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Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types of grading strategies. Rational functions are an exception to this rule: for univariate functions with point singularities, such as branch points, rational approximations exist with root-exponential convergence in the rational degree. This is typically enabled by the clustering of poles near the singularity. Both the theory and computational practice of rational functions for function approximation have focused on the univariate case, with extensions to two dimensions via identification with the complex plane. Multivariate rational functions, i.e., quotients of polynomials of several variables, are relatively unexplored in comparison. Yet, apart from a steep increase in theoretical complexity, they also offer a wealth of opportunities. A first observation is that singularities of multivariate rational functions may be continuous curves of poles, rather than isolated ones. By generalizing the clustering of poles from points to curves, we explore constructions of multivariate rational approximations to functions with curves of singularities. [ABSTRACT FROM AUTHOR]

Celik and Senoglu (Communications in Statistics-Simulation and Computation 48 (1):287–301, 2019) proposed a two-way analysis of variance (ANOVA) model when the distribution of the error terms is skew t. They obtained the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies. They also proposed test statistics based on these estimators for testing the main hypotheses in a two-way ANOVA model. They finally compared the efficiencies of the ML and the MML estimators and the power of the test statistics with the corresponding normal theory results via a Monte Carlo simulation study. Unfortunately, they wrongly presented one of the maximum likelihood equations and consequently, some parts of the simulation results and conclusions are questionable. [ABSTRACT FROM AUTHOR]

This paper demonstrates the mixed formulation of the Brinkman problem using linear equal-order finite element methods in porous media modelling. We introduce Galerkin least-squares (GLS) and least-squares (LS) finite element methods to address the incompatibility of finite element spaces, treating velocity, pressure, and vorticity as independent variables. Theoretical analysis examines coercivity and continuity, providing error estimates. Demonstrating resilience in theoretical findings, these methods achieve optimal convergence rates in the $ L^2 $ L 2 norm by incorporating stabilization terms with low-order basis functions. Numerical experiments validate theoretical predictions, showing the effectiveness of the GLS method and addressing finite element space incompatibility. Additionally, the GLS method exhibits promising capabilities in handling the Brinkman equation at low permeability compared to the LS method. The study reveals an increase in the average pressure difference in the Brinkman problem compared to the Stokes equations as the inlet velocity rises, providing insights into the behaviour of Brinkman equations. [ABSTRACT FROM AUTHOR]

Summary: In industrial process control systems, parameter estimation is crucial for controller design and model analysis. This article examines the issue of identifying parameters in continuous‐time models. This article presents a stochastic gradient estimation algorithm and a recursive least squares estimation algorithm for identifying the parameters of continuous systems. It derives the parameter identification model of linear continuous‐time systems based on the Laplace transforms of the input and output of the systems. To prove that the techniques given here work, we have included a simulated example. [ABSTRACT FROM AUTHOR]

Given sample data points $ \{(x_j,f_j)\}_{j = 1}^N $, in [Brubeck, Nakatsukasa, and Trefethen, SIAM Review, 63 (2021), pp. 405-415], an Arnoldi-based procedure is proposed to accurately evaluate the best fitting polynomial, in the least squares sense, at new nodes $ \{s_j\}_{j = 1}^M $, based on the Vandermonde basis. Numerical tests indicated that this procedure can in general achieve high accuracy. The main purpose of this paper is to perform a forward rounding error analysis in finite precision. Our result establishes sensitivity factors regarding the accuracy of the algorithm, and provides a theoretical justification for why the algorithm works. For least-squares approximation on an interval, we propose a variant of this Arnoldi-based evaluation by using the Chebyshev polynomial basis. Numerical tests are reported to demonstrate our forward rounding error analysis. [ABSTRACT FROM AUTHOR]

In the studies of unit commitment or optimal power flow, to formulate a mixed-integer linear programming model that can be efficiently solved with commercial solvers, it is necessary to approximate the quadratic cost curves of thermal units as piecewise linear (PWL) functions. The conventional approach involves evenly spaced piecewise linear (ES-PWL) interpolation, which often results in relatively large approximation errors. In order to reduce the error, this paper proposes a more accurate PWL method for approximating the quadratic cost functions of thermal units. The method employs a linear least-squares fit instead of linear interpolation within each subinterval and introduces a one-terminal-constraint approach to ensure the continuity of the piecewise function. Subsequently, a straightforward equation is derived, applicable to the widely used ES-PWL interpolation, with the potential to enhance the accuracy of the approximation. Mathematical verification attests that the proposed method substantially diminishes the squared 2-norm error, less than 37.5% of the error associated with ES-PWL interpolation. Subsequent numerical investigations are carried out on a 10-unit system, the IEEE RTS-79, and a real industrial system. The findings validate that all the approximation errors of the proposed method are within 37.5% of the errors associated with the ES-PWL interpolation, meaning that a unit commitment solution that closely approximates the outcome of the quadratic function is obtained. The computational time is also acceptable. [ABSTRACT FROM AUTHOR]

This paper aims to find a maximum likelihood least squares-based iterative algorithm to solve the identification issues of closed-loop input nonlinear equation-error systems. By adopting the key term separation technique, the parameters of the forward channel are identified separately from the parameters of the feedback channel to address the cross-product terms. The hierarchical identification principle is introduced to decompose the original system into two subsystems for reduced computational complexity. The iterative estimation theory and the maximum likelihood principle are applied to design a new least-squares algorithm with high estimation accuracy by taking full use of all the measured input-output data at each iterative computation. Compared with the recursive least-squares (RELS) method. The simulation results verify theoretical findings, and the proposed algorithm can generate more accurate parameter estimates than the RELS algorithm. [ABSTRACT FROM AUTHOR]

To enhance the positioning accuracy of standalone GNSS receivers in environments unable to provide precise ephemeris and clock offset, such as undeveloped forest areas that lack network communication and power supply, this study employed the Time Difference Carrier Phase (TDCP) technology to improve the positioning accuracy of Standard Point Positioning (SPP), where the Least-Squares (LS) and the extended Multi-Epoch Least Squares (MELS) method were applied in the position domain filtering for a single GNSS receiver and compare its performance with the existing observation domain filtering method. Firstly, the simulated data sets with various positioning accuracies were used to verify the effectiveness and convergence of the LS filtering methods. The results indicate that the LS filtering method produces a lower root mean square (RMS) error than the original strategy. Secondly, this study uses two kinematic GNSS data sets to evaluate the performance of the observation and position domain filtering, with an emphasis on the MELS method. The numerical experiment results show that the position domain LS filtering method outperforms the other two methods. The open environment experiments result shows that the positioning domain filtering method achieved positioning accuracies of 0.202 m, 0.843 m, and 2.036 m in the E, N, and U directions, respectively, with improvements of 68.0%, 21.6%, and 24.0%, compared to the original algorithm which achieved positioning accuracies of 0.631 m, 1.076 m, and 2.680 m. It also achieved improvements of 24.0%, 4.0%, and 18.3%, respectively, compared to the observation domain filtering method with positioning accuracies of 0.353 m, 0.886 m, and 2.526 m. The forest scenes experiments result shows that the positioning domain filtering method achieved positioning accuracies of 1.308 m, 1.375 m, and 2.133 m in the E, N, and U directions, respectively, with improvements of 42.4%, 36.2%, and 27.6%, compared to original algorithm which achieved positioning accuracies of 1.863 m, 1.873 m, and 2.722 m, and also achieved improvements of 27.0%, 19.4% and 10.6%, respectively, comparing to observation domain filtering method with positioning accuracies of 1.661 m, 1.642 m and 2.359 m. Moreover, the examination of the LS method results based on different epochs reveals that the filtering accuracy increases as more epochs are incorporated into the position domain integration and the enhancement value reaches a few millimeters. [ABSTRACT FROM AUTHOR]

Single-output regression is a widely used statistical modeling method to predict an output based on one or more features of a datapoint. If a dataset has multiple outputs, they can be predicted independently from each other although this disregards potential correlations and thus may negatively affect the predictive performance. Therefore, multi-output regression methods predict multiple outputs simultaneously. One way to approach single-output regression is by using methods based on support vectors such as support vector regression (SVR) or least-squares SVR (LS-SVR). Based on these two, previous works have devised multi-output support vector regression methods. In this review, we introduce a unified notation to summarize the single-output support vector regression methods SVR and LS-SVR as well as state-of-the-art multi-output support vector regression methods. Furthermore, we implemented a workflow for subject- and record-wise bootstrapping and nested cross-validation experiments, which we used for an exhaustive evaluation of all single- and multi-output support vector regression methods on synthetic and non-synthetic datasets. Although the reviewed papers claim that their multi-output methods improve regression performance, we find that none of them outperform both single-output methods SVR and LS-SVR for various reasons. Due to these results, we reflected about the general concept of support vector regression and then concluded that support vector regression methods do not appear to be suitable for the task of multi-output regression. • We summarize single- and multi-output support vector regression methods with unified notation. • We provide a workflow for resampling experiments to test multi-output regression methods in a standardized way. • We find that multi-output support vector regression is unable to improve predictions and give reasons supporting this claim. [ABSTRACT FROM AUTHOR]

In this paper, a least-squares virtual element method on polygonal meshes is proposed for the stress-velocity formulation of the linear Stokes problem. The H (div) -and H 1 -conforming virtual elements are used to approximate the stress and velocity variables, respectively. Benefiting from the virtual element method and the least-squares formulation, our method allows the use of general polygonal meshes and leads to a symmetric and positive definite system. The a priori error estimates are established for the stress in H (div) norm and for the velocity in H 1 norm. Additionally, the least-squares functional naturally offers an a posteriori error estimator without extra effort, which together with the great flexibility of mesh can guide the adaptive mesh refinement to resolve the singularity. We also extend the present method to the nonlinear Stokes problem and show the corresponding least-squares virtual element method. A series of numerical examples supporting the theoretical results are presented. [ABSTRACT FROM AUTHOR]

Stochastic models play an important role in achieving high accuracy in positioning, the ideal estimator in the least-squares (LS) can be obtained only by using the suitable stochastic model. This study investigates the role of variance component estimation (VCE) in the LS method for Precise Point Positioning (PPP). This estimation is performed by considering the ionospheric-free (IF) functional model for code and the phase observation of Global Positioning System (GPS). The strategy for estimating the accuracy of these observations was evaluated to check the effect of the stochastic model in four modes: a) antenna type, b) receiver type, c) the tropospheric effect, and d) the ionosphere effect. The results show that using empirical variance for code and phase observations in some cases caused erroneous estimation of unknown components in the PPP model. This is because a constant empirical variance may not be suitable for various receivers and antennas under different conditions. Coordinates were compared in two cases using the stochastic model of nominal weight and weight estimated by LS-VCE. The position error difference for the east-west, north-south, and height components was 1.5 cm, 4 mm, and 1.8 cm, respectively. Therefore, weight estimation with LS-VCE can provide more appropriate results. Eventually, the convergence time based on four elevation-dependent models was evaluated using nominal weight and LS-VCE weight. According to the results, the LS-VCE has a higher convergence rate than the nominal weight. The weight estimation using LS-VCE improves the convergence time in four elevation-dependent models by 11, 13, 12, and 9 min, respectively.

Function approximation has gained significant attention in reinforcement learning research as it effectively addresses problems with large-scale, continuous state, and action space.Although the function approximation algorithm based on gradient descent method is one of the most widely used methods in reinforcement learning, it requires careful tuning of the step size parameter as an inappropriate value can lead to slow convergence, unstable convergence, or even divergence.To address these issues, an improvement was made around the temporal-difference (TD) algorithm based on function approximation.The weight update method was enhanced using both the least squares method and gradient descent, resulting in the proposed weights gradient descent (WGD) method.The least squares were used to calculate the weights, combining the ideas of TD and gradient descent to find the error between the weights.And this error was used to directly update the weights.By this method, the weights were updated in a new manner, effectively reducing the consumption of computing resources by the algorithm enhancing other gradient descent-based function approximation algorithms.The WGD method is widely applicable in various gradient descent-based reinforcement learning algorithms.The results show that WGD method can adjust parameters within a wider space, effectively reducing the possibility of algorithm divergence.Additionally, it achieves better performance while improving the convergence speed of the algorithm.

This article shows how to develop an efficient solver for a stabilized numerical space-time formulation of the advection-dominated diffusion transient equation. At the discrete space-time level, we approximate the solution using higher-order continuous B-spline basis functions in both spatial and temporal dimensions. This problem is very difficult to solve numerically using the standard Galerkin finite element method due to artificial oscillations present when the advection term dominates the diffusion term. However, a first-order constraint least-square formulation allows us to obtain numerical solutions avoiding oscillations. The advantages of space-time formulations are the use of high-order methods and the feasibility of developing space-time mesh adaptive techniques on well-defined discrete problems. We develop a solver for a least-square formulation to obtain a stabilized and symmetric problem on finite element meshes. The computational cost of our solver is bounded by the cost of the inversion of the space-time mass and stiffness (with one value fixed at a point) matrices and the cost of the GMRES solver applied for the symmetric and positive definite problem. We illustrate our findings on an advection-dominated diffusion space-time model problem and present two numerical examples: one with isogeometric analysis discretizations and the second one with an adaptive space-time finite element method. [ABSTRACT FROM AUTHOR]

Nova-C is a lunar lander developed by the private company Intuitive Machines to deliver commercial payloads to the Moon. The IM-1 mission will launch and land the Nova-C near the Moon's south pole. In this paper, Batch Dilution of Precision (DOP) methods are explored to assess Orbit Determination (OD) performance based on the nominal IM-1 trajectory. Utilizing a range to range-rate measurement error ratio to normalize the position uncertainties, this paper lays out the methodology to derive and incorporate a recursive, time-series Batch Least-Squares algorithm. Through the normalization, the Batch Least Squares algorithm produces a unitless Position Dilution of Precision metric. The normalized-weighted DOP method is used within a genetic algorithm optimizer to determine ground station tracking schedules that optimize OD performance. An assessment of OD performance is studied for key mission events including trajectory correction maneuvers, lunar orbit insertion, and descent orbit insertion. The long-term goal of this research is to develop a fully integrated DOP and Linear Covariance Analysis optimization tool to minimize operational costs and maximize OD performance. [ABSTRACT FROM AUTHOR]

The simple linear regression model and the associated goodness-of-fit measure, the coefficient of determination, R2, are only appropriate when all measurement errors are associated with the measurement of the data in the dependent variable. When measurement errors are assumed in both variables, a Deming regression can be used; however, there is no associated R2-type measure for this specific type of regression. In this paper, we propose a measure, R g 2 which utilizes the minimum percentage improvement of the Deming regression over either the horizontal or the vertical line through the centroid of the data. We investigate some properties of this measure and its relation to R2. We also consider other candidate methodologies for a generalized R2 measure for a Deming regression model and investigate strengths and weaknesses of each as a way of beginning the conversation about which measure is the best and for which applications it is most suited. [ABSTRACT FROM AUTHOR]

Results on the rational approximation of functions containing singularities are presented. We build further on the "lightning method," recently proposed by Trefethen and Gopal [SIAM J. Numer. Anal., 57 (2019), pp. 2074-2094], based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or partial fractions with poles clustering toward infinity in order to obtain a robust approximation of the smooth behavior of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of xα on [0, 1], the optimal convergence rate as shown by Stahl [Bull. Amer. Math. Soc., 28 (1993), pp. 116-122] is now achieved simply by least-squares fitting. [ABSTRACT FROM AUTHOR]

We study adaptive control for a family of nonlinear systems, involving unknown and uncertain parameters. The proposed control law estimates the system parameters adaptively and stabilizes the closed-loop system asymptotically for the initial state over any given bounded set of the state-space. Moreover, reconstruction filters are designed to obtain error residue signals and to enable the use of the least-squares algorithm for estimating the parameters, in order to achieve the convergence based on the persistent excitation condition and asymptotic linearization. The proposed methods are applicable to full actuation and under actuation control systems. Simulation studies are carried out for a pendulum system and for a third-order vehicle model, as well as control of vehicle platoons, validating the theoretical results presented in this paper. [ABSTRACT FROM AUTHOR]

Public health awareness programs have been a crucial strategy in mitigating the spread of emerging and re-emerging infectious disease outbreaks of public health significance such as COVID-19. This study adopts an Susceptible–Exposed–Infected–Recovered (SEIR) based model to assess the impact of public health awareness programs in mitigating the extent of the COVID-19 pandemic. The proposed model, which incorporates public health awareness programs, uses ABC fractional operator approach to study and analyze the transmission dynamics of SARS-CoV-2. It is possible to completely understand the dynamics of the model's features because of the memory effect and hereditary qualities that exist in the fractional version. The fixed point theorem has been used to prove the presence of a unique solution, as well as the stability analysis of the model. The nonlinear least-squares method is used to estimate the parameters of the model based on the daily cumulative cases of the COVID-19 pandemic in Nigeria from March 29 to June 12, 2020. Through the use of simulations, the model's best-suited parameters and the optimal ABC fractional-order parameter τ may be determined and optimized. The suggested model is proved to understand the virus's dynamical behavior better than the integer-order version. In addition, numerous numerical simulations are run using an efficient numerical approach to provide further insight into the model's features. [ABSTRACT FROM AUTHOR]

In the realm of multi-class classification, the twin K-class support vector classification (Twin-KSVC) generates ternary outputs { − 1 , 0 , + 1 } by evaluating all training data in a "1-versus-1-versus-rest" structure. Recently, inspired by the least-squares version of Twin-KSVC and Twin-KSVC, a new multi-class classifier called improvements on least-squares twin multi-class classification support vector machine (ILSTKSVC) has been proposed. In this method, the concept of structural risk minimization is achieved by incorporating a regularization term in addition to the minimization of empirical risk. Twin-KSVC and its improvements have an influence on classification accuracy. Another aspect influencing classification accuracy is feature selection, which is a critical stage in machine learning, especially when working with high-dimensional datasets. However, most prior studies have not addressed this crucial aspect. In this study, motivated by ILSTKSVC and the cardinality-constrained optimization problem, we propose ℓ p -norm least-squares twin multi-class support vector machine (PLSTKSVC) with 0 < p < 1 to perform classification and feature selection at the same time. The technique employed to solve the optimization problems associated with PLSTKSVC is user-friendly, as it involves solving systems of linear equations to obtain an approximate solution for the proposed model. Under certain assumptions, we investigate the properties of the optimum solutions to the related optimization problems. Several real-world datasets were tested using the suggested method. According to the results of our experiments, the proposed method outperforms all current strategies in most datasets in terms of classification accuracy while also reducing the number of features. [ABSTRACT FROM AUTHOR]

This paper presents new consistent methods for continuous-time Multiple-Input-Single-Output (MISO) Errors-In-Variables (EIV) systems by fractional models. The proposed idea is to use Higher-Order Statistics (HOS), such as fourth-order cumulants (foc), instead of noisy input and output measurements to obtain unbiased estimates. Firstly, all differentiation orders are assumed to be known a priori and linear coefficients are estimated. The developed estimator is based on minimizing the equation error and it is called fractional fourth-order based-least squares estimator ( f r a c - f o c - l s ). Secondly, the global commensurability of the fractional MISO system is considered. The f r a c - f o c - l s is combined with a non linear technique to estimate the global commensurate order along with linear coefficients. The developed algorithm is based on minimizing the output error and called fractional fourth-order cumulants based-least squares combined with global commensurate order optimization ( f r a c - f o c - g c o o l s ). The consistency of the developed estimators, in presence of high levels of noise corrupting both the input and output measurements, is assessed through a numerical example with the help of Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

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Copyright of Journal of China University of Petroleum is the property of China University of Petroleum and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Wireless positioning and tracking technology are key technologies in applying wireless sensor networks and have vital research significance and application value. In this paper, in terms of positioning multiple non-coincident dynamic targets, aiming at the problem of low precision and large computational complexity when locating multiple dynamic targets by the two-step least squares algorithm and the constrained total least squares algorithm, an improved constrained total least squares algorithm is proposed. This algorithm fully considers the constraints, introduces the Lagrangian multiplier technology and the quasi-Newton BFGS iterative formula, avoids the calculation of the Hessian matrix, reduces the amount of calculation, and improves the positioning accuracy. Simulation experiments show that when the measurement error and sensor position error are moderate, the ICTLS positioning algorithm has a smaller RMSE than the TSWLS and CTLS positioning algorithms, showing higher positioning accuracy and stronger robustness. Secondly, aiming at the problem that the target is close to the reference node or any coordinate axis causes the positioning error of the traditional positioning algorithm to increase sharply, an optimized two-step least squares algorithm is proposed. This algorithm corrects the defects of the TSWLS algorithm by selecting the reference station and rotating the coordinate system again, and improves the positioning performance of the algorithm while reducing the amount of calculation. [ABSTRACT FROM AUTHOR]

The AAA algorithm has become a popular tool for data-driven rational approximation of single-variable functions, such as transfer functions of a linear dynamical system. In the setting of parametric dynamical systems appearing in many prominent applications, the underlying (transfer) function to be modeled is a multivariate function. With this in mind, we develop the AAA framework for approximating multivariate functions where the approximant is constructed in the multivariate barycentric form. The method is data driven, in the sense that it does not require access to the full state-space model and requires only function evaluations. We discuss an extension to the case of matrix-valued functions, i.e., multi-input/multi-output dynamical systems, and provide a connection to the tangential interpolation theory. Several numerical examples illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper focus on the detection for false data injection (FDI) attacks. Under external conditions, the state and the output variables of Nonlinear Cyber‐Physical Systems (NCPSs) cannot be described by a linear relationship. It is difficult to detect the attacked systems. The Laguerre function model has excellent approximation capability to change structural parameters such as system delay and order, making it convenient for online parameter identification and suitable for complex industrial process control. Accordingly, in order to detect FDI attacks for NCPSs, the Laguerre function is adopted to detect FDI attacks and improve the detection rate. Numerical simulations verify the effectiveness of this method. [ABSTRACT FROM AUTHOR]

Multipath effects can lead to delays of centimeters in GNSS phase observations depending on the station environment. Those delays can seriously degrade the accuracy of GNSS positioning and need to be carefully calibrated. Although existing multipath mitigation methods use the spatial autocorrelation feature of the multipath in one way or another, no methods can well express and utilize the spatial autocorrelation, resulting in limited calibration performance. In this contribution, a straightforward approach via least-squares collocation (LSC) based on the covariance function, which can accurately model and utilize the spatial autocorrelation feature, is proposed to model and mitigate GNSS multipath effects. This approach does not need to separate the hemispherical surface by grid but can obtain the multipath value of any point based on the covariance function, such as Markov’s function which is homogeneous and isotropic, i.e., the function depends only on the distance between points and is rotationally symmetric. In the experiment with GPS and Galileo baseline and precise point positioning (PPP) models, the LSC approach can effectively reduce the residuals caused by multipath and has lower standard deviations and higher variance reductions than the widely used modified sidereal filter approach and grid approach. For examples, except for the advantages with 10-day residuals in Galileo data processing of the baseline MAT1_MATE, the mean double difference (DD) residual covariance reduction index of other 9 days by the LSC approach reaches 50.8% with only 5-day residuals in multipath modelling. In comparison with the value of 36.6% obtained by the grid approach with the same residuals for modelling, an improvement of 38.8% is achieved. The mean residual covariance reduction indexes of Galileo PPP of the two approaches with 2-day residuals in multipath modelling for the station MADR are 37.9% and 22.2%, respectively, and for the station REYK are 34.9% and 15.8%, respectively, where the LSC approach shows nearly doubled improvement. [ABSTRACT FROM AUTHOR]

In this article we present the nonconforming spectral element method (NSEM) for linear second-order ordinary differential equations with boundary conditions having analytic solutions, and the interface problem. A fully discrete spectral element method for one-dimensional parabolic problems with smooth solutions and parabolic interface problem also have been considered. The stability and error estimates of NSEM for a two-point boundary value problem with analytic solutions are derived. The numerical formulation is described and it is essentially a least-squares formulation and spectral elements are nonconforming. The normal equations which arise in the minimization of the functional are solved using preconditioned conjugate gradient method without storing the stiffness matrix and load vector. The method is exponentially accurate. In the fully discrete formulation for the parabolic problems, Crank–Nicolson scheme is used in time variable and higher-order spectral elements are used in spatial variable. This method is second-order accurate in time and exponential accurate in spatial variable. Various numerical examples are presented to show the accuracy of the method. Finally, we review the nonconforming spectral element method for various problems in higher dimensions. [ABSTRACT FROM AUTHOR]

Over of the last century, Principal Component Analysis (PCA) has become one of the pillars of modern scientific methods. Although PCA is typically viewed as a statistical tool aiming at finding orthogonal directions on which the variance is maximized, its first introduction by Pearson at 1901 was in the framework of the non-linear least-squares minimization problem of fitting a plane to scattered data points. Since linear least-squares regression also fits a plane to scattered data points, PCA and linear least-squares regression have thus a natural kinship, which we explore in this paper. In particular, we present an iterated linear least-squares approach, yielding a sequence of subspaces that converges to the space spanned by the leading principal components. The key observation, by which we establish our result, is that each iteration of the Power (or Subspace) Iterations, applied to the covariance matrix, can be interpreted as a solution to a linear least-squares problem. [ABSTRACT FROM AUTHOR]

Polynomials are common algebraic structures, which are often used to approximate functions, such as probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically estimated distributions. Polynomial distributions offer great modeling flexibility and mathematical tractability. However, unlike canonical distributions, polynomial functions may have non-negative values in the intervals of support for some parameter values; their parameter numbers are usually much larger than for canonical distributions, and the interval of support must be finite. Hence, polynomial distributions are defined here assuming three forms of a polynomial function. Transformations and approximations of distributions and histograms by polynomial distributions are also considered. The key properties of the polynomial distributions are derived in closed form. A piecewise polynomial distribution construction is devised to ensure that it is non-negative over the support interval. A goodness-of-fit measure is proposed to determine the best order of the approximating polynomial. Numerical examples include the estimation of parameters of the polynomial distributions and generating polynomially distributed samples. [ABSTRACT FROM AUTHOR]

When dealing with complex and redundant data classification problems, many classifiers cannot provide high predictive accuracy and interpretability. We also find that the least-squares support vector classifiers (LSSVCs) hardly identify important instances and features from data, so they cannot give an interpretable prediction. Although the LSSVC has the properties of low bias and high robustness, its high variance often gives a poor predictive performance. In this paper, we propose an alternating minimization-based sparse least-squares classifier (AMSLC) approach in the framework of LSSVCs to address the aforementioned problems. Based on the reconstructed row- and column-wise kernel matrices, the sparsity-induced ℓ 0 -norm approximation function is introduced to the LSSVC model. By alternately solving two unconstrained quadratic optimization problems or two systems of linear equations, AMSLC can predict the class labels of given instances and extract the least number of important instances and features to obtain the interpretable classification. Compared with SVC, LSSVC, ℓ 1 -norm SVC (L1SVC), ℓ 0 -norm SVC (L0SVC), the least absolute shrinkage and selection operator classifier (LASSOC), and multiple kernel learning SVC (MKLSVC) on four real credit datasets, the experimental results show that the proposed AMSLC method generally obtains the best predictive accuracy and the interpretable classification with the minimum number of important instances and features. [ABSTRACT FROM AUTHOR]

We revisit the theory of distributed networks of cooperative agents under a broader perspective of diffusion adaptation, by exploiting proximity concepts. This leads to two main families of algorithms with enhanced convergence rate and mean-square-error performance. Part I of this work considers mainly single-task scenarios, which are based on formulating optimal learning and fusion steps via an adaptive network penalty function. The main recursions, which we refer to as Adapt-and-Fuse (AAF) diffusion, are reminiscent of a reweighted network regularized algorithm, usually seen in standalone formulations. This is in line with early approaches that promote proximity among agents in cooperative networks. The AAF strategy employs exact fusion in the least-squares sense, and outperforms the exact global least-squares solution that ignores the topology of the network. It also suggests simplified LMS-complexity algorithms, and motivates us to develop a normalized version of the relative variance diffusion algorithm, which also learns combination weights. It is verified that even when agents do not share estimates, but only their uncertainties, the simplified AAF improves accuracy over the NLMS-RV algorithm in the presence of intruders, and becomes more robust to noisy links. In order to cope with the computational burden associated with long parameter vectors and correlated inputs, an overlapped block multidelay adaptive frequency-domain (FD) version of each new algorithm is derived. It turns out that for correlated inputs, these FD-LMS versions outperform the exact fullband RLS solutions. In the accompanying Part II of this work, we pursue extensions to the multitask scenario. Extensive simulations illustrate the superiority of the new approaches.

In Part I of this presentation, we have formulated single and multitask quadratic optimization problems, where agents are subject to quadratic, smoothing constraints over a graph. We have focused particularly on single task designs, whereby node uncertainties and their strength relative to un-regularized cost are tackled altogether by means of an adaptive penalty function. In this sequel, we readdress the multitask problem and propose new distributed implementations for their corresponding exact leaky-RLS solutions. We motivate a network formulation from a standalone viewpoint by capitalizing on the fact that 1) for regressors having uncorrelated entries, the performance of an efficient ${\mathcal O}(M^2)$ conjugate-gradient (CG) realization of the leaky LS solution is identical to the one of an RLS filter; 2) a CG implementation does not require inversion of the underlying sample covariance matrix. Simple arguments yield an extended network-CG algorithm that relies on node-level recursions employing distinct step-sizes. Unlike the exponentially-weighted RLS algorithm, which tapers off regularization over time, a persistent penalty strength conforms with the very purpose of the equivalent network trust-region problem, while granting a well-conditioned solution. The approach further yields another family of single-task algorithms in terms of network linearly constrained solutions, which can be contrasted to the ones proposed in Part I. In particular, the exact linearly-constrained network LMS implementation proposed here outperforms the adaptive relative-variance NLMS, under much lower computational requirements.

Modeling the relationship between vehicle speed and density on the road is a fundamental problem in traffic flow theory. Recent research found that using the least-squares (LS) method to calibrate single-regime speed-density models is biased because of the uneven distribution of samples. This paper explains the issue of the LS method from a statistical perspective: the biased calibration is caused by the correlations/dependencies in regression residuals. Based on this explanation, we propose a new calibration method for single-regime speed-density models by modeling the covariance of residuals via a zero-mean Gaussian Process (GP). Our approach can be viewed as a generalized least-squares (GLS) method with a specific covariance structure (i.e., kernel function) and is a generalization of the existing LS and the weighted least-squares (WLS) methods. Next, we use a sparse approximation to address the scalability issue of GPs and apply a Markov chain Monte Carlo (MCMC) sampling scheme to obtain the posterior distributions of the parameters for speed-density models and the hyperparameters (i.e., length scale and variance) of the GP kernel. Finally, we calibrate six well-known single-regime speed-density models with the proposed method. Results show that the proposed GP-based methods (1) significantly reduce the biases in the LS calibration, (2) achieve a similar effect as the WLS method, (3) can be used as a non-parametric speed-density model, and (4) provide a Bayesian solution to estimate posterior distributions of parameters and speed-density functions.

Implicit staggered‐grid finite‐difference methods are attractive for elastic wave modelling due to significantly enhanced spatial accuracy compared to explicit ones. However, the central‐grid finite‐difference operators used to approximate the temporal derivatives result in a limited accuracy in time. Temporal high‐order finite‐difference methods have the ability to weaken the temporal dispersion and improve the modelling stability. It is noted that the previous temporal high‐order and spatial implicit finite‐difference methods are all designed in the space domain for performing acoustic wave propagation. To implement 2‐D elastic wave simulation with high‐order accuracy both in space and time, we propose two time–space domain implicit staggered‐grid finite‐difference schemes, in which the spatial derivatives are approximated by the weighted average of a few extra off‐axial nodes and axial nodes of the conventional cross‐stencil. We derive the P‐ and S‐wave dispersion relations of the whole elastic wave equation and estimate the finite‐difference coefficients via a variable substitution‐based Taylor‐series expansion. Our Taylor‐series expansion‐based new scheme yields high‐order temporal and spatial accuracy. Besides, the spatial accuracy can be further enhanced by our newly proposed linear optimization strategy, which benefits from easy implementation since we only optimize the axial spatial coefficients via a least‐squares strategy and set the off‐axial temporal coefficients the same as the solution of the Taylor‐series expansion method. Besides, the P‐ and S‐wave separation approach is adopted to propagate the P‐ and S‐wavefields with the P‐ and S‐wave dispersion relation‐based finite‐difference operators, respectively. Our two new schemes are more capable of suppressing the numerical dispersion and exhibit better stability performance compared to conventional one, as we will illustrate via a detailed analysis of dispersion, stability and numerical experiments. In addition, a comparison of computation times demonstrates the efficiency advantage of two new schemes since small operator lengths and large time steps are allowed. [ABSTRACT FROM AUTHOR]

Parameter reconstructions are indispensable in metrology. Here, the objective is to explain K experimental measurements by fitting to them a parameterized model of the measurement process. The model parameters are regularly determined by least‐square methods, that is, by minimizing the sum of the squared residuals between the K model predictions and the K experimental observations, χ2. The model functions often involve computationally demanding numerical simulations. Bayesian optimization methods are specifically suited for minimizing expensive model functions. However, in contrast to least‐square methods such as the Levenberg–Marquardt algorithm, they only take the value of χ2 into account, and neglect the K individual model outputs. A Bayesian target‐vector optimization scheme with improved performance over previous developments, that considers all K contributions of the model function and that is specifically suited for parameter reconstruction problems which are often based on hundreds of observations is presented. Its performance is compared to established methods for an optical metrology reconstruction problem and two synthetic least‐squares problems. The proposed method outperforms established optimization methods. It also enables to determine accurate uncertainty estimates with very few observations of the actual model function by using Markov chain Monte Carlo sampling on a trained surrogate model. [ABSTRACT FROM AUTHOR]

This paper describes an accurate model of the Electrical Power Subsystem (EPS) of the Thermal Analysis Support and Environment Characterization Laboratory (TASEC-Lab), a university experiment developed for using in high-altitude balloon missions to measure and analyze the convection heat transfer. A Li-ion battery and two high efficient DC–DC converters have been characterized through laboratory tests and fitting to the experimental results. The developed mathematical models described in this paper are used to carry out accurate EPS simulations and are validated by comparison with the experimental data. • Design and modeling of the electrical power subsystem of a stratospheric balloon experiment. • New Li-lion battery discharging model that describes the three main parts: initial decay, linear, and exponential decay. • Formulation of a new DC/DC converter efficiency model. • Simulation and model validation of the electrical power subsystem. [ABSTRACT FROM AUTHOR]

The increasing complexity of stochastic models used to describe the behavior of asset returns along with the practical difficulty of defining suitable hedging strategies are relevant factors that compromise the soundness and quality of risk measurement models. In this paper we define the risk model as the mispricing a consequence of using an inadequate model to describe asset behavior and we develop a least-squares Monte Carlo methodology to estimate market and model risk simultaneously. The results show that at different confidence levels and time horizons the proposed methodology to estimate the market and model risks has a greater joint explanatory power than the isolated estimate of market risk. [ABSTRACT FROM AUTHOR]

The classical least-squares estimation (LS) has been widely used for parameter estimation in satellite precise orbit determination (POD). However, the classical LS estimation with uniform weighting is very sensitive to outliers in the observations. To overcome this issue, we use the M-estimation on the basis of the LS estimation to improve the POD solutions. To demonstrate the added value of the M-estimation, we use one year of GPS data collected by the GRACE Follow-On mission, and two sets of orbits are produced using both the LS and M-estimation. The obtained orbits are assessed through orbit formal error analysis, orbit comparison, residual analysis of satellite laser ranging (SLR), and K-band ranging (KBR) measurements. Formal error analysis shows that the precision can be improved with the M-estimation notably by 25 and 40% for the dynamic and kinematic orbit, respectively. Orbit comparison reveals that the M-estimation can offer better solution consistency between the dynamic and kinematic orbit, with the differences being reduced by 23% when compared to the LS estimation. KBR validation reports that the relative precision can be improved with the M-estimation significantly by 23 and 40% for the dynamic and kinematic orbit, respectively. Specifically, the KBR residuals are reduced from 0.65 to 0.50 mm for dynamic orbit, and from 3.24 to 1.96 mm for kinematic orbit. These results demonstrate that the M-estimation can improve the orbit precision in both absolute and relative senses. Finally, we find that the current orbit accuracy is likely dominated by systematic errors, which can obscure the contribution of the M-estimation. The SLR measurements, with precision at 5–10 mm level, also seem to be not precise enough to validate the potential contribution. As a result, the improvements as indicated by the SLR validation are shown to be insignificant. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic was impacting the health and economy around the world. All countries have taken measures to control the spread of the epidemic. Because it is not known when the epidemic will end in several countries, then the prediction of the COVID-19 pandemic is a very important challenge. This study has predicted the temporal evolution of the COVID-19 pandemic in several countries using the logistic growth model. This model has analyzed several countries to describe the epidemic situation of these countries. The time interval of the actual data used as a comparison with the prediction results of this model was starting in the firstly confirmed COVID-19 cases to December 2020. This study examined an approach to the complexity spread of the COVID-19 pandemic using the logistic growth model formed from an ordinary differential equation. This model described the time-dependent population growth rate characterized by the three parameters of the analytical solution. The non-linear least-squares method was used to estimate the three parameters. These parameters described the rate growth constant of infected cases and the total number of confirmed cases in the final phase of the epidemic. This model is applied to the spread of the COVID-19 pandemic in several countries. The prediction results show the spread dynamics of COVID-19 infected cases which are characterized by time-dependent dynamics. In this study, the proposed model provides estimates for the model parameters that are good for predicting the COVID-19 pandemic because they correspond to actual data for all analyzed countries. It is based on the coefficient of determination, R2, and the R2 value of more than 95% which is obtained from the non-linear curves for all analyzed countries. It shows that this model has the potential to contribute to better public health policy-making in the prevention of the COVID-19 pandemic.