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For distributed estimation arising in the nonlinear least squares (NLLSs) problems over adaptive networks, where every node has the abilities of data processing and learning, only the incomplete local data are exploited by the traditional noncooperative method, thereby resulting in the degradation on estimation performance. In this article, a cooperative diffusion strategy is proposed by using a Gauss-Newton (GN) method in order to fully utilize the diversity of temporal-spatial data on local updates. The proposed algorithm includes two steps, i.e., aggregate then update (ATU), where the aggregating step collects in real time the global information instead of local information due to the diffusion strategy, and the updating step implements the local GN iteration. The resulting ATU diffusion algorithm is a distributed and cooperative system without any increase on communication cost, as compared with the noncooperative version. Based on the detailed convergence analysis for ATU, which is fundamental to the promotion of this algorithm, the sufficient conditions for convergence are derived and the evidences of faster convergence than the noncooperative version are provided. The simulation results confirm the obtained theoretical derivations by applying the ATU algorithm to an NLLS-based target localization problem and show the cooperation gains in many aspects, such as the convergence rate, steady-state accuracy, and robustness to noisy range, step size, node, and link failures.

When a physical system is modeled by a nonlinear function, the unknown parameters can be estimated by fitting experimental observations by a least-squares approach. Newton's method and its variants are often used to solve problems of this type. In this paper, we are concerned with the computation of the minimal-norm solution of an underdetermined nonlinear least-squares problem. We present a Gauss–Newton type method, which relies on two relaxation parameters to ensure convergence, and which incorporates a procedure to dynamically estimate the two parameters, as well as the rank of the Jacobian matrix, along the iterations. Numerical results are presented.

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations. To fulfill the input requirement of Algorithm I, we develop a numerical GCD-finder containing a successive singular value updating and an iterative GCD refinement as the main engine of Algorithm II that calculates the multiplicity structure and the initial root approximation. While limitations of our algorithm exist in identifying the multiplicity structure in certain situations, the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities. To measure the sensitivity of the multiple roots, a structure-preserving condition number is proposed and error bounds are established. According to our computational experiments and error analysis, a polynomial being ill-conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved, and its multiple roots can be computed with high accuracy.

The conditional, unconditional, or the exact maximum likelihood estimation and the least-squares estimation involve minimizing either the conditional or the unconditional residual sum of squares. The maximum likelihood estimation (MLE) approach and the nonlinear least squares (NLS) procedure involve an iterative search technique for obtaining global rather than local optimal estimates. Several authors have presented brief overviews of algorithms for solving NLS problems. Snezana S. Djordjevic (2019) presented a review of some unconstrained optimization methods based on the line search techniques. Mahaboob et al. (2017) proposed a different approach to estimate nonlinear regression models using numerical methods also based on the line search techniques. Mohammad, Waziri, and Santos (2019) have briefly reviewed methods for solving NLS problems, paying special attention to the structured quasi-Newton methods which are the family of the search line techniques. Ya-Xiang Yuan (2011) reviewed some recent results on numerical methods for nonlinear equations and NLS problems based on online searches and trust regions techniques, particularly on Levenberg-Marquardt type methods, quasi-Newton type methods, and trust regions algorithms. The purpose of this paper is to review some online searches and trust region's more well-known robust numerical optimization algorithms and the most used in practice for the estimation of time series models and other nonlinear regression models. The line searches algorithms considered are: Gradient algorithm, Steepest Descent (SD) algorithm, Newton-Raphson (NR) algorithm, Murray’s algorithm, Quasi-Newton (QN) algorithm, Gauss-Newton (GN) algorithm, Fletcher and Powell algorithm (FP), Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. While the only trust-region algorithm considered is the Levenberg-Marquardt (LM) algorithm. We also give some main advantages and disadvantages of these different algorithms.

[EN] This work presents a hybrid modeling methodology based on dynamic response, filtering and identification techniques, in order to determine a ball mill representative model. In essence, we have provided models for electrical drive, mechanical reduction and load, without the need for physical decoupling. The electrical parameters are determined using state variable filtering, linear regression and recursive least square techniques. The mechanical parameters are identified considering the system acceleration time. A final adjustment stage considering the parameters set, is carried out using the nonlinear least squares method. Based on the ball mill complete model, a load torque estimator is proposed, using high-pass filters, and a load torque estimate. The numerical simulations, under different operating conditions, show suitable approximation with experimental results. Therefore, the proposed hybrid methodology, based on both dynamic modeling and signal analysis, has the potential to assist in the design for supervision and control systems of a ball mill., [ES] Este trabajo presenta una metodología híbrida de modelado basada en técnicas de respuesta dinámica, filtrado e identificación, considerando el dominio del tiempo y la frecuencia, para determinar el modelo representativo de un molino de bolas de acoplamiento fijo. Se proponen modelos para el accionamiento eléctrico, reductor mecánico y carga, sin la necesidad de desacoplamiento físico. Los parámetros eléctricos se determinan utilizando técnicas de filtrado de variable de estado, regresión lineal y mínimos cuadrados recursivos, y los parámetros mecánicos se identifican considerando solo el tiempo de aceleración del sistema. Se realiza un ajuste final del conjunto de parámetros mediante la técnica de mínimos cuadrados no lineales. Basado en el modelo completo del molino, se propone un estimador del par de carga, utilizando filtros de paso alto, y se presenta una estimación de la cantidad de carga del molino. Las simulaciones numéricas del modelo determinado, en diferentes condiciones de operación del molino, muestran una buena aproximación con resultados experimentales. Por lo tanto, la metodología híbrida propuesta, basada tanto en el modelado dinámico como en análisis de señales, presenta potencial para ayudar en el proyecto de procesos de supervisión y control del molino de bolas de acoplamiento fijo.

Parameter estimation problems of mathematical models can often be formulated as nonlinear least squares problems. Typically these problems are solved numerically using iterative methods. The local minimiser obtained using these iterative methods usually depends on the choice of the initial iterate. Thus, the estimated parameter and subsequent analyses using it depend on the choice of the initial iterate. One way to reduce the analysis bias due to the choice of the initial iterate is to repeat the algorithm from multiple initial iterates (i.e. use a multi-start method). However, the procedure can be computationally intensive and is not always used in practice. To overcome this problem, we propose the Cluster Gauss–Newton (CGN) method, an efficient algorithm for finding multiple approximate minimisers of nonlinear-least squares problems. CGN simultaneously solves the nonlinear least squares problem from multiple initial iterates. Then, CGN iteratively improves the approximations from these initial iterates similarly to the Gauss–Newton method. However, it uses a global linear approximation instead of the Jacobian. The global linear approximations are computed collectively among all the iterates to minimise the computational cost associated with the evaluation of the mathematical model. We use physiologically based pharmacokinetic (PBPK) models used in pharmaceutical drug development to demonstrate its use and show that CGN is computationally more efficient and more robust against local minima compared to the standard Levenberg–Marquardt method, as well as state-of-the art multi-start and derivative-free methods.

Background: Recent work showed that the temporal growth of the novel coronavirus disease (COVID-19) follows a sub-exponential power-law scaling whenever effective control interventions are in place. Taking this into consideration, we present a new phenomenological logistic model that is well-suited for such power-law epidemic growth. Methods: We empirically develop the logistic growth model using simple scaling arguments, known boundary conditions and a comparison with available data from four countries, Belgium, China, Denmark and Germany, where (arguably) effective containment measures were put in place during the first wave of the pandemic. A non-linear least-squares minimization algorithm is used to map the parameter space and make optimal predictions. Results: Unlike other logistic growth models, our presented model is shown to consistently make accurate predictions of peak heights, peak locations and cumulative saturation values for incomplete epidemic growth curves. We further show that the power-law growth model also works reasonably well when containment and lock down strategies are not as stringent as they were during the first wave of infections in 2020. On the basis of this agreement, the model was used to forecast COVID-19 fatalities for the third wave in South Africa, which is currently in progress. Conclusions: We anticipate that our presented model will be useful for a similar forecasting of COVID-19 induced infections/deaths in other regions as well as other cases of infectious disease outbreaks, particularly when power-law scaling is observed., Final version published as a journal article in Epidemics

The paper considers an algorithm for the direct search for a nonparametric smooth histogram of the particle size distribution from small-angle X-ray and neutron scattering data. The features and details of the implementation of the method, which consists in the sequential search for several solutions with different degrees of smoothness of the distribution contour, are considered. Methods for evaluating the stability of both the whole distribution contour and its individual parts are discussed. The work of the program is illustrated by examples of the analysis of polydisperse spherical particles in silicasol solutions.

By conducting binding experiments at a range of temperatures T using isothermal titration calorimetry (ITC), one can obtain two estimates of the binding enthalpy - calorimetric (ΔH°cal) from the experiments at each T, and van't Hoff (ΔH°vH) from the T dependence of the binding constant K°. From thermodynamics it is clear that these two must be identical, but early efforts to demonstrate this for ITC data indicated significant inconsistency. In an extensive 2004 study of the Ba2+ + 18-crown-6 ether complexation used in prior comparisons, Mizoue and Tellinghuisen found modest (10-20%) but statistically significant differences, which were tentatively attributed to problems converting the calorimetric estimates to their standard state values, as implied by the superscript ° in the notation. In the present work the 2004 results are reanalyzed using results obtained since then from temperature, heat, and volume calibration of the instrument and a better determination of the data variance function required for the weighted least-squares fitting of the data. The new results show consistency for temperatures 5-30 °C but persistent statistically significant differences from 35-46 °C. Several possible explanations for the remaining discrepancies are examined, with methods that include fitting the K and ΔHcal data together.

Bitumen/styrene-butadiene-styrene (SBS) polymer composite material (also coded as polymer modified bitumen, PMB) is one of the most employed composite materials in the pavement industry. PMB is viscoelastic and exhibits temperature-dependent modulus. Abundant researches have investigated the influence of temperature on the PMB modulus, however few have attempted to establish a model to describe the direct relationship between the PMB modulus and the temperature. This paper aims to use a kinetic model for fiber reinforced composite to describe the inherent relationship between the PMB modulus and temperature. Furthermore, a modified model is proposed to find better fitting results specifically for PMB composite. Considering the fact that PMB has a much higher polydispersity, the modified model uses nonlinear least squares regression instead of the original Coats–Redfern method to determine the kinetic parameters (activation energy and pre-exponential factor). The determination of glassy state modulus and rubbery state modulus is also replaced by numerical optimization instead of visual inspection. The predicted modulus was compared with experimental results obtained by Dynamic Mechanical Analysis (DMA), and a good agreement was found. The method introduced by this study could be a promising approach to study the temperature-dependent properties and state transition behaviors of PMB composite.

The paper presents the partial work done within the framework of the EDA Siramis II project focused on magnetic signature reproduction of ships. Reproduction is understood here as the ability to determine the magnetic anomaly of the local Earth magnetic field in any direction and at any measurement depth due to the presence of the analysed object. The B-91 type hydrographic ship Zodiak was selected as the real case study. The work was divided into two main stages: the development of a measurement campaign taking into account physical measurements, and the development of a mathematical model on the basis of the measured values. The measurement campaign included: preparation of the measuring range, selection of equipment for the measurement of magnetic quantities and geographical location, and data recording while the ship passes the measuring point according to the designated course. As a result of the measurement campaign, magnetic flux density components were collected in different positions in relation to the measuring instruments and the ship's heading. A multi-dipole model was used to build the mathematical model in accordance with the idea of inverse modelling. The effectiveness of this model was previously checked on synthetic data of virtual ships generated using the finite element method. Experiments performed with simulation models were helpful in determining the structure of the model, the nature of the data, and the number of samples needed to properly determine the multi-dipole model parameters. The parameters were determined using the nonlinear least squares method according to the idea of data fitting. The classical Ridge and Lasso regularization methods were applied to prevent the developed multi-dipole model from overfitting. Other regularization methods based on GPS accuracy marks and modification of fitness functions were also considered. The verification was done using real data: the data generated by the model was compared with patterns recorded during the Zodiak measurement campaign. High degree of conformity of the shape of characteristics was obtained. Moreover, the correctness of model execution was confirmed by low values of quantitative indices such as RMSE and MAE representing modelling errors. The methodology presented in the paper is quite universal and can be used to determine the signatures of other ferromagnetic objects.

We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each sub-domain is represented by a local feed-forward neural network, and $C^k$ continuity is imposed on the sub-domain boundaries. Each local neural network consists of a small number of hidden layers, while its last hidden layer can be wide. The weight/bias coefficients in all hidden layers of the local neural networks are pre-set to random values and are fixed, and only the weight coefficients in the output layers are training parameters. The overall neural network is trained by a linear or nonlinear least squares computation, not by the back-propagation type algorithms. We introduce a block time-marching scheme together with the presented method for long-time dynamic simulations. The current method exhibits a clear sense of convergence with respect to the degrees of freedom in the neural network. Its numerical errors typically decrease exponentially or nearly exponentially as the number of degrees of freedom increases. Extensive numerical experiments have been performed to demonstrate the computational performance of the presented method. We compare the current method with the deep Galerkin method (DGM) and the physics-informed neural network (PINN) in terms of the accuracy and computational cost. The current method exhibits a clear superiority, with its numerical errors and network training time considerably smaller (typically by orders of magnitude) than those of DGM and PINN. We also compare the current method with the classical finite element method (FEM). The computational performance of the current method is on par with, and oftentimes exceeds, the FEM performance., Comment: 62 pages, 48 figures, 12 tables

The introduction of high-power white LEDs has revolutionized the lighting industry in the past few decades due to the multiple benefits in terms of high reliability, environmental friendliness and versatile applications. However, challenges have arisen in assessing the reliability and lifetime prediction because it is difficult to record the failure data in a short period of time. Currently, the nonlinear least squares (NLS) regression-based method is used in industry for projecting the lumen maintenance lifetime from degradation data. The model parameters estimated using the NLS regression approach are deterministic and introduce high prediction errors. In this paper, a Bayesian method is proposed to estimate the remaining useful lifetimes (RULs) of both high-power white LED packages and lamps. The accelerated degradation tests conducted for gathering lumen degradation data are used to validate the proposed method. The exponential decay model is used as the degradation model and the parameters are estimated based on Markov Chain Monte Carlo (MCMC) sampling and using the Metropolis-Hasting (MH) algorithm. The lifetime prediction results showed that the Bayesian method has better prediction accuracy compared to the NLS method. Thus, the proposed Bayesian method is shown to be a promising approach to address the lifetime prediction issue for high-power white LEDs with improved prediction accuracy.