Hybrid Nelder-Mead Simplex Search Research Articles

Parameter estimation for time-fractional Black-Scholes equation with S &P 500 index option

This paper aims to estimate the parameters of the time-fractional Black-Scholes (TFBS) partial differential equation with the Caputo fractional derivative by using the real option prices of the S &P 500 index options. First, the numerical solution is obtained by developing a high-order scheme with order (3-alpha ) for the time discretisation. Some theoretical analyses such as stability and convergence are presented in order to verify the efficiency and accuracy of the proposed scheme. Secondly, we employ a modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) to identify the fractional order alpha and implied volatility sigma of the TFBS equation, and explore the financial meanings of alpha under extreme stock market conditions such as the Covid-19 and the 2008 global financial crisis. We analyse the values of alpha and compare the mean squared errors of both the TFBS model and the BS model. Our empirical results show that alpha may be regarded as a market fluctuation indicator for classifying financial environments, and the TFBS model is more capable of fitting real option data compared with the BS model, especially for put options during the economic downturn. In addition, we find and discuss an interesting relation between alpha and sigma from both the TFBS model and the BS model in three expressions, which could be an open problem for further research.

Vehicle-Deployed Off-Axis Integrated Cavity Output Spectroscopic CH4/C2H6 Sensor System for Mobile Inspection of Natural Gas Leakage.

A vehicle-deployed parts-per-billion in volume (ppbv)-level off-axis integrated cavity output spectroscopic (OA-ICOS) CH4/C2H6 sensor system was experimentally presented for mobile inspection of natural gas leakage in urban areas. For the time-division-multiplexing-based dual-gas sensor system, an antivibration 35-cm-long optical cavity with an effective path length of ∼2510 m was fabricated with a high-stability temperature and pressure control design. An Allan deviation analysis yielded a minimum detection limit of 0.2 ppbv for CH4 detection and 10 ppbv for C2H6 detection for a 1 s averaging time. A natural gas leakage source location algorithm was proposed using an improved hybrid Nelder-Mead simplex search method and a particle swarm optimization (NM-PSO) algorithm. For field industrial application, the accuracy of the sensor system and leakage source location algorithm was confirmed through a CH4/C2H6 cylinder leakage experiment on the campus. Furthermore, through natural gas pipeline network inspection measurements in urban areas, three types of leakage sources, including natural gas, biogas, and possible leakage source were respectively located and confirmed using the global positioning system and wind speed and direction measurement system, verifying the reliability and potential application of the vehicle-deployed inspection system for future natural gas pipeline leakage monitoring.

The application of the distributed-order time fractional Bloch model to magnetic resonance imaging

It is now well known that the magnetic resonance imaging (MRI) signal decay deviates from the classical mono-exponential relaxation. This deviation is referred to in the literature as anomalous relaxation. The modelling of this anomalous relaxation can provide a better understanding of MRI magnetization. The purpose of this work is to investigate the utility of the distributed-order time fractional Bloch equations to describe anomalous relaxation processes in human brain MRI data. Two choices of continuous distribution weight functions, which are parameterised by their mean μ and standard deviation σ, are studied to investigate their impact on the model solution behaviour. An implicit numerical method implemented on a graded mesh is proposed to solve the model and the stability and convergence analysis are presented. We also derive semi-analytical solutions of the fully coupled Bloch equations using the Laplace transform technique to assess the accuracy of the numerical scheme. Furthermore, three different voxel models of continuous distribution weight functions, namely a single continuous probability distribution (model 1), two distinct continuous probability distributions (model 2) and a mixture of two continuous probability distributions (model 3), are applied to the in vivo human brain MRI data, and a feasible and reliable parameter estimation method based on a modified hybrid Nelder-Mead simplex search and particle swarm optimization is presented to perform the voxel-level temporal fitting of the MRI data. The application of these distributed-order time fractional Bloch models highlights the validity of the proposed models, and based on the mean square error we conclude that models 2 and 3 might be more suitable than model 1 to characterize anomalous relaxation processes in human brain MRI data.

Parameter Estimation for the One-Term (Multiterm) Fractional-Order SEIAR Models of Norovirus Outbreak

In the paper, we use the Caputo fractional derivative to consider general single-term and multiterm fractional-order SEIAR models for the outbreak of Norovirus. Then, the inverse problem about parameter estimation for these fractional-order SEIAR models of the Norovirus outbreak is studied firstly. To provide the numerical solution of the single-term (or multiterm) fractional-order nonlinear differential equation, the GMMP scheme and Newton method are introduced. Then, we make use of the modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) algorithm to obtain the fractional orders and parameters for these fractional-order SEIAR models of Norovirus outbreak. To guarantee the correctness and effectiveness of the methods, the data of a 2007 Norovirus outbreak in a middle school in one city is used as the real data to solve the inverse problem of the parameter estimation. With the new parameters, all numerical studies illustrate that the numerical solutions fit very well with the real data, which reveals that the single-term and multiterm fractional-order SEIAR models of Norovirus outbreak all can predict the number of the infectious people accurately. And it also shows that the GMMP scheme and the MH-NMSS-PSO method are efficient and valid for estimating the parameters of the single-term (or multiterm) fractional-order nonlinear equations. Then, we research the impact of changes in each parameter on the amount of infected humans I t when the remaining parameters are unchanged. All results of numerical simulation reveal that the single-term and multiterm fractional-order SEIAR model of Norovirus can provide better results than other models. And we also study the effect of the isolation on different days. The conclusion is obtained that the earlier the isolation is taken, the less the infected people are. Hence, for a fractional-order application in the SEIAR model of Norovirus outbreak, we establish the effective parameter estimation methods.

Numerical simulation and parameters estimation of the time fractional dual-phase-lag heat conduction in femtosecond laser heating

In this paper, considering the heat flux and the temperature gradient approximated by the fractional Taylor's series expansions with different orders in dual-phase-lag (DPL) model, we propose the time fractional DPL heat conduction model for thin gold films heating by femtosecond laser pulses. The solutions of the model are investigated analytically and numerically. Firstly, the semi-analytical solution is presented by the Laplace transform method and the Fourier cosine transform method. With the L1 approximation of the Caputo fractional derivative, the finite difference scheme is derived to describe the effects of femtosecond laser pulses on the temperature distributions of the gold films. The effectiveness of the numerical algorithm is examined by the comparison with the semi-analytical solution. And then based on the numerical algorithm and the experimental datasets of femtosecond laser pulses heating on gold films of various thicknesses, the estimations of the model parameters are considered by the modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS-PSO). Using the optimal values of the parameters estimation, we verify the consistencies between the semi-analytical solutions of the model and the datasets. It proves the accuracy of the time fractional DPL heat conduction model in characterizing the temperature distributions of thin gold films. Finally, the varieties of temperature distribution with time and position are considered and then the effects of the model parameters on the temperature distribution are also discussed.

Numerical simulation of a two‐compartmental fractional model in pharmacokinetics and parameters estimation

Compartmental model is the classic and widely used approach in pharmacokinetics. Recently, fractional calculus is introduced to describe the time course of drugs in human body which follow the anomalous diffusion mechanism. To consider the different fractional order transmission process, the two‐compartmental fractional model has been proposed and studied. And it will be of great significance to find out a simple and efficient numerical method to solve the model and estimate model parameters. This work investigates two numerical methods of the two‐compartmental fractional model using finite difference schemes, which are based on the shifted Grünwald–Letnikov approximate formula and L1 formula, respectively. The hybrid Nelder–Mead simplex search and particle swarm optimization, denoted as NMSS–PSO, is provided to estimate the fractional model parameters. Comparison between the numerical solution and the solution by the numerical inverse Laplace transform method establishes the validity of the developed numerical algorithms. Then the influence of the order of fractional derivative on the drug amount in human body is also discussed. Finally, the two‐compartmental fractional model is applied to fit the amiodarone pharmacokinetics data based on the NMSS–PSO algorithms. This work will be of importance for the development of fractional pharmacokinetics.

Estimation of parameter of fractional order COVID-19 SIQR epidemic model

Epidemic model have been broadly used in different forms for studying and forecasting epidemiological processes the spread of dengue, zika virus , HIV, SARS and recently , the 2019–20 corona virus which is an ongoing pandemic of corona virus disease (COVID-19). In the present paper, an inverse problem to find the parameters for the single term (multi term) fractional order system of an outbreak of COVID-19 is considered. In the starting, we propose a numerical method for fractional order corona virus system based on the Gorenflo-Mainardi-Moretti-Paradisi (GMMP) scheme, and then to find the parameters we use GMMP method and the modified hybrid Nelder-Mead Simplex search and particle swarm optimization algorithm. With the new fractional orders and parameters our fractional order corona virus system is capable to providing numerical results that agree well with the real data.

Novel parameter estimation techniques for a multi-term fractional dynamical epidemic model of dengue fever

An inverse problem to identify parameters for the single-term (and multi-term) fractional-order system of an outbreak of dengue fever is considered. Firstly, we propose a numerical method for the fractional-order dengue fever system based on the Gorenflo-Mainardi-Moretti-Paradisi (GMMP) scheme and the Newton method. Secondly, two methods, the modified grid approximation method (MGAM) and the modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) algorithm are expanded to estimate the fractional orders and coefficients for fractional differential equations. Then, we use GMMP and MH-NMSS-PSO to estimate the parameters of the fractional-order dengue fever system. With the new fractional orders and parameters, our fractional-order dengue fever system is capable of providing numerical results that agree very well with the real data. Furthermore, for searching a better dengue fever system, a multi-term fractional-order epidemic system of dengue fever is proposed. We also use the MGAM and MH-NMSS-PSO to estimate the fractional orders and coefficients of the multi-term fractional-order system. To verify the efficiency and accuracy of the proposed methods in dealing with the fractional inverse problem, a numerical example with real data is investigated. Using the statistics from the 2009 outbreak of the disease in the Cape Verde islands, we are able to predict the fractional orders and parameters of the fractional dengue fever system. With the new fractional orders and parameters, our multi-term fractional-order dengue fever system is capable of providing numerical results that agree better with the real data than other integer-order models.

Some novel numerical techniques for an inverse problem of the multi-term time fractional partial differential equation

In this paper, an inverse problem to identify parameters for the multi-term time fractional partial differential equation with Caputo fractional derivative is considered. The numerical solution of the direct problem is obtained by the modified fractional predictor–corrector method, while the inverse problem is conducted by the modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS–PSO) algorithm, which is expanded to estimate parameters for fractional differential equations. To verify the efficiency and accuracy of the proposed methods, three numerical examples with experimental data are conducted. Furthermore, in order to improve the practical significance of the research on inverse problems, the predictive value of parameter estimation is also testified. This paper provides effective numerical methods for parameter estimation in practical applications involving multi-term time fractional constitutive equations.

Multi-term time-fractional Bloch equations and application in magnetic resonance imaging

Magnetic resonance imaging can reveal exquisite details about the complex structure and function of human tissue. However, magnetic resonance imaging signal behaviour at high or ultra-high field has shown increased deviation from the classically expected mono-exponential relaxation. The underlying mechanism of anomalous relaxation can contribute to a better understanding of the interaction of spins with their surroundings. The purpose of this work is to explore the utility of the multi-term time-fractional Bloch equations in relation to the anomalous relaxation processes. We proposed an effective predictor–corrector method to solve the multi-term equations. Voxel-level temporal fitting of the magnetic resonance imaging signal was performed based on the model developed from the multi-term time-fractional Bloch equations. A feasible parameter estimation method based on hybrid Nelder–Mead simplex search and particle swarm optimisation was introduced to perform the curve fitting. The extra time-fractional terms provide flexibility in the relaxation process.

Solar cell parameters identification using hybrid Nelder-Mead and modified particle swarm optimization

Solar cell modelling primarily involves the formulation of the non-linear current versus voltage (I-V) curve. Determination of parameters plays an important role in solar cell accurate modelling. This paper presents an application of the hybrid Nelder-Mead simplex search method and modified Particle Swarm Optimization technique for identifying the parameters of solar cell and photovoltaic module models. The proposed technique is used to identify the unknown model parameters, namely, the generated photocurrent, saturation current, series resistance, shunt resistance, and ideality factor that govern the current-voltage relationship of a solar cell. A measurement data of 57 mm diameter commercial (R.T.C. France) silicon solar cell and a module consisting of 36 polycrystalline silicon cells (Photowatt-PWP 201) are used to test and verify the consistency of accurately estimating various parameters. Comparative study among different parameter estimation techniques is presented to demonstrate the effectiveness of the proposed approach. Furthermore, to verify the accurateness of the proposed technique, an experimental test bench unit is built using 72 mono-crystalline silicon cells (HIP-210NKHB5-2 module) on different environment conditions.

Realization of circuit breaker condition-based maintenance using optimal contact wear equation by the modified NM-PSO algorithm

The electrical circuit breaker (CB) is one of the absolutely essential devices in the electrical power system. The evaluation of CB condition and its reliability gain on importance. The contact wear of the CB monitored by Intelligent Electronic Device plays a crucial role in the condition-based maintenance (CBM). The present contact wear maintenance of the CB is based on the maintenance data provided by original manufacturers to compute the value of the accumulated contact wear. However, the evaluation of the contact wear always has a quite error. It may lead to over-maintenance or under-maintenance commonly found in maintenance planning and scheduling. Little literature has been down on the issue. By utilizing the modified hybrid Nelder–Mead simplex search and particle swarm optimization (PSO) algorithm, the paper proposes a curve-fitting technology to obtain an optimal contact wear equation for realizing the CBM of the CBs. The presented optimal contact wear equation minimizes the sum of squared errors that can be programmed in Intelligent Electronic Device. An effective CBM planning can enhance electricity substation reliability and cut maintenance costs. Compared with the conventional Nelder–Mead simplex search and PSO and PSO algorithms, two case studies are given to verify the effectiveness of the proposed method. The work may provide some valuable references for the related workers and researchers. Copyright © 2015 John Wiley & Sons, Ltd.

A Modified NM‐PSO Method for Parameter Estimation Problems of Models

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.

Parameter identification of chaotic systems by hybrid Nelder–Mead simplex search and differential evolution algorithm

Parameter identification of chaotic systems is an important issue in nonlinear science and has attracted increasing interest from a variety of research and application fields. Essentially, parameter identification can be formulated as a multi-dimensional optimization problem. By combining differential evolution (DE) and Nelder–Mead (NM) simplex search, an effective hybrid algorithm named NMDE is proposed in this paper. By suitably fusing the DE-based evolutionary search and NM simplex-based local search, exploration and exploitation abilities can be well balanced and satisfactory optimization performances can be achieved. The NMDE hybrid algorithm is applied to parameter identification of several typical chaotic systems. Numerical simulation and comparisons with some typical existing algorithms demonstrate the effectiveness and robustness of the proposed hybrid NMDE algorithm. Moreover, the effects of noise and population size on the performances of NMDE are investigated as well.

Hybrid Nelder–Mead simplex search and particle swarm optimization for constrained engineering design problems

Constrained optimization problems are very important in that they frequently appear in the real world. A constrained optimization problem consists of the optimization of a function subject to constraints, in which both the function and constraints may be nonlinear. Constraint handling is one of the major concerns when solving constrained optimization problems by hybrid Nelder–Mead simplex search method and particle swarm optimization, denoted as NM–PSO. This paper proposes embedding constraint handling methods, which include the gradient repair method and constraint fitness priority-based ranking method, in NM–PSO as a special operator to deal with satisfying constraints. Experiments using three benchmark function and three engineering design problems are presented and compared with the best known solutions reported in the literature. The comparison results with other evolutionary optimization methods demonstrate that NM–PSO with the embedded constraint operator proves to be extremely effective and efficient at locating optimal solutions.