Bidirectional irradiance transposition based on the Perez model

The Perez irradiance model offers a practical representation of solar irradiance by considering the sky hemisphere as a three-part geometrical framework, namely, the circumsolar disc, the horizon band and the isotropic background. Furthermore, the simplified Perez diffuse irradiance model, commonly known as the Perez transposition model, is one of the most widely adopted models in tilted irradiance modeling. Although the set of model coefficients reported by Perez et al. (1990) is considered to be at an asymptotic level of optimization, later analyses have shown that coefficients which are adjusted to local conditions may perform better than the original set.<p></p> The model coefficients can be adjusted locally based on multiple datasets of diffuse and global irradiance on tilted and horizontal planes. In this paper, we present a different approach to adjust the coefficients, by using only measurements of global irradiance on tilted and horizontal planes from a tropical climate site, Singapore. A complete set of mathematical solutions to the inverse problem, i.e., irradiance transposition from tilt to horizontal, is also proposed. The data can then be used to generate irradiance maps from in-plane irradiance measurements at photovoltaics (PV) systems. Such maps provide relevant information for PV grid integration.<p></p&gt

Approximating Pareto frontier using a hybrid line search approach

This is the post-print version of the final paper published in Information Sciences. The published article is available from the link below. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Copyright @ 2010 Elsevier B.V.The aggregation of objectives in multiple criteria programming is one of the simplest and widely used approach. But it is well known that this technique sometimes fail in different aspects for determining the Pareto frontier. This paper proposes a new approach for multicriteria optimization, which aggregates the objective functions and uses a line search method in order to locate an approximate efficient point. Once the first Pareto solution is obtained, a simplified version of the former one is used in the context of Pareto dominance to obtain a set of efficient points, which will assure a thorough distribution of solutions on the Pareto frontier. In the current form, the proposed technique is well suitable for problems having multiple objectives (it is not limited to bi-objective problems) and require the functions to be continuous twice differentiable. In order to assess the effectiveness of this approach, some experiments were performed and compared with two recent well known population-based metaheuristics namely ParEGO and NSGA II. When compared to ParEGO and NSGA II, the proposed approach not only assures a better convergence to the Pareto frontier but also illustrates a good distribution of solutions. From a computational point of view, both stages of the line search converge within a short time (average about 150 ms for the first stage and about 20 ms for the second stage). Apart from this, the proposed technique is very simple, easy to implement and use to solve multiobjective problems.CNCSIS IDEI 2412, Romani

Optimization of Biochemical Systems Production Using Combination of Newton Method and Particle Swarm Optimization

In the presented paper, an improved method that combines the Newton method with Particle Swarm Optimization (PSO) algorithm to optimize the production of biochemical systems was discussed and presented in detail. The optimization of the biochemical system's production became difficult and complicated when it involves a large size of biochemical systems that have many components and interaction between chemical. Also, two objectives and several constraints make the optimization process difficult. To overcome these situations, the proposed method was proposed by treating the biochemical systems as a nonlinear equations system and then optimizes using PSO. The proposed method was proposed to improve the biochemical system's production and at the same time reduce the total of chemical concentration involves. In the proposed method, the Newton method was used to deal with nonlinear equations system, while the PSO algorithm was utilized to fine-tune the variables in nonlinear equations system. The main reason for using the Newton method is its simplicity in solving the nonlinear equations system. The justification of choosing PSO algorithm is its direct implementation and effectiveness in the optimization process. In order to evaluate the proposed method, two biochemical systems were used, which were E.coli pathway and S. cerevisiae pathway. The experimental results showed that the proposed method was able to achieve the best result as compared to other works

A study of the local convergence of a fifth order iterative method

[EN] We present a local convergence study of a fifth order iterative method to approximate a locally unique root of nonlinear equations. The analysis is discussed under the assumption that first order Frechet derivative satisfies the Lipschitz continuity condition. Moreover, we consider the derivative free method that obtained through approximating the derivative with divided difference along with the local convergence study. Finally, we provide computable radii and error bounds based on the Lipschitz constant for both cases. Some of the numerical examples are worked out and compared the results with existing methods.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C21-C22.Singh, S.; Martínez Molada, E.; Maroju, P.; Behl, R. (2020). A study of the local convergence of a fifth order iterative method. Indian Journal of Pure and Applied Mathematics. 51(2):439-455. https://doi.org/10.1007/s13226-020-0409-5S439455512A. Constantinides and N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications, Prentice Hall PTR, New Jersey, (1999).J. M. Douglas, Process Dynamics and Control, Prentice Hall, Englewood Cliffs, (1972).M. Shacham, An improved memory method for the solution of a nonlinear equation, Chem. Eng. Sci., 44 (1989), 1495–1501.J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New-York, (1970).J. R. Sharma and H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algorithms, 67 (2014), 917–933.I. K. Argyros, A. A. Magreńan, and L. Orcos, Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation, J. Math. Chem., 54 (2016), 1404–1416.E. L. Allgower and K. Georg, Lectures in Applied Mathematics, American Mathematical Society (Providence, RI) 26, 723–762.A. V. Rangan, D. Cai, and L. Tao, Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics, J. Comput. Phys., 221 (2007), 781–798.A. Nejat and C. Ollivier-Gooch, Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations, J. Comput. Phys., 227 (2008), 2366–2386.C. Grosan and A. Abraham, A new approach for solving nonlinear equations systems, IEEE Trans. Syst. Man Cybernet Part A: System Humans, 38 (2008), 698–714.F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algorithms, 54 (2010), 395–409.I. G. Tsoulos and A. Stavrakoudis, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Anal. Real World Appl., 11 (2010), 2465–2471.E. Martínez, S. Singh, J. L. Hueso, and D. K. Gupta, Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces, Appl. Math. Comput., 281 (2016), 252–265.S. Singh, D. K. Gupta, E. Martínez, and J. L. Hueso, Semi local and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition, Appl. Math. Comput., 276 (2016), 266–277.I. K. Argyros and S. George, Local convergence of modified Halley-like methods with less computation of inversion, Novi. Sad.J. Math., 45 (2015), 47–58.I. K. Argyros, R. Behl, and S. S. Motsa, Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative Algorithms 2015, 8, 1076–1087; doi:https://doi.org/10.3390/a8041076.A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett., 25 (2012), 2369–2374.I. K. Argyros and A. A. Magreñán, A study on the local convergence and dynamics of Chebyshev- Halley-type methods free from second derivative, Numer. Algorithms71 (2016), 1–23.M. Grau-Sánchez, Á Grau, asnd M. Noguera, Frozen divided difference scheme for solving systems of nonlinear equations, J. Comput. Appl. Math., 235 (2011), 1739–1743.M. Grau-Sánchez, M. Noguera, and S. Amat, On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. Appl. Math., 237 (2013), 363–372

Highly efficient iterative algorithms for solving nonlinear systems with arbitrary order of convergence p+3, p>5

[EN] It is known that the concept of optimality is not defined for multidimensional iterative methods for solving nonlinear systems of equations. However, usually optimal fourth order schemes (extended to the case of several variables) are employed as starting steps in order to design higher order methods for this kind of problems. In this paper, we use a non optimal (in scalar case) iterative procedure that is specially efficient for solving nonlinear systems, as the initial steps of an eighth-order scheme that improves the computational efficiency indices of the existing methods, as far as the authors know. Moreover, the method can be modified by adding similar steps, increasing the order of convergence three times per step added. This kind of procedures can be used for solving big-sized problems, such as those obtained by applying finite differences for approximating the solution of diffusion problem, heat conduction equations, etc. Numerical comparisons are made with the same existing methods, on standard nonlinear systems and Fisher's equation by transforming it in a nonlinear system by using finite differences. From these numerical examples, we confirm the theoretical results and show the performance of the proposed schemes. (C) 2017 Elsevier B.V. All rights reserved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P, MTM2015-64013-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Jordan-Lluch, C.; Sanabria-Codesal, E.; Torregrosa Sánchez, JR. (2018). Highly efficient iterative algorithms for solving nonlinear systems with arbitrary order of convergence p+3, p>5. Journal of Computational and Applied Mathematics. 330:748-758. https://doi.org/10.1016/j.cam.2017.02.032S74875833

Remarks on Solving Methods of Nonlinear Equations

Abstract: In the field of mechanical engineering, many practical problems can be converted into nonlinear problems, such as the meshing problem of mechanical transmission. So the solution of nonlinear equations has important theoretical research and practical application significance. Whether the traditional Newton iteration method or the intelligent optimization algorithm after the popularization of computers, both them have been greatly enriched and developed through the continuous in-depth research of scholars at home and abroad, and a series of improved algorithms have emerged. This paper mainly reviews the research status of solving nonlinear equations from two aspects of traditional iterative method and intelligent optimization algorithm, systematically reviews the research achievements of domestic and foreign scholars, and puts forward prospects for future research directions

Design and multidimensional extension of iterative methods for solving nonlinear problems

[EN] In this paper, a three-step iterative method with sixth-order local convergence for approximating the solution of a nonlinear system is presented. From Ostrowski¿s scheme adding one step of Newton with ¿frozen¿ derivative and by using a divided difference operator we construct an iterative scheme of order six for solving nonlinear systems. The computational efficiency of the new method is compared with some known ones, obtaining good conclusions. Numerical comparisons are made with other existing methods, on standard nonlinear systems and the classical 1D-Bratu problem by transforming it in a nonlinear system by using finite differences. From this numerical examples, we confirm the theoretical results and show the performance of the presented scheme.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Artidiello, S.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). Design and multidimensional extension of iterative methods for solving nonlinear problems. Applied Mathematics and Computation. 293:194-203. https://doi.org/10.1016/j.amc.2016.08.034S19420329

Research on Solving Systems of Nonlinear Equations Based on Improved PSO

Solving systems of nonlinear equations is perhaps one of the most difficult problems in all of numerical computations, especially in a diverse range of engineering applications. The convergence and performance characteristics can be highly sensitive to the initial guess of the solution for most numerical methods such as Newton’s method. However, it is very difficult to select reasonable initial guess of the solution for most systems of nonlinear equations. Besides, the computational efficiency is not high enough. Aiming at these problems, an improved particle swarm optimization algorithm (imPSO) is proposed, which can overcome the problem of selecting reasonable initial guess of the solution and improve the computational efficiency. The convergence and performance characteristics of this method are demonstrated through some standard systems. The results show that the improved PSO for solving systems of nonlinear equations has reliable convergence probability, high convergence rate, and solution precision and is a successful approach in solving systems of nonlinear equations

Preserving the order of convergence: Low-complexity Jacobian-free iterative schemes for solving nonlinear systems

[EN] In this paper, a new technique to construct a family of divided differences for designing derivative-free iterative methods for solving nonlinear systems is proposed. By using these divided differences any kind of iterative methods containing a Jacobian matrix in its iterative expression can be transformed into a "Jacobian-free" scheme preserving the order of convergence. This procedure is applied on different schemes, showing theoretically their order and error equation. Numerical experiments confirm the theoretical results and show the efficiency and performance of the new Jacobian-free schemes. (C) 2018 Published by Elsevier B.V.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P, MTM2015-64013-P and Generalitat Valenciana PROMETEO/2016/089.Amiri, A.; Cordero Barbero, A.; Darvishi, M.; Torregrosa Sánchez, JR. (2018). Preserving the order of convergence: Low-complexity Jacobian-free iterative schemes for solving nonlinear systems. Journal of Computational and Applied Mathematics. 337:87-97. https://doi.org/10.1016/j.cam.2018.01.004S879733