Nelder-Mead Simplex Algorithm Research Articles

Calculation of the ideal isochronous field for the SC200 cyclotron using the Nelder-Mead simplex algorithm

ABSTRACT In this paper, we propose a numerical method for ideal isochronous field calculation for the SC200 isochronous cyclotron based on the Nelder-Mead simplex algorithm. In this method, the radial field distribution is represented by an nth-order polynomial, an objective function is defined to describe the relative error of the orbital frequency, and the ideal isochronous field is calculated by optimizing the coefficients of the polynomial to minimize the objective function using the Nelder-Mead simplex algorithm. A comparative analysis of the proposed method and conventional methods indicates that the proposed method provides a more accurate ideal isochronous field, especially in the vicinity of the extraction region where the field flutter is large.

A minimization approach to depth and shape determination of mineralized zones from potential field data using the Nelder-Mead simplex algorithm

We have developed a generalized minimization approach to depth and shape determination from potential field data (gravity, self-potential, or magnetic) due to sources of simple geometry (e.g., spheres, cylinders, dikes, contacts) that resemble mineralized bodies. The Nelder-Mead simplex algorithm is applied to solve a nonlinear equation in depth for each fixed shape factor using the anomaly values at few points on anomaly profile. The algorithm is achieved through two steps. First, it computes the standard deviation of the depths determined using different characteristic distances for each value of the shape factor. Second, it chooses the optimum shape and depth with minimum standard deviation. Procedures are also formulated to determine the other model parameters. The technique is applied to noisy synthetic data as well as three real cases of different mineralization from Sweden, Germany, and Canada.

Peak factor optimization of multi-harmonic signals using artificial bee colony algorithm

Multi-harmonic signals are useful in many applications to estimate the frequency response of a device. To minimize noise effects, signal power should be maximized within the amplitude limitations of the signal generation and acquisition hardware. Signal power depends only on the harmonic amplitudes, while the signal amplitude also depends on the harmonic phases. Careful choice of these phases allows the minimization of the peak factor which corresponds to the minimization of the signal’s amplitude for a given signal power. Multiple methods have been proposed to minimize a signal’s peak factor, including closed form expressions and iterative algorithms. However, the existence of many local extrema suggests the use of global search algorithms. This paper proposes the use of the artificial bee colony algorithm. The solution is further improved by using a Nelder-Mead Simplex algorithm. The artificial bee colony algorithm results are compared with those of previously published methods.

Optimizing noisy CNLS problems by using Nelder-Mead algorithm: A new method to compute simplex step efficiency

Nelder-Mead “simplex” algorithm (NMA) is a derivative-free algorithm that can be used to solve complex nonlinear least-squared (CNLS) problems. NMA can fit equivalent electrical circuit models to noisy electrochemical impedance spectroscopy (EIS) data. The ability of NMA simplex to adapt itself onto the local landscape of mathematical functions is governed by the simplex steps (reflection, expansion, inside and outside contraction and shrink steps). However, according to EIS literature, a method to compute the simplex step efficiency has not been reported yet.Herein, we provide and recommend a new method to compute the simplex step efficiency. The new method was used to evaluate the adaptive and standard NMA modifications (SNMA and ANMA). Be advised that by using the new method we detected an unknown property of a more successful ANMA (vs. SNMA), i.e. the application of adaptive parameters decreased only the inside contraction step efficiency. However, the aforementioned drawback was resolved by modifying and boosting the existing ANMA, which was a process that demonstrated the usefulness of the new method. Thereby, SNMA, ANMA, and the modified ANMA fitting engines were embedded in the free (MIT licensed) software by using Python programming language.

Stochastic Dual Simplex Algorithm: A Novel Heuristic Optimization Algorithm.

A new heuristic optimization algorithm is presented to solve the nonlinear optimization problems. The proposed algorithm utilizes a stochastic method to achieve the optimal point based on simplex techniques. A dual simplex is distributed stochastically in the search space to find the best optimal point. Simplexes share the best and worst vertices of one another to move better through search space. The proposed algorithm is applied to 25 well-known benchmarks, and its performance is compared with grey wolf optimizer (GWO), particle swarm optimization (PSO), Nelder-Mead simplex algorithm, hybrid GWO combined with pattern search (hGWO-PS), and hybrid GWO algorithm combined with random exploratory search algorithm (hGWO-RES). The numerical results show that the proposed algorithm, called stochastic dual simplex algorithm (SDSA), has a competitive performance in terms of accuracy and complexity.

Selection of decline curve analysis model using Akaike information criterion for unconventional reservoirs

Petroleum production from unconventional resources (specifically, tight and shale reservoirs) has become a significant portion of the supply in North America which led operating companies to look for more accurate methods for reserve evaluation and production forecasting. Typically, the operators resort to Decline Curve Analysis (DCA) to determine the Estimated Ultimate Recovery (EUR) of hydrocarbon reserves in both conventional and unconventional reservoirs. In shale wells, the best DCA model fitted to the transient flow regimes followed by Boundary Dominated Flow (BDF). Therefore, in this study, we propose a novel framework for systematic DCA model selection using a quantitative measure of goodness of production data fit from shale wells. For this purpose, we selected seven different, most frequently applied DCA models to evaluate large number of wells from the Montney Formation. To collect production data from the Montney wells, we used the GeoSCOUT package. Then, we fed these data into the DCA model selection workflow powered by the Nelder-Mead simplex algorithm and implemented using a computer programming language. The quantitative measure that helps identify the best fit of the data and the model is derived from the Information Theory (IT) and is known as Akaike Information Criterion (AIC). The obtained results show that our DCA model selection framework can identify the most appropriate DCA model for a given production history. Moreover, it gives a specific quantitative measure (AIC) of the fit that can be used for model ranking. The results show that more accurate values of b are necessary in Arps’ DCA models to match production data from the Montney shale wells, and identify the Logistic Growth Analysis Model (LGM) is the best fit for the majority of the selected wells followed by Extended Exponential Decline Model (EED), Duong, and Power Law Exponential Model (PLE).

Parameter estimation and mathematical modeling for the quantitative description of therapy failure due to drug resistance in gastrointestinal stromal tumor metastasis to the liver.

In this work we develop a general mathematical model and devise a practical identifiability approach for gastrointestinal stromal tumor (GIST) metastasis to the liver, with the aim of quantitatively describing therapy failure due to drug resistance. To this end, we have modeled metastatic growth and therapy failure produced by resistance to two standard treatments based on tyrosine kinase inhibitors (Imatinib and Sunitinib) that have been observed clinically in patients with GIST metastasis to the liver. The parameter identification problem is difficult to solve, since there are no general results on this issue for models based on ordinary differential equations (ODE) like the ones studied here. We propose a general modeling framework based on ODE for GIST metastatic growth and therapy failure due to drug resistance and analyzed five different model variants, using medical image observations (CT scans) from patients that exhibit drug resistance. The associated parameter estimation problem was solved using the Nelder-Mead simplex algorithm, by adding a regularization term to the objective function to address model instability, and assessing the agreement of either an absolute or proportional error in the objective function. We compared the goodness of fit to data for the proposed model variants, as well as evaluated both error forms in order to improve parameter estimation results. From the model variants analyzed, we identified the one that provides the best fit to all the available patient data sets, as well as the best assumption in computing the objective function (absolute or proportional error). This is the first work that reports mathematical models capable of capturing and quantitatively describing therapy failure due to drug resistance based on clinical images in a patient-specific manner.

Theoretical confirmation of an established experimental phase diagram of ternary system (Cu(SO4)–Zn(SO4)–H2O) at 25 °C: Pitzer parameters determination

The isotherm phase diagram of the ternary system (Cu(SO4)–Zn(SO4)–H2O) was firstly established at 25°C using a synthetic method of conductivity measurement. Then, we made use of volumetric dosing, in order to identify solid phases. The obtained isotherm has revealed the presence of two solid phases that having a stable equilibrium with the liquid phase, which are Cu(SO4)·5H2O, and the Zn(SO4)·7H2O. This study was confirmed numerically while going through the calculation of the binary and ternary Pitzer parameters for the studied salts through a Nelder Mead simplex algorithm. Indeed, the comparison of these with the bibliographic data has confirmed the accuracy of our experimental results.

Time and Frequency Domain Simulation, Measurement and Optimization of Log-Periodic Antennas

Log-periodic antenna is a special antenna type utilized with great success in many broadband applications due to its ability to achieve nearly constant gain over a wide frequency range. Such antennas are extensively used in electromagnetic compatibility measurements, spectrum monitoring and TV reception. In this study, a log-periodic dipole array is measured, simulated, and then optimized in the 470–860 MHz frequency band. Two simulations of the antenna are initially performed in time and frequency domain respectively. The comparison between these simulations is presented to ensure accurate modelling of the antenna. The practically measured realized gain is in good agreement with the simulated realized gain. The antenna is then optimized to concurrently improve voltage standing wave ratio, realized gain and front-to-back ratio. The optimization process has been implemented by using various algorithms included in CST Microwave Studio, such as Trusted Region Framework, Nelder Mead Simplex algorithm, Classic Powell and Covariance Matrix Adaptation Evolutionary Strategy. The Trusted Region Framework algorithm seems to have the best performance in adequately optimizing all predefined goals specified for the antenna.

Modeling Phase Behavior of Semi-Clathrate Hydrates of CO2, CH4, and N2 in Aqueous Solution of Tetra-n-butyl Ammonium Fluoride

Abstract Semi-clathrate hydrates are members of the class of clathrate compounds. In comparison with clathrate hydrates, where the networks are formed only by H2O molecules, the networks of semi-clathrate hydrates are formed by mixtures of H2O and quaternary ammonium salts (QASs). The addition of QASs to the solution enables to improve the formation of semi-clathrate hydrates at much milder conditions comparing to clathrate hydrates. In this work, we study the phase equilibria of semi-clathrate hydrates of CH4, CO2, and N2 gas in an aqueous solution of tetra-n-butyl ammonium fluoride (TBAF). An extension of the Chen–Guo model is proposed as a thermodynamic model. The Peng–Robinson equation of state (PREOS) was applied to calculate the fugacity of the gas phase and in order to determine the water activity in the presence of TBAF, a correlation between the system temperature, the TBAF mass fraction, and the nature of the guest molecules has been used. These equations were solved simultaneously and through optimizing tuning parameters via the Nelder–Mead simplex algorithm. The results are compared to experimental data and good agreement is observed.

Parameter estimation of proton exchange membrane fuel cells using eagle strategy based on JAYA algorithm and Nelder-Mead simplex method

Building an accurate mathematical model is vital for the simulation, control, evaluation, management, and optimization of proton exchange membrane fuel cells (PEMFCs). Usually this work is processed through building a mathematical model based on empirical or semi-empirical equations firstly and then estimating the unknown model parameters using optimization technologies. In this study, a simple two stage eagle strategy based on JAYA algorithm and Nelder-Mead simplex algorithm is proposed for effectively estimating the unknown model parameters of PEMFCs. In the proposed strategy, JAYA algorithm is employed for the coarse global exploration, and Nelder–Mead simplex search algorithm is employed for the intensive local search. The effectiveness of the proposed strategy is verified through estimation experiments with 7 and 9 unknown parameters. Compared with the basic JAYA algorithm and four other newly reported excellent meta-heuristic algorithms including Grey Wolf Optimizer, Grasshopper Optimization Algorithm, Salp Swarm Optimizer, and Multi-Verse Optimizer, the proposed strategy possesses better performance in terms of accuracy, convergence speed, and stability, thus it is a promising approach for parameter estimation of PEMFC.

Real-time adaptive optical self-interference cancellation system for in-band full-duplex transmission

A real-time adaptive optical self-interference cancellation (RTA-OSIC) system is proposed and demonstrated for in-band full-duplex transmission. The RTA-OSIC system is controlled automatically by adaptive algorithm running on a real-time microcontroller STM32. Modified Hooke–Jeeves (MHJ) algorithm is proposed to search the optimal optical parameters more rapidly. Experimental results show that the MHJ algorithm requires 15–25 samples to find the optimal point. The RTA-OSIC system achieves more than 22 dB cancellation depth within 0-700 MHz frequency band. Compared with the existing adaptive OSIC system which uses the Nelder–Mead Simplex algorithm, there are 60% less samples for our RTA-OSIC system with MHJ algorithm.

Orbital properties of binary post-AGB stars

Binary post-asymptotic giant branch (post-AGB) stars are thought to be the products of a strong but poorly understood interaction during the AGB phase. The aim of this contribution is to update the orbital elements of a sample of galactic post-AGB binaries observed in a long-term radial-velocity monitoring campaign by analysing these systems in a homogeneous way. Radial velocities are computed from high signal-to-noise spectra via a cross-correlation method. The radial-velocity curves are fitted by using both a least-squares algorithm and a Nelder–Mead simplex algorithm. We use a Monte Carlo method to compute uncertainties on the orbital elements. The resulting mass functions are used to derive a companion mass distribution by optimising the predicted to the observed cumulative mass-function distributions, after correcting for observational bias. As a result, we derive and update orbital elements for 33 galactic post-AGB binaries, among which 3 are new orbits. The orbital periods of the systems range from 100 to about 3000 days. Over 70% (23 out of 33) of our binaries have significant non-zero eccentricities ranging over all periods. Their orbits are non-circular even though the Roche-lobe radii are smaller than the maximum size of a typical AGB star and tidal circularisation should have been strong when the objects were on the AGB. We derive a distribution of companion masses that is peaked around 1.09 M⊙ with a standard deviation of 0.62 M⊙. The large spread in companion masses highlights the diversity of post-AGB binary systems. Post-AGB binaries are often chemically peculiar, showing in their photospheres the result of an accretion process of circumstellar gas devoid of refractory elements. We find that only post-AGB stars with high effective temperatures (> 5500 K) in wide orbits are depleted in refractory elements, suggesting that re-accretion of material from a circumbinary disc is an ongoing process. It appears, however, that depletion is inefficient for the closest orbits irrespective of the actual surface temperature.

Hybrid revised weighted fuzzy c-means clustering with Nelder-Mead simplex algorithm for generalized multisource Weber problem

PurposeThe purpose of this paper is to propose hybrid revised weighted fuzzy c-means (RWFCM) clustering and Nelder–Mead (NM) simplex algorithm, called as RWFCM-NM, for generalized multisource Weber problem (MWP).Design/methodology/approachAlthough the RWFCM claims that there is no obligation to sequentially use different methods together, NM’s local search advantage is investigated and performance of the proposed hybrid algorithm for generalized MWP is tested on well-known research data sets.FindingsTest results state the outstanding performance of new hybrid RWFCM and NM simplex algorithm in terms of cost minimization and CPU times.Originality/valueProposed approach achieves better results in continuous facility location problems.

Model Based Optimum Pid Gain Design of Adaptive Front Lighting System

Adaptive Front-Lighting System (AFLS) is a system which assists driver's field of vision by automatically controlling its brightness and illumination angle to adapt various driving conditions such as climate, traffic, road changes and so forth. This paper aims to propose novel model-based PID gain design method to improve the performances of Dynamic Bending Light (DBL) module that change horizontal angle of a system by applying Brent-Dekker algorithm that finds the root of nonlinear function and implementing Nelder-Mead simplex algorithm to the system reduction process. Along with the linear system model-based control theory, motor dynamics were modeled with frequency response. Validation of the prototype resulted in having less than 3 % error from the simulation, where position initialization and the real-time status monitoring function is available due to the closed loop control which enables over 3 times faster response than the conventional open-loop system.

The Nelder–Mead simplex algorithm with perturbed centroid for high-dimensional function optimization

It is a well known fact that the widely used Nelder–Mead (NM) optimization method soon becomes inefficient as the dimension of the problem grows. It has been observed that part of the reason for the inefficiency is that the search direction becomes practically perpendicular to the direction of the local downhill gradient. In this paper, we identify which operations are responsible for the increase of the angle between the search direction and the direction of the local downhill gradient. We show that it is possible to decrease this angle by randomly perturbing the centroid of the n best vertices around which the original NM algorithm moves the worst vertex. Using a perturbed centroid, the algorithm outperforms the standard NM method as well as the NM method with adaptive parameters for problems with higher dimensions, and works well for problems with dimensions of well over 100.

Performance comparison of exponential, Lambert W function and Special Trans function based single diode solar cell models

Accurate modeling plays an important role in solar cell simulation. In order to reveal the applicability and superiority of Special Trans function based single diode model (SBSDM), this paper presents a comprehensive comparison of SBSDM, Lambert W function based single diode model (LBSDM) and exponential-type single diode model (SDM). The performance difference of SBSDM, LBSDM and SDM is verified and compared in two aspects: (1) different fitness to the measured I–V data of solar cells and (2) different parameter extraction performance. To be objective and reproducible, the reported parameter values of standard datasets and measured datasets are employed to validate the fitness difference of the three models. The comparison results indicate that SBSDM always exhibits better fitness than LBSDM and SDM in representing the I–V characteristics of various solar cells and can provide a closer prediction to actual maximum power points. With the help of a ranking based branch selection strategy, a modified Nelder-Mead simplex (MNMS) algorithm is proposed to test the parameter extraction performance of SBSDM, LBSDM and SDM. The comparison results reveal that the time computational efficiency of SBSDM is inferior to SDM but superior to LBSDM. SBSDM always achieves superior accuracy and convergence speed than LBSDM and SDM, although lacking enough statistical robustness. Due to these superiorities, SBSDM is quite promising and envisaged to be the most valuable model for solar cell parameter extraction and PV system simulation.

Dynamic Hassan Nelder Mead with Simplex Free Selectivity for Unconstrained Optimization

We propose a free selective simplex for the downhill Nelder Mead simplex algorithm (1965), rather than the determinant simplex that forces its elements to perform a single operation, such as reflection. Unlike the Nelder-Mead algorithm, the elements of the proposed simplex select various operations of the algorithm to form the next simplex. In this way, we allow non-isometric reflections similar to that of the Nelder Mead, triangle simplex, but with rotation through an angle, permitting the proposed algorithm to have more control over the simplex, to change its size and direction for better performance. As a consequence, the solution that comes from the proposed simplex is always dynamic adaptive in size and orientation to different landscapes of mathematical functions. The proposed algorithm is examined in a large collection of different structures and classes of optimization problems. Additionally, comparisons are made with two enhanced, up-to-date versions of the Nelder-Mead algorithm. The numerical results show that Hassan Nelder Mead is stable due to non-dependence on the number of parameters processed. It also performs a higher accuracy for high dimensions compared with the other algorithms and a faster convergence rate toward global minima with respect to the number of simplex gradient estimates.

A sinusoidal function and the Nelder-Mead simplex algorithm applied to growth data from broiler chickens.

There has been much recent interest in mathematical developments for the analysis of growth in poultry. In this paper, we present a sinusoidal function to describe the evolution of growth as a function of time based on real life experiments. The function was evaluated with regard to its ability to describe the relationship between body weight and age in broilers and was compared to 4 standard growth functions: Gompertz, logistic, Lopez, and Richards. In order to estimate the model parameters, we adopted a global optimization method based on a direct search method instead of using gradient-based techniques. The results of this study show that both the sinusoidal function and the direct search method precisely describe the growth dynamics of broiler chickens. Fitting the growth functions to different data profiles nearly always led to the same or less maximized log-likelihood values for the sinusoidal equation, which is an indication of its superiority in describing growth data from broiler chickens.

Defense against primary user emulation attacks from the secondary user throughput perspective

The maximization of Secondary user (SU) throughput has been studied extensively in honest cooperative spectrum sensing (CSS). However, when primary user emulation attacks (PUEAs) are launched, the model of CSS changes. This also necessitates the redesigning of the SU throughput maximisation scheme. In this paper, such redesigning of the SU throughput maximisation scheme under PUEAs has been carried out. Our objective is to suppress the damages caused by the PUEAs on the SU throughput. To serve this purpose, an optimally weighted CSS has been proposed. The optimal weights are obtained by maximising the SU throughput while protecting the primary user (PU) from interference in a network facing the PUEAs. Considering the significance of simplicity and speed in CSS, we apply the Nelder Mead Simplex Algorithm to solve the problem. The experiments carried out endorse the effectiveness of the proposed scheme compared to the existing combination schemes.