Optimization Method In Order Research Articles

Design and Analysis of a New Class of Derivative-Free Optimal Order Methods for Nonlinear Equations

We develop a general class of derivative free iterative methods with optimal order of convergence in the sense of Kung–Traub hypothesis for solving nonlinear equations. The methods possess very simple design, which makes them easy to remember and hence easy to implement. The Methodology is based on quadratically convergent Traub–Steffensen scheme and further developed by using Padé approximation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Numerical examples are provided to confirm the theoretical results and to show the good performance of new methods.

A TP-LPV-LMI based control for Tumor Growth Inhibition

Abstract The paper investigates the applicability of an advanced modern control method related to control of tumor growth under angiogenic inhibition. In order to describe the physiological process, a simple mathematical model was applied consisting of two states, the volume of the tumor and the inhibitor value. Extended Kalman Filter (EKF) was applied to estimate the unmeasurable state (inhibitor level). Linear Parameter Varying (LPV) models are used both at controller design (difference based control oriented LPV model) and EKF development (LPV model) level as well. We have used the Tensor Product (TP) model transformation accompanied by Linear Matrix Inequality (LMI) based optimization method in order to design a Parallel Distributed Compensator (PDC) kind TP-LPV-LMI controller considering additive disturbances (on both states) and sensor noise as well. Despite the assumed unfavorable effects the TP-LPV-LMI controller performed well achieving low final tumor volume and less totally injected inhibitor level.

Proposal of heuristic regression method applied in descriptive data analysis: case studies

The purpose of this paper is to use the hybridized optimization method in order to find mathematical structures for analysis of experimental data. The heuristic optimization method will be hybridized with deterministic optimization method in order to that structures found require not knowledge about data generated experimentally. Five case studies are proposed and discussed to validate the results. The proposed method has viable solution for the analysis of experimental data and extrapolation, with mathematical expression reduced.

Numerical Solution of Fuzzy linear Regression using Fuzzy Neural Network Based on Probability Function

In this work, we consider the development of a fuzzy neural network based on probability function for Estimated output of fuzzy regression models with test real input and fuzzy output. The proposed approach is a fuzzification of the outputs and weights of conventional fuzzy neural network based on probability function. The error of the proposed method is based on total square error is minimized by optimization method in order to be able to obtain the optimal weights of the neural network. The advantage of the proposed approach is its simplicity and computation as well as its performance. To compare the performance of the proposed method with the other traditional methods given in the literature several numerical examples are presented.

Handling obstacles in pedestrian simulations: Models and optimization

In this paper we are concerned with the simulation of crowds in built environments, where obstacles play a role in the dynamics and in the interactions among pedestrians. First of all, we review the state-of-the-art of the techniques for handling obstacles in numerical simulations. Then, we introduce a new modeling technique which guarantees both impermeability and opacity of the obstacles, and does not require ad hoc runtime interventions to avoid collisions. Most important, we solve a complex optimization problem by means of the Particle Swarm Optimization method in order to exploit the so-called Braess’s paradox. More precisely, we reduce the evacuation time from a room by adding in the walking area multiple obstacles optimally placed and shaped.

A novel simulation model for PV panels based on datasheet parameter tuning

Circuit simulation model for photovoltaic array is very useful in simulating and evaluating the performance of power conditioning system. Among various programs, PSIM is widely used in power simulation and its physical model can handle changing output according to temperature and irradiance. However, accuracy of the model heavily depends on user-defined information. Specifically, construction of the conventional PSIM model involves calculation of tangential slope at open circuit voltage point in the I-V curve and estimation of ideality factor, both of which are obtained by trial and error and thus strongly influence the model accuracy.In this paper, a new PSIM simulation model for PV panels using online parameter tuning is presented. The proposed model utilizes only datasheet values measured under STC and excludes the necessity of the error-prone processes. Furthermore, all of the five parameters in single-diode model are successively tuned using Powell’s optimization method in order to guarantee good accuracy and high speed. Comparisons with conventional model are performed using crystalline PV panel sample and the results shows that the proposed model provides more accurate and uniform performance according to EN50530 standard.

Structural and parametric data representation using the second order optimization method

Working with various data sources in real-time requires approaches capable of adaptive parameters tuning. We propose algorithms that represent dynamic data streams in apriori defined structures. The algorithms are based on the certain error minimization. The used method is Newton's method, which is appropriate because of its high convergence. At every step, when the new data are received we make corrections to the unknown parameters vector by solving differential equations systems. Initial values are selected using estimates obtained from the practical stability theory. The computational experiment was conducted to compare models based on the first and second order optimization approaches. It confirms the effectiveness of our approach.

Synthesis and Implementation of a Fixed Low Order Controller on an Electronic System

The development of microelectronic field and software allows researchers to implement some control law in miniaturized devices such as Field Programmable Gate Arrays (FPGA) and microcontroller. These control laws may be used in industrial applications. The key of this work is the design and the implementation of a fixed low order controller on a STM32 microcontroller in order to control an electronic system. The main objective of this controller is to ensure some time response performances as the settling time and the overshoot. The controller parameters are obtained by resolving a non convex optimization problem while considering the desired closed loop specifications. So, the use of a classical optimization method to resolve such kind of problems may lead to a local solution and then the obtained solution is not optimal. Therefore, it is suggested to apply a global optimization method in order to get an optimal control law that can ensure the specified time response performances. The proposed method in this work is the Generalized Geometric Programming (GGP) method. This method consists on transforming, by some mathematical transformations, a non convex optimization problem to a convex one. The implementation of a Proportional Integral (PI) controller, a Proportional Integral Derivate (PID) and a fixed low order controller, on a real electronic system, shows the efficiency of the latter one.

Recovering Incomplete Data using Tucker Model for Tensor with Low-n-rank

Tensor with missing or incomplete values is a ubiquitous problem in various fields such as biomedical signal processing, image processing, and social network analysis. In this paper, we considered how to reconstruct a dataset with missing values by using tensor form which is called tensor completion process. We applied Tucker factorization to solve tensor completion which was built base on optimization problem. We formulated the optimization objective function using components of Tucker model after decomposing. The weighted least square matric contained only known values of the tensor with low rank in its modes. A first order optimization method, namely Nonlinear Conjugated Gradient, was applied to solve the optimization problem. We demonstrated the effectiveness of the proposed method in EEG signals with about 70% missing entries compared to other algorithms. The relative error was proposed to compare the difference between original tensor and the process output.

Order-optimal methods for integro-differential equations in the singular case

We study a linear integro-differential equation with a coefficient that has zeros of finite orders. For its approximate solution in the space of generalized functions, we suggest and justify special generalized versions of spline methods. Their optimization in the accuracy order is carried out.

A GPU accelerated adjoint-based optimizer for inverse modeling of the two-dimensional shallow water equations

The increasing computational capacity of current computing technologies makes feasible the application of predictive high-resolution mathematical models for studying physical phenomena. Additionally, this can be enhanced if those methods are incorporated to some optimization method in order to perform inverse modeling. In this paper, an optimization procedure based on the adjoint equations is used for the reconstruction of information in a 2D Shallow Water model previously developed and proved to be fast, robust and accurate. The continuous adjoint approach used for the evaluation of the gradient that is introduced in a gradient-based optimizer. Furthermore, the computation of both physical and adjoint systems is accelerated by the use of GPU programming. Even though notable speed-ups are achieved with this technique they are only possible in small to medium size grids due to memory limitations. A novel checkpointing strategy is proposed to allow data handling in these devices hence offering the possibility to overcome that limitation.

Micro-genetic algorithms for detecting and classifying electric power disturbances

The power quality analysis represents an important aspect in the overall society welfare. The analysis of power disturbances in electrical systems is typically performed in two steps: disturbance detection and disturbance classification. Disturbance detection is usually made through space transform techniques, and their classification is usually performed through artificial intelligence methods. The problem with those approaches is the adequate selection of parameters for these techniques. Due to the advantages of a variant scheme known as the micro-genetic algorithms, in this investigation, a new methodology to directly detect and classify electrical disturbances in one step is developed. The proposed approach is validated through synthetic signals and experimental test on real data, and the obtained results are compared with the particle swarm optimization method in order to show the effectiveness of this methodology.

Iterative Project Quasi-Newton Algorithm for Training RBM

The restricted Boltzmann machine (RBM) has been used as building blocks for many successful deep learning models, e.g., deep belief networks (DBN) and deep Boltzmann machine (DBM) etc. The training of RBM can be extremely slow in pathological regions. The second order optimization methods, such as quasi-Newton methods, were proposed to deal with this problem. However, the non-convexity results in many obstructions for training RBM, including the infeasibility of applying second order optimization methods. In order to overcome this obstruction, we introduce an em-like iterative project quasi-Newton (IPQN) algorithm. Specifically, we iteratively perform the sampling procedure where it is not necessary to update parameters, and the sub-training procedure that is convex. In sub-training procedures, we apply quasi-Newton methods to deal with the pathological problem. We further show that Newton's method turns out to be a good approximation of the natural gradient (NG) method in RBM training. We evaluate IPQN in a series of density estimation experiments on the artificial dataset and the MNIST digit dataset. Experimental results indicate that IPQN achieves an improved convergent performance over the traditional CD method.

New developments on reconstruction of high resolution chlorophyll-a vertical profiles

We present a methodology to vertical profiles of chlorophyll-a pigment concentration in open-ocean waters based on radiance values at different depths. The inverse problem is formulated here as an optimization problem and iteratively solved by an Ant Colony System (ACS) meta-heuristic. An objective function is given by the square difference between computed and experimental radiances at each iteration. The Laplace transform discrete ordinate (LTSN) method is used to solve the radiative transfer equation (direct problem) in order to compute the radiances. In a first approach, this methodology did not allow the reconstruction of profiles with two or more peaks of chlorophyll-a concentration. This limitation can be partially explained by the relatively low number (11) of sampling points at different depths, which limits the spatial resolution of the vertical profile to be reconstructed. Alternatively, we propose the such reconstruction of the profile by increasing the vertical resolution, in order to evaluate the ability of identifying any peak in the chlorophyll-a concentration. A hybrid methodology is adopted: initially, the original inverse ACS methodology is employed to retrieve high resolution profiles (41 and 81 points), and then these results are used as an initial guesses for the Lavenberg-Marquardt deteriministic optimization method in order to refine the vertical profiles.

An Optimal Order Method for Multiple Roots in Case of Unknown Multiplicity

In the literature, recently, some three-step schemes involving four function evaluations for the solution of multiple roots of nonlinear equations, whose multiplicity is not known in advance, are considered, but they do not agree with Kung–Traub’s conjecture. The present article is devoted to the study of an iterative scheme for approximating multiple roots with a convergence rate of eight, when the multiplicity is hidden, which agrees with Kung–Traub’s conjecture. The theoretical study of the convergence rate is investigated and demonstrated. A few nonlinear problems are presented to justify the theoretical study.

Considering Wind Speed Characteristics in the Design of a Coreless AFPM Synchronous Generator

This paper presents an improved design procedure of a coreless Axial Flux Permanent Magnet Synchronous Generator (AFPMSG) for wind turbine application. An analytical design approach is proposed based on sizing equations and a nonlinear Magnetic Equivalent Circuit (MEC) model of AFPMSG. The reluctance of rotor back iron core is separated into two parts, which are calculated through the nonlinear iterative algorithm. An optimal design of the coreless AFPMSG is performed using Partial Swarm Optimization (PSO) method in order to increase the Annual Energy Yield (AEY) of the generator. The statistical model of wind speed distribution and wind turbine characteristic are used for considering the generator performance over the whole operating wind speed range. In order to obtain the optimal values of design variables, the optimization procedure is provided for various number of pole pairs and coil numbers. The proposed procedure is used to design a double rotor-one coreless stator AFPMSG. Finally, 3-D Finite Element Analysis (FEA) is used to validate the design algorithm. Also, the generator performance is evaluated for different values of wind speed and load currents.

Novel optimal control of semi-active suspension considering a hysteresis model for MR damper

In this paper, a new form of the Bouc–Wen model is identified. This model, which includes hysteresis loop, is suitable to use in controller design. The modified Bouc–Wen model is employed as an experimental model to identify the new model. Evaluation of the proposed model illustrated that this model has good accuracy in small velocity area. A novel approach of model predictive control is employed to derive the control input or magneto-rheological damper current, with respect to constraints. This controller gives analytical control law by means of the Karush–Kuhn–Tucker (KKT) optimization method in order to solve the constrained optimization problem. One of the main advantages of this controller is its analytical form, which allows easy implementation of the controller, and this special form leads to rapid calculation of the control input. Simulations show that the proposed control strategy establishes an appropriate trade-off between suspension criteria, including ride comfort and road holding.

A generalized Suzuki–Trotter type method in optimal control of coupled Schrödinger equations

A generalized Suzuki---Trotter (GST) method for the solution of an optimal control problem for quantum molecular systems is presented in this work. The control of such systems gives rise to a minimization problem with constraints given by a system of coupled Schrodinger equations. The computational bottleneck of the corresponding minimization methods is the solution of time-dependent Schrodinger equations. To solve the Schrodinger equations we use the GST framework to obtain an explicit polynomial approximation of the matrix exponential function. The GST method almost exclusively uses the action of the Hamiltonian and is therefore efficient and easy to implement for a variety of quantum systems. Following a first discretize, then optimize approach we derive the correct discrete representation of the gradient and the Hessian. The derivatives can naturally be expressed in the GST framework and can therefore be efficiently computed. By recomputing the solutions of the Schrodinger equations instead of saving the whole time evolution, we are able to significantly reduce the memory requirements of the method at the cost of additional computations. This makes first and second order optimization methods viable for large scale problems. In numerical experiments we compare the performance of different first and second order optimization methods using the GST method. We observe fast local convergence of second order methods.

An Optimal Biparametric Multipoint Family and Its Self-Acceleration with Memory for Solving Nonlinear Equations

In this paper, a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634–651) that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m-1 . Furthermore, the family of Steffensen-type methods of super convergence is suggested by using arithmetic expressions for the parameters with memory but no additional new evaluation of the function. Their error equations, asymptotic convergence constants and convergence orders are obtained. Finally, they are compared with related root-finding methods in the numerical examples.

Derivation of Continuous Linear Multistep Methods Using Hermite Polynomials as Basis Functions

This paper concerns the derivation of continuous linear multistep methods for solving first-order initial value problems (IVPs) of ordinary differential equations (ODEs) with step number k=3 using Hermite polynomials as basis functions. Adams-Bashforth, Adams-Moulton and optimal order methods are derived through collocation and interpolation technique. The derived methods are applied to solve two first order initial value problems of ordinary differential equations. The result obtained by the optimal order method compared favourably with those of the standard existing methods of Adams-Bashforth and Adams-Moulton.