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-SQUARES METHOD-BASED FEATURE FITTING AND EXTRACTION IN REVERSE ENGINEERING

The main purpose of reverse engineering is to convert discrete data pointsinto piecewise smooth, continuous surface models. Before carrying out model reconstruction it issignificant to extract geometric features because the quality of modeling greatly depends on therepresentation of features. Some fitting techniques of natural quadric surfaces with least-squaresmethod are described. And these techniques can be directly used to extract quadric surfaces featuresduring the process of segmentation for point cloud.

Solution of shallow-water equations using least-squares finite-element method

A least-squares finite-element method （LSFEM） for the non-conservative shallow-water equations is presented. The model is capable of handling complex topography, steady and unsteady flows, subcritical and supercritical flows, and flows with smooth and sharp gradient changes. Advantages of the model include： （1） sources terms, such as the bottom slope, surface stresses and bed frictions, can be treated easily without any special treatment; （2） upwind scheme is no needed; （3） a single approximating space can be used for all variables, and its choice of approximating space is not subject to the Ladyzhenskaya-Babuska-Brezzi （LBB） condition; and （4） the resulting system of equations is symmetric and positive-definite （SPD） which can be solved efficiently with the preconditioned conjugate gradient method. The model is verified with flow over a bump, tide induced flow, and dam-break. Computed results are compared with analytic solutions or other numerical results, and show the model is conservative and accurate. The model is then used to simulate flow past a circular cylinder. Important flow charac-teristics, such as variation of water surface around the cylinder and vortex shedding behind the cylinder are investigated. Computed results compare well with experiment data and other numerical results.

Least-squares finite-element method for shallow-water equations with source terms

Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.

NEGATIVE NORM LEAST-SQUARES METHODS FOR THE INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS

The purpose of this article is to develop and analyze least-squares approximations for the incompressible magnetohydrodynamic equations. The major advantage of the least-squares finite element method is that it is not subjected to the so-called Ladyzhenskaya-Babuska-Brezzi （LBB） condition. The authors employ least-squares functionals which involve a discrete inner product which is related to the inner product in H^-1（Ω）.

ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS

The Galerkin and least-squares methods are two classes of the most popular Krylov subspace methOds for solving large linear systems of equations. Unfortunately, both the methods may suffer from serious breakdowns of the same type: In a breakdown situation the Galerkin method is unable to calculate an approximate solution, while the least-squares method, although does not really break down, is unsucessful in reducing the norm of its residual. In this paper we first establish a unified theorem which gives a relationship between breakdowns in the two methods. We further illustrate theoretically and experimentally that if the coefficient matrix of a lienar system is of high defectiveness with the associated eigenvalues less than 1, then the restarted Galerkin and least-squares methods will be in great risks of complete breakdowns. It appears that our findings may help to understand phenomena observed practically and to derive treatments for breakdowns of this type.

AN ASSESSMENT OF THE MESHLESS WEIGHTED LEAST-SQUARE METHOD

The meshless weighted least-square (MWLS) method was developed based on the weighted least-square method. The method possesses several advantages, such as high accuracy, high stability and high e?ciency. Moreover, the coe?cient matrix obtained is symmetric and semi- positive de?nite. In this paper, the method is further examined critically. The e?ects of several parameters on the results of MWLS are investigated systematically by using a cantilever beam and an in?nite plate with a central circular hole. The numerical results are compared with those obtained by using the collocation-based meshless method (CBMM) and Galerkin-based meshless method (GBMM). The investigated parameters include the type of approximations, the type of weight functions, the number of neighbors of an evaluation point, as well as the manner in which the neighbors of an evaluation point are determined. This study shows that the displacement accuracy and convergence rate obtained by MWLS is comparable to that of the GBMM while the stress accuracy and convergence rate yielded by MWLS is even higher than that of GBMM. Furthermore, MWLS is much more e?cient than GBMM. This study also shows that the instability of CBMM is mainly due to the neglect of the equi- librium residuals at boundary nodes. In MWLS, the residuals of all the governing equations are minimized in a weighted least-square sense.

Complex derivatives valuation: applying the Least-Squares Monte Carlo Simulation Method with several polynomial basis

Background:This article investigates the Least-Squares Monte Carlo Method by using different polynomial basis in American Asian Options pricing.The standard approach in the option pricing literature is to choose the basis arbitrarily.By comparing four different polynomial basis we show that the choice of basis interferes in the option's price.Methods:We assess Least-Squares Method performance in pricing four different American Asian Options by using four polynomial basis:Power,Laguerre,Legendre and Hermite A.To every American Asian Option priced,three sets of parameters are used in order to evaluate it properly.Results:We show that the choice of the basis interferes in the option's price by showing that one of them converges to the option's value faster than any other by using fewer simulated paths.In the case of an Amerasian call option,for example,we find that the preferable polynomial basis is Hermite A.For an Amerasian put option,the Power polynomial basis is recommended.Such empirical outcome is theoretically unpredictable,since in principle all basis can be indistinctly used when pricing the derivative.Conclusion:In this article The Least-Squares Monte Carlo Method performance is assessed in pricing four different types of American Asian Options by using four different polynomial basis through three different sets of parameters.Our results suggest that one polynomial basis is best suited to perform the method when pricing an American Asian option.Theoretically all basis can be indistinctly used when pricing the derivative.However,our results does not confirm these.We find that when pricing an American Asian put option,Power A is better than the other basis we have studied here whereas when pricing an American Asian call,Hermite A is better.

An Adaptive Least-Squares Mixed Finite Element Method for Fourth Order Parabolic Problems

A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.

LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR A CLASS OF STOKES EQUATION

A least-squares mixed finite element method was formulated for a class of Stokes equations in two dimensional domains. The steady state and the time-dependent Stokes' equations were considered. For the stationary equation, optimal H-t and L-2-error estimates are derived under the standard regularity assumption on the finite element partition ( the LBB-condition is not required). Far the evolutionary equation, optimal L-2 estimates are derived under the conventional Raviart-Thomas spaces.

Application of neural network model coupling with the partial least-squares method for forecasting watre yield of mine

Scientific forecasting water yield of mine is of great significance to the safety production of mine and the colligated using of water resources. The paper established the forecasting model for water yield of mine, combining neural network with the partial least square method. Dealt with independent variables by the partial least square method, it can not only solve the relationship between independent variables but also reduce the input dimensions in neural network model, and then use the neural network which can solve the non-linear problem better. The result of an example shows that the prediction has higher precision in forecasting and fitting.

Least-Squares Finite Element Method for the Steady Upper-Convected Maxwell Fluid

In this paper, a least-squares finite element method for the upper-convected Maxell (UCM) fluid is proposed. We first linearize the constitutive and momentum equations and then apply a least-squares method to the linearized version of the viscoelastic UCM model. The L2 least-squares functional involves the residuals of each equation multiplied by proper weights. The corresponding homogeneous functional is equivalent to a natural norm. The error estimates of the finite element solution are analyzed when the conforming piecewise polynomial elements are used for the unknowns.

Least-squares reverse time migration method using the factorization of the Hessian matrix

Least-squares reverse time migration(LSRTM)can eliminate imaging artifacts in an iterative way based on the concept of inversion,and it can restore imaging amplitude step by step.LSRTM can provide a high-resolution migration section and can be applied to irregular and poor-quality seismic data and achieve good results.Steeply dipping refl ectors and complex faults are imaged by using wavefi eld extrapolation based on a two-way wave equation.However,the high computational cost limits the method’s application in practice.A fast approach to realize LSRTM in the imaging domain is provided in this paper to reduce the computational cost signifi cantly and enhance its computational effi ciency.The method uses the Kronecker decomposition algorithm to estimate the Hessian matrix.A low-rank matrix can be used to calculate the Kronecker factor,which involves the calculation of Green’s function at the source and receiver point.The approach also avoids the direct construction of the whole Hessian matrix.Factorization-based LSRTM calculates the production of low-rank matrices instead of repeatedly calculating migration and demigration.Unlike traditional LSRTM,factorization-based LSRTM can reduce calculation costs considerably while maintaining comparable imaging quality.While having the same imaging eff ect,factorization-based LSRTM consumes half the running time of conventional LSRTM.In this regard,the application of factorization-based LSRTM has a promising advantage in reducing the computational cost.Ambient noise caused by this method can be removed by applying a commonly used fi ltering method without signifi cantly degrading the imaging quality.

Least-squares reverse time migration in visco-acoustic media based on symplectic stereo-modeling method

The authors proposed a symplectic stereo-modeling method(SSM)in the Birkhoffian dynam-ics and apply it to the visco-acoustic least-squares reverse time migration(LSRTM).The SSM adopts ste-reo-modeling operator in space and symplectic Runge-Kutta scheme in time,resulting in great ability in suppressing numerical dispersion and long-time computing.These advantages are further confirmed by numerical dispersion analysis,long-time computation test and computational efficiency comparison.After these theoretical analyses and experiments,acoustic and visco-acoustic LSRTM are tested and compared between SSM method and the conventional symplectic method(CSM)using the fault and marmousi models.Meanwhile,dynamic source encoding and exponential decay moving average gradients method are adopted to reduce the computation cost and improve the convergence rate.The imaging results show that LSRTM based on visco-acoustic wave equations effectively takes into account the influence of viscosity can therefore compensate for the amplitude attenuation.Besides,SSM method not only has high numerical accuracy and computational efficiency,but also performs effectively in LSRTM.

Structural Modal Parameter Recognition and Related Damage Identification Methods under Environmental Excitations: A Review

Modal parameters can accurately characterize the structural dynamic properties and assess the physical state of the structure.Therefore,it is particularly significant to identify the structural modal parameters according to the monitoring data information in the structural health monitoring(SHM)system,so as to provide a scientific basis for structural damage identification and dynamic model modification.In view of this,this paper reviews methods for identifying structural modal parameters under environmental excitation and briefly describes how to identify structural damages based on the derived modal parameters.The paper primarily introduces data-driven modal parameter recognition methods(e.g.,time-domain,frequency-domain,and time-frequency-domain methods,etc.),briefly describes damage identification methods based on the variations of modal parameters(e.g.,natural frequency,modal shapes,and curvature modal shapes,etc.)and modal validation methods(e.g.,Stability Diagram and Modal Assurance Criterion,etc.).The current status of the application of artificial intelligence(AI)methods in the direction of modal parameter recognition and damage identification is further discussed.Based on the pre-vious analysis,the main development trends of structural modal parameter recognition and damage identification methods are given to provide scientific references for the optimized design and functional upgrading of SHM systems.

Distributed Least-Squares Iterative Methods in Large-Scale Networks:A Survey

Many science and engineering applications involve solvinga linear least-squares system formed from some field measurements. In the distributed cyber-physical systems(CPS),each sensor node used for measurement often only knowspartial independent rows of the least-squares system. To solve the least-squares all the measurements must be gathered at a centralized location and then perform the computa-tion. Such data collection and computation are inefficient because of bandwidth and time constraints and sometimes areinfeasible because of data privacy concerns. Iterative methods are natural candidates for solving the aforementionedproblem and there are many studies regarding this. However,most of the proposed solutions are related to centralized/parallel computations while only a few have the potential to beapplied in distributed networks. Thus distributed computations are strongly preferred or demanded in many of the realworld applications, e.g. smart-grid, target tracking, etc. Thispaper surveys the representative iterative methods for distributed least-squares in networks.

Least-Squares及Galerkin谱元方法求解环形区域内的泊松方程

为研究基于Least-Squares变分及Galerkin变分两种形式的谱元方法的求解特性,推导了极坐标系中采用两种变分方法求解环形区域内Poisson方程时对应的弱解形式,采用Chebyshev多项式构造插值基函数进行空间离散,得到两种谱元方法对应的代数方程组,由此分析了系数矩阵结构的特点。数值计算结果显示:Least-Squares谱元方法为实现方程的降阶而引入新的求解变量,使得代数方程组形式更为复杂,但边界条件的处理比Galerkin谱元方法更为简单;两种谱元方法均能求解极坐标系中的Poisson方程且能获得高精度的数值解,二者绝对误差分布基本一致;固定单元内的插值阶数时,增加单元数可减小数值误差,且表现出代数精度的特点,误差降低速度较慢,而固定单元数时,在一定范围内数值误差随插值阶数的增加而减小的速度更快,表现出谱精度的特点;单元内插值阶数较高时,代数方程组系数矩阵的条件数急剧增多,方程组呈现病态,数值误差增大,这一特点限制了单元内插值阶数的取值。研究内容对深入了解两种谱元方法在极坐标系中求解Poisson方程时的特点、进一步采用相关分裂算法求解实际流动问题具有参考价值。

Quantitative energy-dispersive X-ray fluorescence analysis for unknown samples using full-spectrum least-squares regression

The full-spectrum least-squares(FSLS) method is introduced to perform quantitative energy-dispersive X-ray fluorescence analysis for unknown solid samples.Based on the conventional least-squares principle, this spectrum evaluation method is able to obtain the background-corrected and interference-free net peaks, which is significant for quantization analyses. A variety of analytical parameters and functions to describe the features of the fluorescence spectra of pure elements are used and established, such as the mass absorption coefficient, the Gi factor, and fundamental fluorescence formulas. The FSLS iterative program was compiled in the C language. The content of each component should reach the convergence criterion at the end of the calculations. After a basic theory analysis and experimental preparation, 13 national standard soil samples were detected using a spectrometer to test the feasibility of using the algorithm. The results show that the calculated contents of Ti, Fe, Ni, Cu, and Zn have the same changing tendency as the corresponding standard content in the 13 reference samples. Accuracies of 0.35% and 14.03% are obtained, respectively, for Fe and Ti, whose standard concentrations are 8.82% and 0.578%, respectively. However, the calculated results of trace elements (only tens of lg/g) deviate from the standard values. This may be because of measurement accuracy and mutual effects between the elements.

Global interpolating meshless shape function based on generalized moving least-square for structural dynamic analysis

A global interpolating meshless shape function based on the generalized moving least-square （GMLS） is formulated by the transformation technique. Both the shape function and its derivatives meet the Kronecker delta function property. With the interpolating GMLS （IGMLS） shape function, an improved element-free Galerkin （EFG） method is proposed for the structural dynamic analysis. Compared with the conven- tional EFG method, the obvious advantage of the proposed method is that the essential boundary conditions including both displacements and derivatives can be imposed by the straightforward way. Meanwhile, it can greatly improve the ill-condition feature of the standard GMLS approximation, and provide good accuracy at low cost. The dynamic analyses of the Euler beam and Kirchhoff plate are performed to demonstrate the feasi- bility and effectiveness of the improved method. The comparison between the numerical results of the conventional method and the improved method shows that the proposed method has better stability, higher accuracy, and less time consumption.

Regularized least-squares migration of simultaneous-source seismic data with adaptive singular spectrum analysis

Simultaneous-source acquisition has been recog- nized as an economic and efficient acquisition method, but the direct imaging of the simultaneous-source data produces migration artifacts because of the interference of adjacent sources. To overcome this problem, we propose the regularized least-squares reverse time migration method （RLSRTM） using the singular spectrum analysis technique that imposes sparseness constraints on the inverted model. Additionally, the difference spectrum theory of singular values is presented so that RLSRTM can be implemented adaptively to eliminate the migration artifacts. With numerical tests on a fiat layer model and a Marmousi model, we validate the superior imaging quality, efficiency and convergence of RLSRTM compared with LSRTM when dealing with simultaneoussource data, incomplete data and noisy data.