Nonlinear Least-squares Technique Research Articles

Non-iterative pulse tail extrapolation algorithms for correcting nuclear pulse pile-up

Radiation detection systems working at high count rates suffer from the overlapping of their output electric pulses, known as pulse pile-up phenomenon, resulting in spectrum distortion and degradation of the energy resolution. Pulse tail extrapolation is a pile-up correction method which tries to restore the shifted baseline of a piled-up pulse by extrapolating the overlapped part of its preceding pulse. This needs a mathematical model which is almost always nonlinear, fitted usually by a nonlinear least squares (NLS) technique. NLS is an iterative, potentially time-consuming method.The main idea of the present study is to replace the NLS technique by an integration-based non-iterative method (NIM) for pulse tail extrapolation by an exponential model. The idea of linear extrapolation, as another non-iterative method, is also investigated. Analysis of experimental data of a NaI(Tl) radiation detector shows that the proposed non-iterative method is able to provide a corrected spectrum quite similar with the NLS method, with a dramatically reduced computation time and complexity of the algorithm. The linear extrapolation approach suffers from a poor energy resolution and throughput rate in comparison with NIM and NLS techniques, but provides the shortest computation time.

A Novel Liquid Water Content Retrieval Method Based on Mass Absorption for Single-Wavelength Cloud Radar

Low-level clouds (LLC), mainly composed of liquid water droplets, cool the climate system by strongly reflecting solar radiation back to space, and thus play an important role in the Earth energy budget. However, the LLC properties and their radiative effects are poorly represented in climate models, leading to the largest source of uncertainty in the climate prediction. Liquid water content (LWC) is a key property of LLC determining cloud extinction characteristics and is a fundamental parameter in the radiative transfer model. To improve the understanding of LWC properties, algorithms have been proposed to retrieve LWC based on millimeter-wavelength radar. However, the traditional retrieval relies on pre-constructed empirical relationship between reflectivity and LWC and have noticeable limitations. Particularly, the retrieval uncertainty is strongly depended on the assumed particle size distribution, the existence of drizzle particle; and on the accuracy of reflectivity measurement. In this study, we develop a new self-consistent algorithm to retrieve LWC by constraining radar reflectivity factor and attenuation in the whole liquid cloud layer. A relationship between the radar measured reflectivity, LWC, and the intrinsic reflectivity is first constructed based on the radiative transfer theory under Rayleigh scattering regime. A nonlinear least-square regression technique is then applied to derive the optimal parameters in the retrieval equations to obtain the LWC. Comparison with the microwave radiometer (MWR) derived liquid water path (LWP) indicates that our proposed method retrieves more accurate LWC products than that from the traditional empirical algorithms.

A Stochastic Mathematical Model for Understanding the COVID-19 Infection Using Real Data

Natural symmetry exists in several phenomena in physics, chemistry, and biology. Incorporating these symmetries in the differential equations used to characterize these processes is thus a valid modeling assumption. The present study investigates COVID-19 infection through the stochastic model. We consider the real infection data of COVID-19 in Saudi Arabia and present its detailed mathematical results. We first present the existence and uniqueness of the deterministic model and later study the dynamical properties of the deterministic model and determine the global asymptotic stability of the system for R0≤1. We then study the dynamic properties of the stochastic model and present its global unique solution for the model. We further study the extinction of the stochastic model. Further, we use the nonlinear least-square fitting technique to fit the data to the model for the deterministic and stochastic case and the estimated basic reproduction number is R0≈1.1367. We show that the stochastic model provides a good fitting to the real data. We use the numerical approach to solve the stochastic system by presenting the results graphically. The sensitive parameters that significantly impact the model dynamics and reduce the number of infected cases in the future are shown graphically.

CAPUTO-BASED MODEL FOR INCREASING STRAINS OF CORONAVIRUS: THEORETICAL ANALYSIS AND EXPERIMENTAL DESIGN

One of the most severe and troubling diseases these days is COVID-[Formula: see text] pandemic. The COVID-[Formula: see text] pandemic’s dangerous effects are extremely rapid, and infection normally results in death within a few weeks. As a consequence, it is important to delve deeper into the complexities of this elusive virus. In this study, we propose a Caputo-based model for increasing COVID-19 strains. The memory effect and hereditary properties of the fractional variant for the model enable us to fully comprehend the dynamics of the model’s features. The existence of unique solution using the fixed-point theorem and Arzelá–Ascoli principle as well as the stability analysis of the model by means of Ulam–Hyer stability (UHS) and generalized Ulam–Hyer stability (GUHS) have been discussed. Furthermore, the parameters of the model are estimated using [Formula: see text] months data points chosen from Nigeria using the nonlinear least-squares technique. The best-suited parameters and the optimized Caputo fractional-order parameter [Formula: see text] are obtained by running simulations for both models. The proposed model is shown to comprehend the dynamical behavior of the virus better than the integer-order version. In addition, to shed more light on the model’s characteristics, various numerical simulations are performed using an efficient numerical scheme.

Theoretical Study of Hybrid Solar Cell Parameters Evaluation From I-V Characteristics

Photovoltaics, which convert directly solar energy into electricity, provide a practical and sustainable solution to the challenge of meeting the increasing global energy demand. Computer simulation is an important tool for investigating solar cell device’s behavior and optimizing their performance. This work develops a new approach to retrieve the five parameters of the single diode equivalent solar cell/module model using the measured current-voltage data and its derivative (G=dI/dV). A nonlinear least-square technique based on the Newton-Raphson method under MATLAB Program is applied to determine the five parameters of the hybrid solar cell including under different temperature.

A mechanistic model for airborne and direct human-to-human transmission of COVID-19: effect of mitigation strategies and immigration of infectious persons.

The COVID-19 pandemic is the most significant global crisis since World War II that affected almost all the countries of our planet. To control the COVID-19 pandemic outbreak, it is necessary to understand how the virus is transmitted to a susceptible individual and eventually spread in the community. The primary transmission pathway of COVID-19 is human-to-human transmission through infectious droplets. However, a recent study by Greenhalgh et al. (Lancet 397:1603–1605, 2021) demonstrates 10 scientific reasons behind the airborne transmission of SARS-COV-2. In the present study, we introduce a novel mathematical model of COVID-19 that considers the transmission of free viruses in the air beside the transmission of direct contact with an infected person. The basic reproduction number of the epidemic model is calculated using the next-generation operator method and observed that it depends on both the transmission rate of direct contact and free virus contact. The local and global stability of disease-free equilibrium (DFE) is well established. Analytically it is found that there is a forward bifurcation between the DFE and an endemic equilibrium using central manifold theory. Next, we used the nonlinear least-squares technique to identify the best-fitted parameter values in the model from the observed COVID-19 mortality data of two major districts of India. Using estimated parameters for Bangalore urban and Chennai, different control scenarios for mitigation of the disease are investigated. Results indicate that the vaccination of susceptible individuals and treatment of hospitalized patients are very crucial to curtailing the disease in the two locations. It is also found that when a vaccine crisis is there, the public health authorities should prefer to vaccinate the susceptible people compared to the recovered persons who are now healthy. Along with face mask use, treatment of hospitalized patients, and vaccination of susceptibles, immigration should be allowed in a supervised manner so that economy of the overall society remains healthy.

Caputo SIR model for COVID-19 under optimized fractional order

Everyone is talking about coronavirus from the last couple of months due to its exponential spread throughout the globe. Lives have become paralyzed, and as many as 180 countries have been so far affected with 928,287 (14 September 2020) deaths within a couple of months. Ironically, 29,185,779 are still active cases. Having seen such a drastic situation, a relatively simple epidemiological SIR model with Caputo derivative is suggested unlike more sophisticated models being proposed nowadays in the current literature. The major aim of the present research study is to look for possibilities and extents to which the SIR model fits the real data for the cases chosen from 1 April to 15 March 2020, Pakistan. To further analyze qualitative behavior of the Caputo SIR model, uniqueness conditions under the Banach contraction principle are discussed and stability analysis with basic reproduction number is investigated using Ulam–Hyers and its generalized version. The best parameters have been obtained via the nonlinear least-squares curve fitting technique. The infectious compartment of the Caputo SIR model fits the real data better than the classical version of the SIR model (Brauer et al. in Mathematical Models in Epidemiology 2019). Average absolute relative error under the Caputo operator is about 48% smaller than the one obtained in the classical case (nu =1). Time series and 3D contour plots offer social distancing to be the most effective measure to control the epidemic.

Simulation of geoelectrical responses of structures buried in freshwater and brackish water environments

Marine geoelectrical simulation offers the opportunity to predict matrix type, interstitial fluid and geometry of buried structures. In this research, simulation of geoelectrical responses over buried structures in freshwater and brackish water environments were tested. A model tank made of acrylic and housed within an iron frame was built for the experiment. The tank was filled with water and sieved sediment having grain sizes ≤1.18 mm, corresponding to very coarse sand to colloid. Four models were designed namely freshwater static model (FSM), brackish water static model (BSM), freshwater model with buried structures (FMBS), and brackish water model with buried structures (BMBS). A bamboo pipe, metal pipe and granite block were buried in the FMBS and BMBS. Physico-chemical parameters of the water which includes temperature (22.74 – 26.06 °C), salinity (0.07 – 15.72 psu), conductivity (153 – 25,420 μS/cm and resistivity (5.6 x 103 – 6.6 x 103 Ω-cm) were measured. Dipole-dipole array using inter-electrode separation (a) of 5 cm and 1≤n≤5 was adopted. Resistivity measurements obtained were processed and inverted using non-linear least-square optimization technique. We present obtained results as contoured 2D resistivity structures. FSM and FMBS were characterized by relatively high resistivity values compared with the BSM and BMBS. The buried materials were effectively resolved in the freshwater models, compared to the brackish water model due to higher contrast in resistivity between the material and saturating fluid. The bamboo pipe was not effectively delineated in the brackish water medium, as a result of the low resistivity contrast which exists between the medium and the material. Zones with high compaction are characterized by high resistivity values, while concealed channel structures were characterized by low compaction. Estimated depth were exact for the freshwater models but was over estimated by `∼12.5% in the brackish water models. Marine electrical resistivity method shows great potential in mapping structures buried in medium with high contrasting resistivity values, and sedimentary structures with varying degree ofcompaction.

A General Frequency Domain Estimation Method for Gegenbauer Processes

Abstract In this paper a new method for estimation of all the parameters of a k-factor Gegenbauer process is developed using a broadband nonlinear least-squares regression technique in the frequency-domain, with similarities to a Whittle estimator. Simulation studies where the underlying distribution is symmetric suggest that while the new method may have a slightly lower level of accuracy than existing methods (Whittle, conditional sum-of-squares), it can improve the accuracy in determining the values for the short-memory parameters of highly skewed non-Gaussian data (e.g., χ 2), while having the added advantage of being evaluated considerably faster. In a supplementary addendum we provide some theoretical results under a Gaussian assumption.

Identification of nonlinear aerodynamic damping of wind-excited flexible structures by curve-fitting non-Gaussian response probability density function

This study presents a new approach for identification of nonlinear aerodynamic damping of wind-excited flexible structures from stochastic response time history that can be modeled as the output of a single-degree-of-freedom system under a white noise excitation. The amplitude-dependent aerodynamic damping ratio is described as a polynomial function of vibration displacement or velocity. Based on the equivalent nonlinear equation approach, the probability density function (PDF) of displacement can be given in terms of damping model parameters, which generally follows a hardening non-Gaussian distribution. It is proposed to identify the damping model parameters by curve-fitting displacement PDF determined from the response data using a nonlinear least-square optimization technique. Firstly, the stochastic response of a tall building with a known nonlinear aerodynamic model is simulated and the performance of the proposed approach is investigated. This method is then applied to a square-shaped tall building model and a bridge deck section using wind tunnel measured data. Finally, it is used for a traffic-signal-support structure using field measurement response data. The results are also compared with those from other existing approaches. The proposed approach permits determination of nonlinear aerodynamic damping from single stochastic response history and is very easy to use in practice.

Fractional modeling for the spread of Hookworm infection under Caputo operator

Abstract It is estimated that, about one billion people mostly from Asia, Sub-Saharan Africa and Latin America are infected with the Hookworm infection. In this paper, we developed and analyzed a model for the transmission dynamics of Hookworm infection in a human population using Caputo fractional order differential operator. Under Caputo operator, existence and uniqueness for the solutions of the new Hookworm infection model have been analyzed using fixed point theorems. Parameters of the model are estimated with the help of real statistics available for the Hookworm infection from a city of Ghana and the best fit is obtained under the nonlinear least-squares curve fitting technique. Further analysis of the proposed model shows that the disease free (infection-absent) equilibrium is locally asymptotically stable whenever a certain reproduction number R 0 1 and the endemic (infection-present) equilibrium point is globally asymptotically stable whenever R 0 1 and unstable if R 0 > 1 . Using forward normalized sensitivity index, the most sensitive parameters are identified that are essential for control of the infection and we obtained different types of simulations for the proposed Hookworm transmission system with the best fitted fractional order parameter (χ). The modelling results show that the chemotherapy treatment, awareness and improvement of personal hygiene are the best measures to be taken for control of the Hookworm infection among vulnerable community.

Comparison of Spectrum Estimation Methods for the Accurate Evaluation of Sea State Parameters.

The monitoring of sea state conditions, either for weather forecasting or ship seakeeping analysis, requires the reliable assessment of the sea spectra encountered by the ship, either as a final result or intermediate step for the measurement of the relevant wave-motion parameters. In current analyses, different spectrum estimation methods, namely the Welch, Thomson and ARMA models, have been applied and compared based on a set of random wave signals, with different durations, representative of several sea state conditions. Subsequently, two sea spectrum reconstruction techniques were described and applied in order to detect the main sea state parameters, namely the significant wave height, the mean wave period and the spectrum peak enhancement factor. The performances of both spectral analysis and sea state reconstruction methods are discussed in order to provide some preliminary guidelines for practical application purposes. In this respect, based on current results, the Welch and Thomson methods seem to be the most promising techniques, combined with the nonlinear least-square reconstruction technique.

DebyeFit: a simple tool to obtain an appropriate model of atomic vibrations in solids from atomic displacement parameters obtained at different temperatures

DebyeFit is a simple tool to calculate the Debye or Einstein characteristic temperature of thermal vibrations in crystals from the equivalent atomic displacement parameters (ADPs) of any atom obtained at several temperatures. The ADP values are separated into static and dynamic components to get a best fit to the Debye, Einstein or mixed model. The static term is added to account for possible static disorder. A nonlinear least-squares technique is used to refine the parameters of the model for sets of ADPs observed in multi-temperature structural studies. The program provides a good fit between theoretical and observed ADP values.

Pressure broadening coefficients in the ν6 band of CH3F in collision with He

In this study we report the first measurements of the pressure line broadening coefficients of 219 lines of CH3F in the PP, RP, RQ, PQ, and RR sub-branches of the ν6 band (8.5 µm-spectral region) at room temperature using a high-resolution Fourier transform spectrometer. The collisional line widths are obtained from the non-linear least-square multi-pressure fitting techniques. A Voigt line shape model was used in the present analysis. Measured values are compared with the results of theoretical calculation based on semi-classical formalism of Robert-Bonamy (RB) model with exact isotropic trajectories. Even with a Smith-Giraud-Cooper intermolecular potential model the calculated values show a quite good agreement with experimental results.

Complex Brillouin Optical Time-Domain Analysis

We demonstrate a novel complex Brillouin optical time-domain analysis (BOTDA) approach for distributed fiber sensing that detects the complex Brillouin spectrum. Unlike other approaches, we utilize the complex nonlinear least-square regression technique to determine the Brillouin frequency shift (BFS) directly in the complex domain. The reduction of BFS uncertainty with our approach is shown through theoretical analysis and is validated by both simulations and experiments. Compared with either the gain-based or the phase-based BOTDA approaches, our complex BOTDA can improve the sensing accuracy by a factor of $\sqrt {2} $ , which is equivalent to a 3 dB improvement in the power signal-to-noise ratio. Distributed temperature sensing is demonstrated over 40 km standard single-mode fiber. The experimental results agree well with the theoretical predictions of BFS uncertainty that confirms this considerable improvement.

Impact of fitting algorithms on errors of parameter estimates in dynamic contrast-enhanced MRI

Parameter estimation in dynamic contrast-enhanced MRI (DCE MRI) is usually performed by non-linear least square (NLLS) fitting of a pharmacokinetic model to a measured concentration-time curve. The two-compartment exchange model (2CXM) describes the compartments ‘plasma’ and ‘interstitial volume’ and their exchange in terms of plasma flow and capillary permeability. The model function can be defined by either a system of two coupled differential equations or a closed-form analytical solution. The aim of this study was to compare these two representations in terms of accuracy, robustness and computation speed, depending on parameter combination and temporal sampling. The impact on parameter estimation errors was investigated by fitting the 2CXM to simulated concentration-time curves. Parameter combinations representing five tissue types were used, together with two arterial input functions, a measured and a theoretical population based one, to generate 4D concentration images at three different temporal resolutions. Images were fitted by NLLS techniques, where the sum of squared residuals was calculated by either numeric integration with the Runge–Kutta method or convolution. Furthermore two example cases, a prostate carcinoma and a glioblastoma multiforme patient, were analyzed in order to investigate the validity of our findings in real patient data. The convolution approach yields improved results in precision and robustness of determined parameters. Precision and stability are limited in curves with low blood flow. The model parameter ve shows great instability and little reliability in all cases. Decreased temporal resolution results in significant errors for the differential equation approach in several curve types. The convolution excelled in computational speed by three orders of magnitude. Uncertainties in parameter estimation at low temporal resolution cannot be compensated by usage of the differential equations. Fitting with the convolution approach is superior in computational time, with better stability and accuracy at the same time.

Parameter Extraction of PV Module using NLS Algorithm with Experimental Validation

Photovoltaic (PV) module parameters act an important task in PV system design and simulation. Most popularly used single diode R<sub>sh</sub> model has five unknown electrical parameters such as series resistance (R<sub>se</sub>), shunt resistance (R<sub>sh</sub>), diode quality factor (a), photo-generated current (I<sub>pg</sub>) and dark saturation current (I<sub>s</sub>) in the mathematical model of PV module. The PV module output voltage and current relationship is represented by a transcendental equation and is not possible to solve analytically. This paper proposes nonlinear least square (NLS) technique to extract five unknown parameters. The proposed technique is compared with other two popular techniques available in the literature such as Villalva’s comprehensive technique and modified Newton-Raphson (N-R) technique. Only two parameters R<sub>se</sub> and R<sub>sh</sub> are estimated by Villalva’s technique, but all single diode unknown electrical parameters can be estimated by the NLS technique. The accuracy of different estimation techniques is compared in terms of absolute percentage errors at MPP and is found the minimum for the proposed technique. The elapsed time for parameter estimation for NLS technique is minimum and much less compared to other two techniques. Extracted parameters of polycrystalline ELDORA-40 PV panel by the proposed technique have been validated through simulation and experimental current-voltage (I-V) and power-voltage (P-V) characteristics.

The effects of ionic strength and organic matter on virus inactivation at low temperatures: general likelihood uncertainty estimation (GLUE) as an alternative to least-squares parameter optimization for the fitting of virus inactivation models

This study examined how the inactivation of bacteriophage MS2 in water was affected by ionic strength (IS) and dissolved organic carbon (DOC) using static batch inactivation experiments at 4 °C conducted over a period of 2 months. Experimental conditions were characteristic of an operational managed aquifer recharge (MAR) scheme in Uppsala, Sweden. Experimental data were fit with constant and time-dependent inactivation models using two methods: (1) traditional linear and nonlinear least-squares techniques; and (2) a Monte-Carlo based parameter estimation technique called generalized likelihood uncertainty estimation (GLUE). The least-squares and GLUE methodologies gave very similar estimates of the model parameters and their uncertainty. This demonstrates that GLUE can be used as a viable alternative to traditional least-squares parameter estimation techniques for fitting of virus inactivation models. Results showed a slight increase in constant inactivation rates following an increase in the DOC concentrations, suggesting that the presence of organic carbon enhanced the inactivation of MS2. The experiment with a high IS and a low DOC was the only experiment which showed that MS2 inactivation may have been time-dependent. However, results from the GLUE methodology indicated that models of constant inactivation were able to describe all of the experiments. This suggested that inactivation time-series longer than 2 months were needed in order to provide concrete conclusions regarding the time-dependency of MS2 inactivation at 4 °C under these experimental conditions.

Selection and Configuration of Sorption Isotherm Models in Soils Using Artificial Bees Guided by the Particle Swarm

A precise estimation of isotherm model parameters and selection of isotherms from the measured data are essential for the fate and transport of toxic contaminants in the environment. Nonlinear least-square techniques are widely used for fitting the isotherm model on the experimental data. However, such conventional techniques pose several limitations in the parameter estimation and the choice of appropriate isotherm model as shown in this paper. It is demonstrated in the present work that the classical deterministic techniques are sensitive to the initial guess and thus the performance is impeded by the presence of local optima. A novel solver based on modified artificial bee-colony (MABC) algorithm is proposed in this work for the selection and configuration of appropriate sorption isotherms. The performance of the proposed solver is compared with the other three solvers based on swarm intelligence for model parameter estimation using measured data from 21 soils. Performance comparison of developed solvers on the measured data reveals that the proposed solver demonstrates excellent convergence capabilities due to the superior exploration-exploitation abilities. The estimated solutions by the proposed solver are almost identical to the mean fitness values obtained over 20 independent runs. The advantages of the proposed solver are presented.

A new model for the viscosity of highly concentrated multi-sized particulate Bingham slurries

ABSTRACTBench scale tests were carried out to investigate the rheological properties of multi-sized particulate Bingham slurries at high solid concentrations ranging from 50% to 70% by weight. In addition, rheological data from Biswas et al. (2000) and Chandel et al. (2009, 2010) have also been considered. Based on these extensive amount of rheological data, an empirical model is proposed for viscosity as a function of solid volume fraction (ɸ), maximum solid volume fraction (ɸm), median particle diameter (d50), and coefficient of uniformity (Cu) using optimization and nonlinear least-square curve-fitting technique. The proposed model shows good agreement with the experimental data considered in the present study and is found to be much better than the previously developed models in predicting the viscosity of multi-sized particulate Bingham slurries at high solid concentrations.