Algorithms and statistical analysis for linear structured weighted total least squares problem

Weighted total least squares(WTLS)have been regarded as the standard tool for the errors-in-variables(EIV)model in which all the elements in the observation vector and the coefficient matrix are contaminated with random errors.However,in many geodetic applications,some elements are error-free and some random observations appear repeatedly in different positions in the augmented coefficient matrix.It is called the linear structured EIV(LSEIV)model.Two kinds of methods are proposed for the LSEIV model from functional and stochastic modifications.On the one hand,the functional part of the LSEIV model is modified into the errors-in-observations(EIO)model.On the other hand,the stochastic model is modified by applying the Moore-Penrose inverse of the cofactor matrix.The algorithms are derived through the Lagrange multipliers method and linear approximation.The estimation principles and iterative formula of the parameters are proven to be consistent.The first-order approximate variance-covariance matrix(VCM)of the parameters is also derived.A numerical example is given to compare the performances of our proposed three algorithms with the STLS approach.Afterwards,the least squares(LS),total least squares(TLS)and linear structured weighted total least squares(LSWTLS)solutions are compared and the accuracy evaluation formula is proven to be feasible and effective.Finally,the LSWTLS is applied to the field of deformation analysis,which yields a better result than the traditional LS and TLS estimations.

Numerical simulation of inlet placement on sewage characteristics in the rounded square aquaculture tank with single inlet

To improve the self-cleaning ability of aquaculture tank and the efficiency of circulating water,physical and numerical experiments were conducted on the influence of inlet structure on sewage discharge in a rounded square aquaculture tank with a single inlet.Based on the physical model of the tank,analysis of how inlet structure adjustment affects sewage discharge efficiency and flow field characteristics was conducted to provide suitable flow field conditions for sinkable solid particle discharge.In addition,an internal flow field simulation was conducted using the RNG k-εturbulence model in hydraulic drive mode.Then a solid-fluid multiphase model was created to investigate how the inlet structure affects sewage collection in the rounded square aquaculture tank with single inlet and outlet.The finding revealed that the impact of inlet structure is considerably affecting sewage collection.The conditions of C/B=0.07-0.11(the ratio of horizontal distance between the center of the inlet pipe and the tank wall(C)to length of the tank(B))andα=25°(αis the angle between the direction of the jet and the tangential direction of the arc angle)resulted in optimal sewage collection,which is similar to the flow field experiment in the rounded square aquaculture tank with single inlet and outlet.An excellent correlation was revealed between sewage collection and fluid circulation stability in the aquaculture tank.The present study provided a reference for design and optimization of circulating aquaculture tanks in aquaculture industry.

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

Least Squares One-Class Support Tensor Machine

One-class classification problem has become a popular problem in many fields, with a wide range of applications in anomaly detection, fault diagnosis, and face recognition. We investigate the one-class classification problem for second-order tensor data. Traditional vector-based one-class classification methods such as one-class support vector machine (OCSVM) and least squares one-class support vector machine (LSOCSVM) have limitations when tensor is used as input data, so we propose a new tensor one-class classification method, LSOCSTM, which directly uses tensor as input data. On one hand, using tensor as input data not only enables to classify tensor data, but also for vector data, classifying it after high dimensionalizing it into tensor still improves the classification accuracy and overcomes the over-fitting problem. On the other hand, different from one-class support tensor machine (OCSTM), we use squared loss instead of the original loss function so that we solve a series of linear equations instead of quadratic programming problems. Therefore, we use the distance to the hyperplane as a metric for classification, and the proposed method is more accurate and faster compared to existing methods. The experimental results show the high efficiency of the proposed method compared with several state-of-the-art methods.

Numerical Simulation of Oil-Water Two-Phase Flow in Low Permeability Tight Reservoirs Based on Weighted Least Squares Meshless Method

In response to the complex characteristics of actual low-permeability tight reservoirs,this study develops a meshless-based numerical simulation method for oil-water two-phase flow in these reservoirs,considering complex boundary shapes.Utilizing radial basis function point interpolation,the method approximates shape functions for unknown functions within the nodal influence domain.The shape functions constructed by the aforementioned meshless interpolation method haveδ-function properties,which facilitate the handling of essential aspects like the controlled bottom-hole flow pressure in horizontal wells.Moreover,the meshless method offers greater flexibility and freedom compared to grid cell discretization,making it simpler to discretize complex geometries.A variational principle for the flow control equation group is introduced using a weighted least squares meshless method,and the pressure distribution is solved implicitly.Example results demonstrate that the computational outcomes of the meshless point cloud model,which has a relatively small degree of freedom,are in close agreement with those of the Discrete Fracture Model(DFM)employing refined grid partitioning,with pressure calculation accuracy exceeding 98.2%.Compared to high-resolution grid-based computational methods,the meshless method can achieve a better balance between computational efficiency and accuracy.Additionally,the impact of fracture half-length on the productivity of horizontal wells is discussed.The results indicate that increasing the fracture half-length is an effective strategy for enhancing production from the perspective of cumulative oil production.

Comparative Study of Probabilistic and Least-Squares Methods for Developing Predictive Models

This article explores the comparison between the probability method and the least squares method in the design of linear predictive models. It points out that these two approaches have distinct theoretical foundations and can lead to varied or similar results in terms of precision and performance under certain assumptions. The article underlines the importance of comparing these two approaches to choose the one best suited to the context, available data and modeling objectives.

Least Square Finite Element Model for Analysis of Multilayered Composite Plates under Arbitrary Boundary Conditions

Laminated composites are widely used in many engineering industries such as aircraft, spacecraft, boat hulls, racing car bodies, and storage tanks. We analyze the 3D deformations of a multilayered, linear elastic, anisotropic rectangular plate subjected to arbitrary boundary conditions on one edge and simply supported on other edge. The rectangular laminate consists of anisotropic and homogeneous laminae of arbitrary thicknesses. This study presents the elastic analysis of laminated composite plates subjected to sinusoidal mechanical loading under arbitrary boundary conditions. Least square finite element solutions for displacements and stresses are investigated using a mathematical model, called a state-space model, which allows us to simultaneously solve for these field variables in the composite structure’s domain and ensure that continuity conditions are satisfied at layer interfaces. The governing equations are derived from this model using a numerical technique called the least-squares finite element method (LSFEM). These LSFEMs seek to minimize the squares of the governing equations and the associated side conditions residuals over the computational domain. The model is comprised of layerwise variables such as displacements, out-of-plane stresses, and in- plane strains, treated as independent variables. Numerical results are presented to demonstrate the response of the laminated composite plates under various arbitrary boundary conditions using LSFEM and compared with the 3D elasticity solution available in the literature.

PSR-SQUARES:基于程序空间约简器的SQL逆向合成系统

针对SQUARES程序空间增长过快,导致程序合成效率偏低的问题,在SQUARES的基础上,增加了以深度神经网络为核心的程序空间约简器,将给定的<被查询表,查询结果>示例表示成二维张量,作为深度神经网络的输入,网络的输出是关于目标SQL语句合成规则的相关性标记向量。约简器根据神经网络的输出结果,采用末N位淘汰策略,删除与目标SQL语句相关性弱的合成规则,以减少候选SQL语句的生成和验证,提升系统合成效率。对约简器中深度神经网络的结构设计、训练样本集的生成方法和网络训练过程进行了详细描述。同时将PSR-SQUARES与当前有代表性SQL逆向合成系统进行实验对比,实验结果表明,PSR-SQUARES的综合性能不同程度地优于其他合成系统,平均合成时间由SQUARES的251 s降低至130 s,目标程序合成成功率由80%提升至89%。

Factor-square-full半模

作为factor-square-full模的真推广,引入了factor-square-full半模的概念.设S是半环,M是左S-半模.称M的真subtractive子半模X是d-square,若存在正规满同态f:M→(M/X)2;称M的真subtractive子半模X是quasi-d-square,若存在M的d-square子半模Y使得X≤Y;称M是factor-square-full,若M的任意真subtractive子半模是quasi-d-square.讨论了这些子半模类及factor-square-full半模的性质.

Experimental Study of Submergence Ratio on Local Scour Around a Square Pile in Steady Flow

Scour around a submerged square pile was realized experimentally in a steady flow to study the effects of flow depth on local scour.Flow depth to pile height ratios ranging from 1.5 to 5 in uniform sand and 2 to 5 in non-uniform sand were tested in the approaching flow velocity to critical velocity(larger than which the sediment particle is motivated)ratios of 0.56 and 1.03,respectively.The influences of flow depth were investigated on the basis of analysis of the three-dimensional topography,temporal maximum scour depth,bed profile development,and equilibrium scour depth.Results showed that the maximum scour depth was at the upstream corners of the pile other than at the stagnation point.The evolutions of the maximum scour depth data in non-uniform sand were well fitted with a recent exponential function,which characterized the initial,developing,and equilibrium stages of scour depth.The scour hole slopes upstream of the pile were found to be parallel to each other in the process of each test and were mainly governed by the sediment repose underwater.The equilibrium scour depth varied slightly with flow depth when the submergence ratio was larger than 1 in uniform sand while it was 2 in non-uniform sand.The armoring effects of coarse sediment particles markedly reduced the sediment transport in non-uniform sand despite the 0.34 increment in non-uniformity.

Robust least squares projection twin SVM and its sparse solution

Least squares projection twin support vector machine(LSPTSVM)has faster computing speed than classical least squares support vector machine(LSSVM).However,LSPTSVM is sensitive to outliers and its solution lacks sparsity.Therefore,it is difficult for LSPTSVM to process large-scale datasets with outliers.In this paper,we propose a robust LSPTSVM model(called R-LSPTSVM)by applying truncated least squares loss function.The robustness of R-LSPTSVM is proved from a weighted perspective.Furthermore,we obtain the sparse solution of R-LSPTSVM by using the pivoting Cholesky factorization method in primal space.Finally,the sparse R-LSPTSVM algorithm(SR-LSPTSVM)is proposed.Experimental results show that SR-LSPTSVM is insensitive to outliers and can deal with large-scale datasets fastly.

Revisiting Akaike’s Final Prediction Error and the Generalized Cross Validation Criteria in Regression from the Same Perspective: From Least Squares to Ridge Regression and Smoothing Splines

In regression, despite being both aimed at estimating the Mean Squared Prediction Error (MSPE), Akaike’s Final Prediction Error (FPE) and the Generalized Cross Validation (GCV) selection criteria are usually derived from two quite different perspectives. Here, settling on the most commonly accepted definition of the MSPE as the expectation of the squared prediction error loss, we provide theoretical expressions for it, valid for any linear model (LM) fitter, be it under random or non random designs. Specializing these MSPE expressions for each of them, we are able to derive closed formulas of the MSPE for some of the most popular LM fitters: Ordinary Least Squares (OLS), with or without a full column rank design matrix;Ordinary and Generalized Ridge regression, the latter embedding smoothing splines fitting. For each of these LM fitters, we then deduce a computable estimate of the MSPE which turns out to coincide with Akaike’s FPE. Using a slight variation, we similarly get a class of MSPE estimates coinciding with the classical GCV formula for those same LM fitters.

Two-Stage Procrustes Rotation with Sparse Target Matrix and Least Squares Criterion with Regularization and Generalized Weighting

In factor analysis, a factor loading matrix is often rotated to a simple target matrix for its simplicity. For the purpose, Procrustes rotation minimizes the discrepancy between the target and rotated loadings using two types of approximation: 1) approximate the zeros in the target by the non-zeros in the loadings, and 2) approximate the non-zeros in the target by the non-zeros in the loadings. The central issue of Procrustes rotation considered in the article is that it equally treats the two types of approximation, while the former is more important for simplifying the loading matrix. Furthermore, a well-known issue of Simplimax is the computational inefficiency in estimating the sparse target matrix, which yields a considerable number of local minima. The research proposes a new rotation procedure that consists of the following two stages. The first stage estimates sparse target matrix with lesser computational cost by regularization technique. In the second stage, a loading matrix is rotated to the target, emphasizing on the approximation of non-zeros to zeros in the target by least squares criterion with generalized weighing that is newly proposed by the study. The simulation study and real data examples revealed that the proposed method surely simplifies loading matrices.

Numerical Study of Natural Convective Heat Transfer in an Air Filled Square Cavity Heated from Below and Symmetrically Cooled from the Sides with a Partition in the Hot Wall

A two-dimensional numerical study of laminar natural convection in a square enclosure filled with air with a wall partially heated on the bottom is presented.The heat source is located on the lower wall with different heated widths varied from 20 to 80%(ε=0.2–0.8)of the total width of the lower wall and different heights h=H/4 and H/2 of the partition.The effect of the partition height on the main system dynamics is investigated through solution of the two-dimensional Navier-Stokes equations and the energy equation by means of a finite volume method based on the SIMPLE algorithm.The influence of the Rayleigh number(Ra=10^(3) to 10^(6))and the hot wall length is also examined.It is shown that the average Nusselt number grows whenεincreases and when h decreases.For a given value ofεand h,the average Nusselt number increases as Ra increases.It is concluded that the partition height causes a decrease in the average Nusselt number.

Least square method based on Haar wavelet to solve multi-dimensional stochastic Ito-Volterra integral equations

This paper proposes a method combining blue the Haar wavelet and the least square to solve the multi-dimensional stochastic Ito-Volterra integral equation.This approach is to transform stochastic integral equations into a system of algebraic equations.Meanwhile,the error analysis is proven.Finally,the effectiveness of the approach is verified by two numerical examples.

Modeling of Mixed Convection in a Lid Driven Wavy Enclosure with Two Square Blocks Placed at Different Positions

The purpose of this paper is to investigate the simulation of mixed convection in a lid-driven wavy enclosure with blocks positioned at various positions. This study also examined the impact of the longitudinal position of the heated block on heat transfer enhancement. The Galerkin weighted residual finite element method is employed to computationally solve the governing equations of Navier-Stokes, thermal energy, and mass conservation. The enclosure consists of two square heated blocks strategically placed at different heights—firstly, one set is closer to the bottom surface;secondly, one set is nearer to the middle area and finally, one set is closer to the upper undulating surface of the enclosure. The wavy top wall’s thermal insulation, along with active heating of the bottom wall and blocks, generates a dynamic convective atmosphere. In addition, the left wall ascends as the right wall falls, causing the flow formed by the lid. The study investigates the impact of the Richardson number on many factors, such as streamlines, isotherms, dimensionless temperature, velocity profiles, and average Nusselt numbers. These impacts are depicted through graphical illustrations. In all instances, two counter-rotating eddies were generated within the cage. Higher rotating speed consistently leads to improved performance, irrespective of other characteristics. Furthermore, an ideal amalgamation of the regulating factors would lead to increased heat transmission.

Secondary steady-state and time-periodic flows from a basic flow with square array of odd number of vortices

In a magnetohydrodynamic(MHD)driven fluid cell,a plane non-parallel flow in a square domain satisfying a free-slip boundary condition is examined.The energy dissipation of the flow is controlled by the viscosity and linear friction.The latter arises from the influence of the Hartmann bottom boundary layer in a three-dimensional(3D)MHD experiment in a square bottomed cell.The basic flow in this fluid system is a square eddy flow exhibiting a network of N~2 vortices rotating alternately in clockwise and anticlockwise directions.When N is odd,the instability of the flow gives rise to secondary steady-state flows and secondary time-periodic flows,exhibiting similar characteristics to those observed when N=3.For this reason,this study focuses on the instability of the square eddy flow of nine vortices.It is shown that there exist eight bi-critical values corresponding to the existence of eight neutral eigenfunction spaces.Especially,there exist non-real neutral eigenfunctions,which produce secondary time-periodic flows exhibiting vortices merging in an oscillatory manner.This Hopf bifurcation phenomenon has not been observed in earlier investigations.

Comparative Analysis of Equal and Unequal Grounding Grid Configurations by Compression Ratio and Least Square Curve Fitting Techniques

The primary aim of the power system grounding is to safeguard the person and satisfying the performance of the power systemtomaintain reliable operation.With equal conductor spacing grounding grid design,the distribution of the current in the grid is not uniform.Hence,unequal grid conductor span in which grid conductors are concentrated more at the periphery is safer to practice than equal spacing.This paper presents the comparative analysis of two novel techniques that create unequal spacing among the grid conductors:the least-square curve fitting technique and the compression ratio techniquewith equal grid configuration for both square and rectangular grids.Particle Swarm Optimization(PSO)is adopted for finding out one optimal feasible solution among many feasible solutions of equal grid configuration for both square and rectangular grids.Comparative analysis is also carried out between square and rectangular grids using the least square curve fitting technique as it results in only one unequal grid configuration.Simulation results are obtained by theMATLAB software developed.Percentage of improvement in ground potential rise,step voltage,touch voltage,and grid resistancewith variation in compression ratios are plotted.

Multi-physical fields distribution in billet during helical electromagnetic stirring:A numerical simulation research

Electromagnetic stirring is one of the widely applied techniques to modify the quality of casting billets.Different from conventional rotate stirring,the helical stirring is more professional in assisting multi-dimensional flow of molten metal and eliminating solidification defects.In this study,the single-winding helical stirring(SWHS)was introduced,offering advantages such as smaller volume and lower electromagnetic shielding compared to traditional helical stirring methods.Following a comprehensive numerical simulation,the stirring parameters of SWHS were adjusted to yoke inclination angle of 43°and frequency of 12 Hz.The higher electromagnetic force and flow velocity in drawing direction,as well as the lower temperature gradient induced by the SWHS,are positive factors for homogeneous solidification of billet.The experimental results on Al-8%Si alloy and 0.4%C-1.1%Mn steel demonstrate that compared to rotate stirring,the SWHS process can induce better billet quality and is more effective in accelerating the equiaxed expansion and reducing element segregation.The SWHS process can enhance the equiaxed ratio of the billet by 58.3%and reduce segregation degree of carbon element by 10.97%.Consequently,SWHS holds great promise as a potential approach for improving the quality of continuous casting billets.

A high resolution inversion method for fluid factor with dynamic dryrock V_(P)/V_(S) ratio squared

As an important indicator parameter of fluid identification,fluid factor has always been a concern for scholars.However,when predicting Russell fluid factor or effective pore-fluid bulk modulus,it is necessary to introduce a new rock skeleton parameter which is the dry-rock VP/VS ratio squared(DVRS).In the process of fluid factor calculation or inversion,the existing methods take this parameter as a static constant,which has been estimated in advance,and then apply it to the fluid factor calculation and inversion.The fluid identification analysis based on a portion of the Marmousi 2 model and numerical forward modeling test show that,taking the DVRS as a static constant will limit the identification ability of fluid factor and reduce the inversion accuracy.To solve the above problems,we proposed a new method to regard the DVRS as a dynamic variable varying with depth and lithology for the first time,then apply it to fluid factor calculation and inversion.Firstly,the exact Zoeppritz equations are rewritten into a new form containing the fluid factor and DVRS of upper and lower layers.Next,the new equations are applied to the four parameters simultaneous inversion based on the generalized nonlinear inversion(GNI)method.The testing results on a portion of the Marmousi 2 model and field data show that dynamic DVRS can significantly improve the fluid factor identification ability,effectively suppress illusion.Both synthetic and filed data tests also demonstrate that the GNI method based on Bayesian deterministic inversion(BDI)theory can successfully solve the above four parameter simultaneous inversion problem,and taking the dynamic DVRS as a target inversion parameter can effectively improve the inversion accuracy of fluid factor.All these results completely verified the feasibility and effectiveness of the proposed method.