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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]

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]

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]

Monotone single index models have gained increasing popularity over the past decades due to their flexibility and versatile use in diverse areas. Semi‐parametric estimators such as the least squares and maximum likelihood estimators of the unknown index and monotone ridge function were considered to make inference in such models without having to choose some tuning parameter. Description of the asymptotic behavior of those estimators crucially depends on acquiring a good understanding of the optimization problems associated with the corresponding population criteria. In this paper, we give several insights into these criteria by proving existence of minimizers thereof over general classes of parameters. In order to describe these minimizers, we prove different results which give the direction of variation of the population criteria in general and in the special case where the common distribution of the covariates is Gaussian. A complementary simulation study was performed and whose results give support to our main theorems. [ABSTRACT FROM AUTHOR]

If both the coefficient matrix and the observation vector are affected by noise, a total least-squares algorithm should be applied to obtain the solution. However, if they are also contaminated by outliers, the solution of the algorithm will seriously deviate from the true values. Therefore, the effect of outliers needs to be eliminated. For this purpose, the robust estimation is introduced into the total least-squares algorithm, developing a new robust weighted total least-squares algorithm. Simultaneously, considering the robustness of structure space, the standardized residuals are utilized to construct the weight function. Finally, the robustness and efficiency of this algorithm are verified by three experiments involving line-fitting and 2D affine transformation. [ABSTRACT FROM AUTHOR]

We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns. [ABSTRACT FROM AUTHOR]

Least squares migration uses the assumption that, if we have an operator that can create data from a reflectivity function, the optimal image will predict the actual recorded data with minimum square error. For this assumption to be true, it is also required that: (a) the prediction operator must be error‐free, (b) model elements not seen by the operator should be constrained by other means and (c) data weakly predicted by the operator should make limited contribution to the solution. Under these conditions, least squares migration has the advantage over simple migration of being able to remove interference between different model components. Least squares migration does that by de‐convolving or inverting the so‐called Hessian operator. The Hessian is the cascade of forward modelling and migration; for each image point, it computes the effects of interference from other image points (point‐spread function) given the actual recording geometry and the subsurface velocity model. Because the Hessian contains illumination information (along its diagonal), and information about the model cross‐correlation produced by non‐orthogonality of basis functions, its inversion produces illumination compensation and increases resolution. In addition, sampling deficiencies in the recording geometry map to the Hessian (both diagonal and non‐diagonal elements), so least squares migration has the potential to remove sampling artefacts as well. These (illumination compensation, resolution and mitigating recording deficiencies) are the three main goals of least squares migration, although the first one can be achieved by cheaper techniques. To invert the Hessian, least squares migration relies on the residual errors during iterations. Iterative algorithms, like conjugate gradient and others, use the residuals to calculate the direction and amplitudes (gradient and step size) of the necessary corrections to the reflectivity function or model. Failure of conditions (a), (b) or (c) leads the inversion to calculate incorrect model updates, which translate to noise in the final image. In this paper, we will discuss these conditions for Kirchhoff migration and reverse time migration. [ABSTRACT FROM AUTHOR]

Least squares (LS) fitting, the most widespreadly used approach for ellipses, operates by minimizing the sum of squares of some error term measured at each data point. It is not an easy task to measure the accuracy of any fitting method in fitting elliptical sections since they rely on different error criteria. There is no unique criterion valid for any fitting method for quantifying for ellipse fitting. For this reason, there is a need for a general measure that can be used to compare the accuracy of fitted ellipses using different methods. In this work, an error measure is proposed which can be used both to measure the accuracy of any ellipse fitting method and to compare the accuracy of the ellipses fitted with different (i.e. algebraic or geometric) methods. This measure is generated from the widely known orthogonal least squares fitting (OLSF) method revising the computation scheme of initial values for the orthogonal contacting points for an ellipse in the study. This is a flexible error measure since it always computes orthogonal distance residuals between data points and the optimal ellipse and can then be used to compare the performance of different ellipse fitting methods. By computing this measure it is possible to obtain the precision of the ellipse parameters with respect to the orthogonal distance residuals. This measure is applied to the measurement of the outer section of a piston and results indicate the effectiveness of the criterion. [ABSTRACT FROM AUTHOR]

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This paper considers the parameter estimation problems for a class of multivariable equation-error systems with colored noises. By using the decomposition technique, a multivariable system is transformed into several subsystems to reduce the computational burden, and a maximum likelihood-based recursive least-squares identification algorithm is developed for estimating the parameters of each subsystem. As a comparison, a multivariable recursive extended least-squares algorithm is presented. The analysis indicates that the proposed algorithm has lower computational complexity than the multivariable recursive extended least-squares algorithm, and the numerical simulation results demonstrate that the proposed method is effective. [ABSTRACT FROM AUTHOR]

The posed question arises for instance in regional gravity field modelling using weighted least-squares techniques if the gravity field functionals are synthesised from the spherical harmonic coefficients of a satellite-only global gravity model (GGM), and are used as one of the noisy datasets. The associated noise covariance matrix, appeared to be extremely ill-conditioned with a singular value spectrum that decayed gradually to zero without any noticeable gap. We analysed three methods to deal with the ill-conditioned noise covariance matrix: Tihonov regularisation of the noise covariance matrix in combination with the standard formula for the weighted least-squares estimator, a formula of the weighted least-squares estimator, which does not involve the inverse noise covariance matrix, and an estimator based on Rao's unified theory of least-squares. Our analysis was based on a numerical experiment involving a set of height anomalies synthesised from the GGM GOCO05s, which is provided with a full noise covariance matrix. We showed that the three estimators perform similar, provided that the two regularisation parameters each method knows were chosen properly. As standard regularisation parameter choice rules do not apply here, we suggested a new parameter choice rule, and demonstrated its performance. Using this rule, we found that the differences between the three least-squares estimates were within noise. For the standard formulation of the weighted least-squares estimator with regularised noise covariance matrix, this required an exceptionally strong regularisation, much larger than one expected from the condition number of the noise covariance matrix. The preferred method is the inversion-free formulation of the weighted least-squares estimator, because of its simplicity with respect to the choice of the two regularisation parameters. [ABSTRACT FROM AUTHOR]

Highlights • A novel reliability mask of a 0–1 mask is presented and generated according to the reliability of each pixel in the wrapped phase, which is calculated based on the second differences of each pixel. It is able to detect the inconsistencies of the phase map effectively. • Phase unwrapping algorithm based on reliability mask and weighted minimum least-squares method is proposed. • The proposed algorithm has the advantages of not only suppressing the propagation of errors effectively but also guaranteeing the continuity and smoothness of the unwrapped results in the noisy area. • Comparison experimental results show that the proposed algorithm can obtain a more reliable and more accurate result even if the wrapped phase map contains severe noisy or discontinuous region. Abstract A robust phase unwrapping algorithm based on reliability mask and weighted minimum least-squares method is proposed. The reliability mask is a 0–1 mask generated according to the reliability map of a wrapped phase map, which is calculated based on the second differences of each pixel. This mask not only ensures the reliability of weighting coefficients but also retains the ability of the 0–1 mask to prevent the noise values from influencing the results. The principle and realization of the proposed algorithm are described in detail. Experiments involving both simulated and actual wrapped phase images are performed to verify the feasibility and effectiveness of the proposed algorithm under various noise conditions. The results show that the proposed method can provide a more reliable and more accurate unwrapped phase, which indicates that our algorithm is more suitable for high-precision phase unwrapping of the actual wrapped phases containing various and complex noises. [ABSTRACT FROM AUTHOR]

ABSTRACT: Seismic time‐lapse surveys are susceptible to repeatability errors due to varying environmental conditions. To mitigate this problem, we propose the use of interferometric least‐squares migration to estimate the migration images for the baseline and monitor surveys. Here, a known reflector is used as the reference reflector for interferometric least‐squares migration, and the data are approximately redatumed to this reference reflector before imaging. This virtual redatuming mitigates the repeatability errors in the time‐lapse migration image. Results with synthetic and field data show that interferometric least‐squares migration can sometimes reduce or eliminate artifacts caused by non‐repeatability in time‐lapse surveys and provide a high‐resolution estimate of the time‐lapse change in the reservoir. [ABSTRACT FROM AUTHOR]

In order to reduce over-fitting, a new polynomial least-squares technique, named polynomial area least-squares (PALS), was developed. The technique is based on the minimization of the sum of the squared areas defined between the polynomial fitting function and the line segments connecting every two consecutive data points. A relatively simple system of linear equations was developed to compute the free parameters of the polynomial using the available training data. The ability of the PALS technique to generalize was evaluated through the generation of test data and statistical simulations. The PALS technique produces polynomials with reduced over-fitting, with little dependency on the order of the polynomial, especially when the polynomial order is high, so that, in some cases, increasing the order of the polynomial may even contribute to further over-fitting reduction. The major advantage of PALS is that it does not depend on any case-dependent parameter necessary for the data fitting procedure; the major limitation is that it can only be applied to polynomial regression with a single independent variable. [ABSTRACT FROM AUTHOR]

In this article, we discuss a least-squares/fictitious domain method for incompressible viscous flow around obstacles with Navier slip boundary condition. Assuming that Ω and B are two bounded sub-domains of R d , with B ‾ ⊂ Ω , in order to solve the incompressible Navier–Stokes equations with a Navier slip condition on the boundary γ of the obstacle B , we advocate a fictitious domain method where one solves a simpler variant of the original problem on the whole Ω, followed by a well-chosen correction over B . This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. A detailed discussion of the finite element implementation of the above methodology is also provided. Numerical results are given; they suggest optimal order of convergence. [ABSTRACT FROM AUTHOR]

At the ANZIAM conference in Hobart in February 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components. [ABSTRACT FROM AUTHOR]

In this paper, we propose a robust ridge regression model based on self-paced learning (RR-SPL) for the high-dimensional spectroscopic data. The proposed RR-SPL model consists of a weighted least-squares loss term on all training samples, a self-paced regularizer on sample weights, and a smoothness penalty on the model parameter. Designating an explicit form of the self-paced regularizer, the weights that indicate the importance of training samples can be automatically optimized in an augmented ridge regression framework. By increasing the model age, more and more training samples from easy to hard are added into the training set to learn a mature model. As a result, the RR-SPL model can weaken the effect of outliers and obtain an accurate spectra-concentrate relation. Experimental results on simulated data sets and four real near-infrared (NIR) spectra data sets show the effectiveness of the proposed RR-SPL method in a wide range of specific prediction tasks with or without outliers. [ABSTRACT FROM AUTHOR]

To describe the interfacial dynamics between two phases using the phase-field method, the interfacial region needs to be close enough to a sharp interface so as to reproduce the correct physics. Due to the high gradients of the solution within the interfacial region and consequent high computational cost, the use of the phase-field method has been limited to the small-scale problems whose characteristic length is similar to the interfacial thickness. By using finer mesh at the interface and coarser mesh in the rest of computational domain, the phase-field methods can handle larger scale of problems with realistic interface thicknesses. In this work, a C 1 continuous h -adaptive mesh refinement technique with the least-squares spectral element method is presented. It is applied to the Navier–Stokes-Cahn–Hilliard (NSCH) system and the isothermal Navier–Stokes–Korteweg (NSK) system. Hermite polynomials are used to give global differentiability in the approximated solution, and a space–time coupled formulation and the element-by-element technique are implemented. Two refinement strategies based on the solution gradient and the local error estimators are suggested, and they are compared in two numerical examples. [ABSTRACT FROM AUTHOR]

Least-squares migration (LSM) is applied to image subsurface structures and lithology by minimizing the objective function of the observed seismic and reverse-time migration residual data of various underground reflectivity models. LSM reduces the migration artifacts, enhances the spatial resolution of the migrated images, and yields a more accurate subsurface reflectivity distribution than that of standard migration. The introduction of regularization constraints effectively improves the stability of the least-squares offset. The commonly used regularization terms are based on the L2-norm, which smooths the migration results, e.g., by smearing the reflectivities, while providing stability. However, in exploration geophysics, reflection structures based on velocity and density are generally observed to be discontinuous in depth, illustrating sparse reflectance. To obtain a sparse migration profile, we propose the super-resolution least-squares Kirchhoff prestack depth migration by solving the L0-norm-constrained optimization problem. Additionally, we introduce a two-stage iterative soft and hard thresholding algorithm to retrieve the super-resolution reflectivity distribution. Further, the proposed algorithm is applied to complex synthetic data. Furthermore, the sensitivity of the proposed algorithm to noise and the dominant frequency of the source wavelet was evaluated. Finally, we conclude that the proposed method improves the spatial resolution and achieves impulse-like reflectivity distribution and can be applied to structural interpretations and complex subsurface imaging. [ABSTRACT FROM AUTHOR]

In this paper, we propose a maximum likelihood estimation based regression (MLER) model for multivariate calibration. The proposed MLER method seeks for the maximum likelihood estimation (MLE) solution of the least-squares problem, and it is much more robust to noise or outliers and accurate than the traditional least-squares method. An efficient iteratively reweighted least squares technique is proposed to solve the MLER model. As a result, our model can obtain accurate spectra–concentrate relations. Experimental results on three real near-infrared (NIR) spectra data sets demonstrate that the proposed MLER model is much more efficacious and effective than state-of-the-art partial least squares (PLS) methods. [ABSTRACT FROM AUTHOR]

In this paper, we deal with distributed estimation problems in diffusion networks with heterogeneous nodes, i.e., nodes that either implement different adaptive rules or differ in some other aspect such as the filter structure or length, or step size. Although such heterogeneous networks have been considered from the first works on diffusion networks, obtaining practical and robust schemes to adaptively adjust the combiners in different scenarios is still an open problem. In this paper, we study a diffusion strategy specially designed and suited to heterogeneous networks. Our approach is based on two key ingredients: 1) the adaptation and combination phases are completely decoupled, so that network nodes keep purely local estimations at all times and 2) combiners are adapted to minimize estimates of the network mean-square-error. Our scheme is compared with the standard adapt-then-combine scheme and theoretically analyzed using energy conservation arguments. Several experiments involving networks with heterogeneous nodes show that the proposed decoupled adapt-then-combine approach with adaptive combiners outperforms other state-of-the-art techniques, becoming a competitive approach in these scenarios. [ABSTRACT FROM PUBLISHER]

In full-duplex orthogonal frequency-division multiplexing systems with IQ imbalances, a frequency-domain least-squares (FD-LS) channel estimator is proposed to estimate both source-to-destination (intended) and destination-to-destination (self-interference) channels. Subsequently, an optimal closed-form pilot matrix is derived to minimize the sum of mean square errors (sum-MSE) of the proposed FD-LS channel estimator. Then, an improved FD-LS estimator is presented and proved to further improve the performance of the FD-LS by exploiting the time-domain property of the channel. In the presence of channel estimation error, an optimal low-complexity maximum likelihood (ML) detector is developed by using eigenvalue and singular value decompositions and whitening the residual self-interference plus noise. Simulation results show that given the proposed FD-LS estimator, the optimal pilot matrix performs much better than those nonsingular pilot matrices with larger conditional numbers (CN). To be specific, the former achieves about 10-dB signal-to-noise ratio (SNR) gain over the latter with CN = 10, the improved FD-LS channel estimator provides about 6-dB SNR gain over the FD-LS estimator at a fixed bit error rate (BER), and the proposed whitening-filter ML detector performs at least 0.6 dB better than the conventional ML detector at a given BER of  10^-3 in the medium-and high-SNR regions. [ABSTRACT FROM AUTHOR]

We analyze, both experimentally and numerically, two EVA foam specimens with densities ρ = 120 kg / m 3 and ρ = 220 kg / m 3 . In the numerical analysis, we use our recent finite strain element formulations based on least-squares strains (in both solid-shells and full three-dimensional discretizations). The Ogden–Hill hyperelastic model is coupled with a one-term standard solid containing a Maxwell element. Compression experiments are performed to evaluate the Ogden–Hill properties for the two EVA foams. Viscous properties are obtained from interpolation of creep results reported by R. Verdejo. An exponential integration of the internal variables (which have the role of back-stresses) is proposed along with the complete description of the constitutive system. The classical analysis of the acoustic tensor is also performed for the measured properties. Four benchmark examples are introduced. [ABSTRACT FROM AUTHOR]

A new immersed boundary interpolation for discrete forcing methods is presented. The method decreases spurious oscillations in the pressure field and consequently in the body force calculations which are a common issue in several immersed boundary methods. The method applies a continuity constraint in the least-squares interpolation, and guarantees that adjacent interpolation polynomials are continuous between each other. This approach strictly enforces continuity in the flow reconstruction domain, reducing time discontinuities caused by the boundary conditions applied at the immersed boundary. Several moving body problems test cases are simulated, including a three dimensional one, to demonstrate the new method's capability of computing stable pressure fields, even for coarse grids and small CFL numbers, which are known to increase the pressure oscillations. [ABSTRACT FROM AUTHOR]

The downlink communication channel from high-altitude platform (HAP) to high-speed train (HST) in the Ka-band is a slowly time-varying Rician distributed flat fading channel with 10-25 dB Rician K factor. In this respect, the received signal is mainly affected by the Doppler shift of the line-of-sight (LOS) link. In order to increase receiver performance, we propose to firstly compensate the Doppler shift of the received signal before least-squares (LS) adaptive filtering is pursued. Implementing the proposed method requires a priori knowledge of the time-varying phase of the LOS component. This is justified since signalling between the train and the controller exists such that the train velocity and location are predictable. Implementing the proposed method to the recursive LS (RLS) received beamforming algorithm shows reduction of mean square error (MSE) and bit error rate (BER). [ABSTRACT FROM AUTHOR]

In this paper we address the identification of linear time-invariant single-input multi-output (SIMO) systems. In particular, we assess the performance of the prediction error method by quantifying the variance of the parameter estimates. Using an orthonormal representation for the modules composing the SIMO structure, we show that the parameter estimate of a module depends on the model structure of the other modules, and on the correlation structure of the output disturbances. We provide novel results which quantify the variance-error of the parameter estimates for finite model orders, where the effects of noise correlation structure, model structure and input spectrum are visible. In particular, we show that a sensor does not increase the accuracy of a module if common dynamics have to be estimated. When a module is identified using less parameters than the other modules, we derive the noise correlation structure that gives the minimum total variance. The implications of our results are illustrated through numerical examples and simulations. [ABSTRACT FROM AUTHOR]

The numerical information in the calibration reports for indicating instruments is typically sparse, often comprising a simple table of corrections or indicated values against a small number of reference values. Users are left to interpolate between the tabulated values using one of several well-known interpolation algorithms, including straight-line, spline, Lagrange, and least-squares interpolation. Although these algorithms are well known, there has apparently been no comparison of their performance in respect of uncertainty propagation. This paper provides an overview of the advantages and disadvantages of the most common interpolation algorithms with respect to uncertainty propagation, immunity to interpolation error, and sensitivity to data spacing. Secondly, the paper illustrates an unconventional method for the uncertainty analysis. The method exploits the linear dependence of the interpolations on measurements of the interpolated quantity, and is easily applied to any linear functional interpolation. In many respects, the best all-round interpolation scheme is a polynomial fitted by least-squares methods, which has a low propagated uncertainty, continuity to the chosen order of the fitted polynomial, and a good immunity to large gaps in the data. [ABSTRACT FROM AUTHOR]

The least-squares criterion is widely used in the multivariate calibration models. Rather than using the conventional linear least-squares metric, we employ a nonlinear correntropy-based metric to describe the spectra-concentrate relations and propose a maximum correntropy criterion based regression (MCCR) model. To solve the correntropy-based model, a half-quadratic optimization technique is developed to convert a non-convex and nonlinear optimization problem into an iteratively re-weighted least-squares problem. Finally, MCCR can provide an accurate estimation of the regression relation by alternatively updating an auxiliary vector represented as a nonlinear Gaussian function of fitted residuals and a weight computed by a regularized weighted least-squares model. The proposed method is compared to some modified PLS algorithms and robust regression methods on four real near-infrared (NIR) spectra data sets. Experimental results demonstrate the efficacy and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We present a thickness-extensible finite strain quadrilateral element based on least-squares in-plane shear strains and assumed transverse-shear strains. At each node, two thickness parameters are connected to the constitutive laws by a linear system. The zero out-of-plane normal stress condition is satisfied at the constitutive level using the normal strain as unknown in all integration points. Assumed in-plane strains based on least-squares are introduced as an alternative to the enhanced-assumed-strain (EAS) formulations and, contrasting with these, the result is an element satisfying ab-initio both the in-plane and the transverse Patch tests. There are no additional degrees-of-freedom, as it is the case with EAS, even by means of static condensation. Least-squares fit allows the derivation of invariant finite strain elements which are shear-locking free and amenable to be incorporated in commercial codes. With that goal, we use automatically generated code produced by AceGen and Mathematica. Full assessment of the element formulation and the two-parameter thickness variation methodology is accomplished. Alternative thickness variation algorithms are tested. All benchmarks show very competitive results, similar to the best available enriched shell elements. [ABSTRACT FROM AUTHOR]

We present an iterative algorithm, called the symmetric tensor eigen-rank-one iterative decomposition (STEROID), for decomposing a symmetric tensor into a real linear combination of symmetric rank-1 unit-norm outer factors using only eigendecompositions and least-squares fitting. Originally designed for a symmetric tensor with an order being a power of two, STEROID is shown to be applicable to any order through an innovative tensor embedding technique. Numerical examples demonstrate the high efficiency and accuracy of the proposed scheme even for large scale problems. Furthermore, we show how STEROID readily solves a problem in nonlinear block-structured system identification and nonlinear state-space identification. [ABSTRACT FROM AUTHOR]

Brain tumor is one of the leading causes of death making tumor detection very important and challenging in the medical field. This paper describes tumor detection in medical images using immune feature weighted least squares-support vector machine (IFWLS-SVM). The challenge in brain tumor detection in magnetic resonance (MR) images is the existence of non-linearity in real data. Least squares-support vector machine (LS-SVM) is a conventional algorithm that has been applied to diagnose the detection problems in MR images and non-linear distribution in brain tumors. LS-SVM solves a linear system for a training algorithm instead of using quadratic programming in SVM. In conventional LS-SVM, each sample feature taken has equal importance for classification results, which does not give accurate results in real applications. In addition, parameters of LS-SVM and their kernel function prominently affect the classification result. An IFWLS-SVM has been used to optimize the kernel and tune the parameters of LS-SVM in this paper. Theoretical analysis and experimental results showed that IFWLS-SVM has better performance than other conventional algorithms. [ABSTRACT FROM PUBLISHER]

A finite-strain solid–shell element is proposed. It is based on least-squares in-plane assumed strains, assumed natural transverse shear and normal strains. The singular value decomposition (SVD) is used to define local (integration-point) orthogonal frames-of-reference solely from the Jacobian matrix. The complete finite-strain formulation is derived and tested. Assumed strains obtained from least-squares fitting are an alternative to the enhanced-assumed-strain (EAS) formulations and, in contrast with these, the result is an element satisfying the Patch test. There are no additional degrees-of-freedom, as it is the case with the enhanced-assumed-strain case, even by means of static condensation. Least-squares fitting produces invariant finite strain elements which are shear-locking free and amenable to be incorporated in large-scale codes. With that goal, we use automatically generated code produced by AceGen and Mathematica. All benchmarks show excellent results, similar to the best available shell and hybrid solid elements with significantly lower computational cost. [ABSTRACT FROM AUTHOR]

Combining subspace correction method with least-squares finite element procedure, we construct a new overlapping domain decomposition parallel algorithm for solving the first-order time-dependent convection-diffusion system. This algorithm is fully parallel. We analyze the convergence of approximate solution, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration number and sub-domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to given accuracy at each time step. [ABSTRACT FROM AUTHOR]

This article compares the mean-squared error (or ℓ2risk) of ordinary least squares (OLS), James–Stein, and least absolute shrinkage and selection operator (Lasso) shrinkage estimators in simple linear regression where the number of regressors is smaller than the sample size. We compare and contrast the known risk bounds for these estimators, which shows that neither James–Stein nor Lasso uniformly dominates the other. We investigate the finite sample risk using a simple simulation experiment. We find that the risk of Lasso estimation is particularly sensitive to coefficient parameterization, and for a significant portion of the parameter space Lasso has higher mean-squared error than OLS. This investigation suggests that there are potential pitfalls arising with Lasso estimation, and simulation studies need to be more attentive to careful exploration of the parameter space. [ABSTRACT FROM AUTHOR]

In linear least squares fitting of B-spline surfaces, the choice of knot vector is essentially important to the quality of the approximating surface. In this paper, a heuristic criterion for optimal knot positions in the fitting problem is formulated as an optimization problem according to the geometric feature distribution of the input data. Then, the coordinate descent algorithm is used for the optimal knot computation. Based on knot position optimization, an iterative surface fitting framework is developed, which adaptively introduces more knot isolines passing through the regions with more complex geometry or large fitting errors. Hence, the approximation quality of the reconstructed surface is progressively improved up to a pre-specified threshold. We test several models to demonstrate the efficacy of our method in fitting surface with distinct geometric features. Different from the knot placement technique (NKTP method) proposed in Piegl and Tiller [1] and the dominant-column-based fitting method (DOM-based method) (Park [2]) which require input data in semi-grid or grid form, our algorithm takes more general data points as input, i.e., any scattered data sets with parameterization. Comparing to NKTP method and DOM-based method, our method efficiently produces more accurate results by using the same number of knots. [ABSTRACT FROM AUTHOR]

The multipath effect is one of the main error sources in the Global Satellite Navigation Systems (GNSSs). The code correlation reference waveform (CCRW) technique is an effective multipath mitigation algorithm for the binary phase shift keying (BPSK) signal. However, it encounters the false lock problem in code tracking, when applied to the binary offset carrier (BOC) signals. A least-squares approximation method of the CCRWdesign scheme is proposed, utilizing the truncated singular value decomposition method. This algorithm was performed for the BPSK signal, BOC(1,1) signal, BOC(2,1) signal, BOC(6,1) and BOC(7,1) signal. The approximation results of CCRWs were presented. Furthermore, the performances of the approximation results are analyzed in terms of the multipath error envelope and the tracking jitter. The results show that the proposed method can realize coherent and non-coherent CCRW discriminators without false lock points. Generally, there is performance degradation in the tracking jitter, if compared to the CCRW discriminator. However, the performance promotions in the multipath error envelope for the BOC(1,1) and BPSK signals makes the discriminator attractive, and it can be applied to high-order BOC signals. [ABSTRACT FROM AUTHOR]

Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3] ), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7] ). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximated by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. We also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties. [ABSTRACT FROM AUTHOR]

We present local energy type error estimates for first-order system div least-squares (LS) finite element methods. The estimate shows that the local energy norm error is bounded by the local best approximation and weaker norms which account for the pollution. The estimate is similar to the one for the standard Galerkin methods. However, our estimate needs to consider the effect of error of dual (flux) variables since LS methods approximate the primary and dual variables simultaneously. The effect of error of the dual variables is shown to be of higher order. Moreover, our estimate shows the convergence behavior when locally enriched approximation spaces are used in the area of interest. As an elementary consequence of the estimate, asymptotically exact a posteriori error estimator is constructed for the local area of interest under mild assumptions. [ABSTRACT FROM AUTHOR]

Least-squares (LS) solutions to the 3D Helmert and symmetric Helmert transformations with rotational invariant covariance structure are studied in a unified framework. This is an extension of the 3D Helmert transformation with naïve identity covariance and the counterpart of the 2D symmetric Helmert transformation with rotational invariant covariance. It is found that the closed form LS solution still exists and that the rotation parameters are still the same between the Helmert and symmetric Helmert transformations. [ABSTRACT FROM AUTHOR]

In this paper, we present the weighted least-squares finite element approximations to the steady Phan-Thien–Tanner (PTT) viscoelastic fluid. We approximate the nonlinear viscoelastic model by linearizing the constitutive equation and apply a least-squares method to solve the linearized system. The least-squares functional involves the residuals of each equation in L 2 -norm multiplied by mesh dependent weights. The method is tested on a planar channel flow problem. Numerical results exhibit the convergence rates of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, a family of stencil selection algorithms is presented for WENO schemes on unstructured meshes. The associated freedom of stencil selection for unstructured meshes, in the context of WENO schemes present a plethora of various stencil selection algorithms. The particular focus of this paper is to assess the performance of various stencil selection algorithm, investigate the parameters that dictate their robustness, accuracy and computational efficiency. Ultimately, efficient and robust stencils are pursued that can provide significant savings in computational performance, while retaining the non-oscillatory character of WENO schemes. This is achieved when making the stencil selection algorithms adaptive, based on the quality of the cells for unstructured meshes, that can in turn reduce the computational cost of WENO schemes. For assessing the performance of the developed algorithms well established test problems are employed. These include the least square approximation of polynomial functions, linear advection equation of smooth functions and solid body rotation test problem. Euler and Navier-Stokes equations test problems are also pursued such as the Shu-Osher test problem, the Double Mach Reflection, the supersonic Forward Facing step, the Kelvin-Helmholtz instability, the Taylor-Green Vortex, and the flow past a transonic circular cylinder. • Various central and directional stencils algorithms presented. • Uniqueness, Cover, Width and Compactness the most dominant parameters for directional stencils. • Algorithms suitability is dependent on the numerical framework. • Adaptive stencils reduce the computational cost of the WENO schemes. [ABSTRACT FROM AUTHOR]

Phylogenies provide a useful way to understand the evolutionary history of genetic samples, and data sets with more than a thousand taxa are becoming increasingly common, notably with viruses (e.g., human immunodeficiency virus (HIV)). Dating ancestral events is one of the first, essential goals with such data. However, current sophisticated probabilistic approaches struggle to handle data sets of this size. Here,we present very fast dating algorithms, based on a Gaussian model closely related to the Langley-Fitch molecular-clock model.We show that this model is robust to uncorrelated violations of the molecular clock. Our algorithms apply to serial data, where the tips of the tree have been sampled through times. They estimate the substitution rate and the dates of all ancestral nodes. When the input tree is unrooted, they can provide an estimate for the root position, thus representing a new, practical alternative to the standard rooting methods (e.g., midpoint). Our algorithms exploit the tree (recursive) structure of the problem at hand, and the close relationships between least-squares and linear algebra. We distinguish between an unconstrained setting and the case where the temporal precedence constraint (i.e., an ancestral node must be older that its daughter nodes) is accounted for. With rooted trees, the former is solved using linear algebra in linear computing time (i.e., proportional to the number of taxa), while the resolution of the latter, constrained setting, is based on an active-set method that runs in nearly linear time. With unrooted trees the computing time becomes (nearly) quadratic (i.e., proportional to the square of the number of taxa). In all cases, very large input trees (>10,000 taxa) can easily be processed and transformed into time-scaled trees.We compare these algorithms to standard methods (root-to-tip, r8s version of Langley-Fitch method, and BEAST). Using simulated data, we show that their estimation accuracy is similar to that of the most sophisticated methods, while their computing time is much faster.We apply these algorithms on a large data set comprising 1194 strains of Influenza virus from the pdm09 H1N1 Human pandemic. Again the results show that these algorithms provide a very fast alternative with results similar to those of other computer programs. These algorithms are implemented in the LSD software (least-squares dating), which can be downloaded from http://www.atgc-montpellier.fr/LSD/, along with all our data sets and detailed results. An Online Appendix, providing additional algorithm descriptions, tables, and figures can be found in the Supplementary Material available on Dryad at http://dx.doi.org/10.5061/dryad.968t3. [ABSTRACT FROM AUTHOR]

An ordinary least squares regression estimate for the slope, regardless of its strength, can have its sign reversed through adjustment for a random confounding vector of data. The assumption of a rotationally invariant distribution, on the space of centered, random, confounding vectors of data, makes calculation of probabilities for these reversals possible. Here, as the sample size increases, these probabilities are shown to decrease exponentially. This analytic result leads to some asymptotic comparison between regular sampling error and the error due to a mis-specified model. [ABSTRACT FROM AUTHOR]

Graphical ANOVA is a simple and effective tool for visualizing evidence of differences between treatment means for data coming from factorial experiments. The purpose of the present article is to propose an analogous method for the visualization of the significance of regression, using the QR decomposition. Two graphical tests are proposed and compared with the classical $$F$$ test, by simulation. It is found that when the number of candidate predictors is small relative to the sample size, the classical test has slightly higher power than the graphical tests. When the number of predictors is large, the graphical tests remain powerful. while the classical $$F$$ test exhibits poor power properties. [ABSTRACT FROM AUTHOR]