evolutionary algorithms with deterministic mutation operators used for the optimization of the trajectory of a four-bar mechanism

In this paper, we propose an Evolutionary Algorithm (EA) with a deterministic mutation operator which is a combination of EA with the Broyden, Fletcher, Goldfarb and Shanno (BFGS) method. The advantages of both optimization algorithms are retained and interconnected. The proposed algorithm shows faster convergence as well as increased reliability in the search for the global optimum. Results referring to the Fletcher and Powel test function in comparison with EA (Evolution Strategies, Evolutionary Programming, and Genetic Algorithms), provide sufficient indication for the performance of the new method. Finally, the proposed method is successfully implemented for the trajectory optimization of a four-bar mechanism.

Determining optimal parameters for stereolithography processes via genetic algorithm

In this paper, we propose an Evolutionary Algorithm (EA) with a deterministic mutation operator which is a combination of EA with the Broyden, Fletcher, Goldfarb and Shanno (BFGS) method. The advantages of both optimization algorithms are retained and interconnected. The proposed algorithm shows faster convergence as well as increased reliability in the search for the global optimum. Results referring to the Fletcher and Powel test function in comparison with EA (Evolution Strategies, Evolutionary Programming, and Genetic Algorithms), provide sufficient indication for the performance of the new method. Finally, the proposed method is successfully implemented for the trajectory optimization of a four-bar mechanism.

Limited-Memory BFGS Diagonal Preconditioners for a Data Assimilation Problem in Meteorology

This paper uses two simple variational data assimilation problems with the 1D viscous Burgers' equation on a periodic domain to investigate the impact of various diagonal-preconditioner update and scaling strategies, both on the limited-memory BFGS (Broyden, Fletcher, Goldfarb and Shanno) inverse Hessian approximation and on the minimization performance. These simple problems share some characteristics with the large-scale variational data assimilation problems commonly dealt with in meteorology and oceanography.

Parallel processing architecture for the quasi‐Newton method based on BFGS using one‐dimensional systolic arrays

This paper presents a parallel algorithm and its systolic array architecture for the BFGS (Broyden, Fletcher, Goldfarb, and Shanno) quasi-Newton method of minimizing an n-vector function. The calculation of search direction vectors and the update of approximation to Hessian matrices by the BFGS updating formula both have O(n2) complexity. We first introduce an algorithm suitable for parallel processing by employing UD factorization of the approximation to the Hessian. Then, systolic arrays with some FIFO and LIFO memories are proposed for the calculation of search direction vectors and the processing of the BFGS updating formula. The hardware complexity of the proposed parallel processing architecture is O(n), and the parallel time complexity is also O(n). © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 81(10): 73–82, 1998

A computational procedure for the simulation of ductile fracture with large plastic deformation

It is now well-known that the applicability of the single-parameter J-approach is restricted only to high constraint crack geometries and materials of low ductility. Consequently, there is a need to develop a ductile fracture criterion that does not suffer from the geometry dependence exhibited by the J-integral. However, the ability of any analytical model to provide accurate description of fracture initiation and propagation in a ductile material is contingent upon its capability to model the large deformations that occur in the fracture process zone. The purpose of this paper is to describe the formulation and implementation of an efficient finite deformation algorithm that can be used for the prediction of elasto-plastic fracture in ductile materials where J-dominance is violated. An incrementally objective mid-interval integration algorithm is used in conjunction with an efficient polar decomposition scheme for the accurate integration of the plasticity equations in the presence of large deformations. Both rate-dependent and rate-independent plasticity models are included. A sophisticated node release algorithm for the simulation of crack propagation is also implemented. The application of continuum damage models as potential ductile fracture criteria is discussed. The overall efficiency of the code is enhanced by means of the BFGS (Broyden, Fletcher, Goldfarb and Shanno) solution algorithm. Three elasto-plasticity problems that involve finite deformation are analyzed in order to assess the accuracy of the solution procedure. The effectiveness of including material damage in the constitutive behavior to predict ductile failure in a notched tensile specimen is addressed