scholarly journals A Particle Swarm Optimization-Based Method for Numerically Solving Ordinary Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Xian-Ci Zhong ◽  
Jia-Ye Chen ◽  
Zhou-Yang Fan
Keyword(s):  
Particle Swarm Optimization ◽  
Approximate Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Optimization Problem ◽  
Particle Swarm ◽  
Euler Method ◽  
Initial Value ◽  
Swarm Optimization ◽  
Grid Points

The Euler method is a typical one for numerically solving initial value problems of ordinary differential equations. Particle swarm optimization (PSO) is an efficient algorithm for obtaining the optimal solution of a nonlinear optimization problem. In this study, a PSO-based Euler-type method is proposed to solve the initial value problem of ordinary differential equations. In the typical Euler method, the equidistant grid points are always used to obtain the approximate solution. The existing shortcoming is that when the iteration number is increasing, the approximate solution could be greatly away from the exact one. Here, it is considered that the distribution of the grid nodes could affect the approximate solution of differential equations on the discrete points. The adopted grid points are assumed to be free and nonequidistant. An optimization problem is constructed and solved by particle swarm optimization (PSO) to determine the distribution of grid points. Through numerical computations, some comparisons are offered to reveal that the proposed method has great advantages and can overcome the existing shortcoming of the typical Euler formulae.

scholarly journals A Modified NM-PSO Method for Parameter Estimation Problems of Models

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
An Liu ◽  
Erwie Zahara ◽  
Ming-Ta Yang
Keyword(s):  
Genetic Algorithm ◽  
Parameter Estimation ◽  
Particle Swarm Optimization ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Optimization Problems ◽  
Particle Swarm ◽  
Swarm Optimization ◽  
Wide Range ◽  
Estimation Problems

Ordinary differential equations usefully describe the behavior of a wide range of dynamic physical systems. The particle swarm optimization (PSO) method has been considered an effective tool for solving the engineering optimization problems for ordinary differential equations. This paper proposes a modified hybrid Nelder-Mead simplex search and particle swarm optimization (M-NM-PSO) method for solving parameter estimation problems. The M-NM-PSO method improves the efficiency of the PSO method and the conventional NM-PSO method by rapid convergence and better objective function value. Studies are made for three well-known cases, and the solutions of the M-NM-PSO method are compared with those by other methods published in the literature. The results demonstrate that the proposed M-NM-PSO method yields better estimation results than those obtained by the genetic algorithm, the modified genetic algorithm (real-coded GA (RCGA)), the conventional particle swarm optimization (PSO) method, and the conventional NM-PSO method.

scholarly journals Parameter Estimation in Ordinary Differential Equations Modeling via Particle Swarm Optimization

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Devin Akman ◽  
Olcay Akman ◽  
Elsa Schaefer
Keyword(s):  
Parameter Estimation ◽  
Particle Swarm Optimization ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Efficient Method ◽  
Epidemic Model ◽  
Particle Swarm ◽  
Computing Methods ◽  
Swarm Optimization ◽  
Computationally Expensive

Researchers using ordinary differential equations to model phenomena face two main challenges among others: implementing the appropriate model and optimizing the parameters of the selected model. The latter often proves difficult or computationally expensive. Here, we implement Particle Swarm Optimization, which draws inspiration from the optimizing behavior of insect swarms in nature, as it is a simple and efficient method for fitting models to data. We demonstrate its efficacy by showing that it outstrips evolutionary computing methods previously used to analyze an epidemic model.

Nelder-Mead Algorithm in Solving Ordinary Differential Equations Whose Solutions Possess Singularities

2021 ◽  
Vol 9 (1) ◽  
pp. 11-17
Author(s):  
Ashiribo Senapon Wusu ◽  
Olusola Olabanjo
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Rational Function ◽  
Initial Value Problems ◽  
Optimization Problem ◽  
Constrained Optimization Problem ◽  
Initial Value ◽  
Numerical Examples ◽  
Nelder Mead Algorithm

This research considers Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) whose solutions possess singularities. Here, we represent the theoretical solution by a rational function as it is more convenient representing a function close to a singularity by a rational function. The process of transforming the IVP to a constrained optimization problem and application of Nelder-Mead algorithm in obtaining approximate solution is presented in this work. Accuracy and efficiency of this scheme is demonstrated on two numerical examples. The proposed approach produced better results compared with existing methods discussed in literature.

Convergence Analysis of Particle Swarm Optimization via Illustration Styles

2021 ◽  
pp. 29-37
Author(s):  
Wei-Der Chang ◽  
Keyword(s):  
Particle Swarm Optimization ◽  
Convergence Analysis ◽  
Optimization Problem ◽  
Particle Swarm ◽  
Pso Algorithm ◽  
Function Optimization ◽  
Evolutionary Computations ◽  
Swarm Optimization ◽  
Simulation Results ◽  
Population Based Algorithm

Particle swarm optimization (PSO) is the most important and popular algorithm to solving the engineering optimization problem due to its simple updating formulas and excellent searching capacity. This algorithm is one of evolutionary computations and is also a population-based algorithm. Traditionally, to demonstrate the convergence analysis of the PSO algorithm or its related variations, simulation results in a numerical presentation are often given. This way may be unclear or unsuitable for some particular cases. Hence, this paper will adopt the illustration styles instead of numeric simulation results to more clearly clarify the convergence behavior of the algorithm. In addition, it is well known that three parameters used in the algorithm, i.e., the inertia weight w, position constants c1 and c2, sufficiently dominate the whole searching performance. The influence of these parameter settings on the algorithm convergence will be considered and examined via a simple two-dimensional function optimization problem. All simulation results are displayed using a series of illustrations with respect to various iteration numbers. Finally, some simple rules on how to suitably assign these parameters are also suggested

scholarly journals Efficient Computation Offloading in Multi-Tier Multi-Access Edge Computing Systems: A Particle Swarm Optimization Approach

2019 ◽  
Vol 10 (1) ◽  
pp. 203 ◽  
Author(s):  
Luan N. T. Huynh ◽  
Quoc-Viet Pham ◽  
Xuan-Qui Pham ◽  
Tri D. T. Nguyen ◽  
Md Delowar Hossain ◽  
...  
Keyword(s):  
Resource Allocation ◽  
Particle Swarm Optimization ◽  
Optimization Problem ◽  
Particle Swarm ◽  
Edge Computing ◽  
Computation Offloading ◽  
Mixed Integer ◽  
5G Networks ◽  
Swarm Optimization ◽  
Multi Access

In recent years, multi-access edge computing (MEC) has become a promising technology used in 5G networks based on its ability to offload computational tasks from mobile devices (MDs) to edge servers in order to address MD-specific limitations. Despite considerable research on computation offloading in 5G networks, this activity in multi-tier multi-MEC server systems continues to attract attention. Here, we investigated a two-tier computation-offloading strategy for multi-user multi-MEC servers in heterogeneous networks. For this scenario, we formulated a joint resource-allocation and computation-offloading decision strategy to minimize the total computing overhead of MDs, including completion time and energy consumption. The optimization problem was formulated as a mixed-integer nonlinear program problem of NP-hard complexity. Under complex optimization and various application constraints, we divided the original problem into two subproblems: decisions of resource allocation and computation offloading. We developed an efficient, low-complexity algorithm using particle swarm optimization capable of high-quality solutions and guaranteed convergence, with a high-level heuristic (i.e., meta-heuristic) that performed well at solving a challenging optimization problem. Simulation results indicated that the proposed algorithm significantly reduced the total computing overhead of MDs relative to several baseline methods while guaranteeing to converge to stable solutions.

scholarly journals A Novel Application of Improved Marine Predators Algorithm and Particle Swarm Optimization for Solving the ORPD Problem

Energies ◽  
2020 ◽  
Vol 13 (21) ◽  
pp. 5679
Author(s):  
Mohamed A. M. Shaheen ◽  
Dalia Yousri ◽  
Ahmed Fathy ◽  
Hany M. Hasanien ◽  
Abdulaziz Alkuhayli ◽  
...  
Keyword(s):  
Particle Swarm Optimization ◽  
Power Systems ◽  
Electric Power ◽  
Optimization Algorithm ◽  
Optimization Problem ◽  
Reactive Power ◽  
Particle Swarm ◽  
Continuous Optimization ◽  
Swarm Optimization ◽  
Marine Predators

The appropriate planning of electric power systems has a significant effect on the economic situation of countries. For the protection and reliability of the power system, the optimal reactive power dispatch (ORPD) problem is an essential issue. The ORPD is a non-linear, non-convex, and continuous or non-continuous optimization problem. Therefore, introducing a reliable optimizer is a challenging task to solve this optimization problem. This study proposes a robust and flexible optimization algorithm with the minimum adjustable parameters named Improved Marine Predators Algorithm and Particle Swarm Optimization (IMPAPSO) algorithm, for dealing with the non-linearity of ORPD. The IMPAPSO is evaluated using various test cases, including IEEE 30 bus, IEEE 57 bus, and IEEE 118 bus systems. An effectiveness of the proposed optimization algorithm was verified through a rigorous comparative study with other optimization methods. There was a noticeable enhancement in the electric power networks behavior when using the IMPAPSO method. Moreover, the IMPAPSO high convergence speed was an observed feature in a comparison with its peers.

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