On Certain Least-Squares Synthesis Methods Misconceptions

1979 ◽  
Vol 101 (1) ◽  
pp. 47-50 ◽  
Author(s):  
G. H. Sutherland
Keyword(s):  
Least Squares ◽  
Lagrange Multiplier ◽  
Numerical Results ◽  
Synthesis Methods ◽  
Linear Superposition ◽  
Synthesis Techniques ◽  
Lagrange Multiplier Technique ◽  
False Premise

This paper shows a fundamental flaw in previously published linear superposition least-squares synthesis techniques. These techniques have the desirable noniteration characteristic. However, this paper shows, by a reformulation of the method, that they are based on a false premise. Numerical results bear this out. A Lagrange-multiplier technique is suggested as a superior alternative.

scholarly journals A Weighted-Least-Squares Meshless Model for Non-Hydrostatic Shallow Water Waves

Water ◽  
2021 ◽  
Vol 13 (22) ◽  
pp. 3195
Author(s):  
Nan-Jing Wu ◽  
Yin-Ming Su ◽  
Shih-Chun Hsiao ◽  
Shin-Jye Liang ◽  
Tai-Wen Hsu
Keyword(s):  
Least Squares ◽  
Shallow Water ◽  
Water Waves ◽  
Dynamic Pressure ◽  
Weighted Least Squares ◽  
Shallow Water Equation ◽  
Analytic Solutions ◽  
Numerical Results ◽  
Benchmark Problems ◽  
Time Marching

In this paper, an explicit time marching procedure for solving the non-hydrostatic shallow water equation (SWE) problems is developed. The procedure includes a process of prediction and several iterations of correction. In these processes, it is essential to accurately calculate the spatial derives of the physical quantities such as the temporal water depth, the average velocities in the horizontal and vertical directions, and the dynamic pressure at the bottom. The weighted-least-squares (WLS) meshless method is employed to calculate these spatial derivatives. Though the non-hydrostatic shallow water equations are two dimensional, on the focus of presenting this new time marching approach, we just use one dimensional benchmark problems to validate and demonstrate the stability and accuracy of the present model. Good agreements are found in the comparing the present numerical results with analytic solutions, experiment data, or other numerical results.

Prediction of Deviations of the Vertical Using Heterogeneous Data

1976 ◽  
Vol 30 (2) ◽  
pp. 97-108
Author(s):  
Gérard Lachapelle
Keyword(s):  
Least Squares ◽  
Gravity Anomalies ◽  
Heterogeneous Data ◽  
Numerical Results ◽  
Geodetic Data ◽  
West Germany ◽  
Mountainous Areas ◽  
Least Squares Collocation ◽  
Dynamic Information ◽  
Geopotential Coefficients

A method for estimating deviations of the vertical from a combination of topographic-isostatic deviations of the vertical and dynamic information in the form of geopotential coefficients is presented. The method is especially well suited for large areas, either continental or oceanic, where no geodetic measurements, such as gravity anomalies or deviations of the vertical, are available. It is ideally applicable in mountainous areas and along coastlines where the deviations depend greatly on the topography. Numerical results using topographic-isostatic data calculated in Canada, Switzerland and West Germany are presented. Furthermore, if geodetic data such as observed deviations of the vertical and gravity anomalies are available in the area considered, they can be combined with existing estimated deviations by using least squares collocation to achieve a greater accuracy.

A Multigrid Solver based on Distributive Smoother and Residual Overweighting for Oseen Problems

2015 ◽  
Vol 8 (2) ◽  
pp. 237-252 ◽  
Author(s):  
Long Chen ◽  
Xiaozhe Hu ◽  
Ming Wang ◽  
Jinchao Xu
Keyword(s):  
Least Squares ◽  
Correction Method ◽  
Numerical Results ◽  
Staggered Grid ◽  
Defect Correction ◽  
Multigrid Solver ◽  
Mac Scheme

AbstractAn efficient multigrid solver for the Oseen problems discretized by Marker and Cell (MAC) scheme on staggered grid is developed in this paper. Least squares commutator distributive Gauss-Seidel (LSC-DGS) relaxation is generalized and developed for Oseen problems. Residual overweighting technique is applied to further improve the performance of the solver and a defect correction method is suggested to improve the accuracy of the discretization. Some numerical results are presented to demonstrate the efficiency and robustness of the proposed solver.

scholarly journals Modified Preconditioned GAOR Methods for Systems of Linear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xue-Feng Zhang ◽  
Qun-Fa Cui ◽  
Shi-Liang Wu
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Least Squares ◽  
Linear Equations ◽  
Generalized Least Squares ◽  
Numerical Results ◽  
Least Squares Problem ◽  
Comparison Results ◽  
Systems Of Linear Equations ◽  
Better Than

Three kinds of preconditioners are proposed to accelerate the generalized AOR (GAOR) method for the linear system from the generalized least squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned generalized AOR (PGAOR) methods is better than that of the original GAOR methods. Finally, some numerical results are reported to confirm the validity of the proposed methods.

scholarly journals A XFEM Lagrange Multiplier Technique for Stefan Problems

2016 ◽  
Vol 7 ◽  
Author(s):  
Dave Martin ◽  
Hicham Chaouki ◽  
Jean-Loup Robert ◽  
Mario Fafard ◽  
Donald Ziegler
Keyword(s):  
Lagrange Multiplier ◽  
Stefan Problems ◽  
Lagrange Multiplier Technique

scholarly journals On removing blur in images using least squares solutions

Filomat ◽  
2016 ◽  
Vol 30 (14) ◽  
pp. 3855-3866
Author(s):  
Predrag Stanimirovic ◽  
Igor Stojanovic ◽  
Dimitrios Pappas ◽  
Spiros Chountasis ◽  
Zoran Zdravev
Keyword(s):  
Statistical Analysis ◽  
Image Restoration ◽  
Least Squares ◽  
Numerical Results ◽  
Tikhonov Regularization Method ◽  
Truncated Singular Value Decomposition ◽  
Singular Value Decomposition Method ◽  
Restoration Method ◽  
Value Decomposition ◽  
Least Squares Approach

The further investigation of the image restoration method introduced in [19, 20] is presented in this paper. Continuing investigations in that area, two additional applications of the method are investigated. More precisely, we consider the possibility to replace the available matrix in the method by the restoration obtained applying the Tikhonov regularization method or the Truncated Singular Value decomposition method. Additionally, statistical analysis of numerical results generated by applying the proposed improvement of image restoration methods is presented. Previously performed numerical experiments as well as new numerical results and the statistical analysis confirm that the least squares approach can be used as a useful tool for improving restored images obtained by other image restoration methods.

Numerical solutions of the GEW equation using MLS collocation method

2017 ◽  
Vol 28 (01) ◽  
pp. 1750011
Author(s):  
Ayşe Gül Kaplan ◽  
Yılmaz Dereli
Keyword(s):  
Least Squares ◽  
Collocation Method ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Rates Of Convergence ◽  
Numerical Results ◽  
Moving Least Squares ◽  
Test Problems ◽  
Least Squares Collocation ◽  
Energy And Momentum

In this paper, the generalized equal width wave (GEW) equation is solved by using moving least squares collocation (MLSC) method. To test the accuracy of the method some numerical experiments are presented. The motion of single solitary waves, the interaction of two solitary waves and the Maxwellian initial condition problems are chosen as test problems. For the single solitary wave motion whose analytical solution was known [Formula: see text], [Formula: see text] error norms and pointwise rates of convergence were calculated. Also mass, energy and momentum invariants were calculated for every test problems. Obtained numerical results are compared with some earlier works. It is seen that the method is very efficient and reliable due to obtained numerical results are very satisfactorily. Stability analysis of difference equation was done by applying the moving least squares collocation method for GEW equation.

scholarly journals Lagrange Multiplier Test for Spatial Autoregressive Model with Latent Variables

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1375
Author(s):  
Anik Anekawati ◽  
Bambang Widjanarko Otok ◽  
Purhadi Purhadi ◽  
Sutikno Sutikno
Keyword(s):  
Least Squares ◽  
Lagrange Multiplier ◽  
Latent Variables ◽  
Autoregressive Model ◽  
Structural Equation ◽  
Weighted Least Squares ◽  
Equation Modeling ◽  
Spatial Autoregressive Model ◽  
Lm Test ◽  
Spatial Autoregressive

The focus of this research is to develop a Lagrange multiplier (LM) test of spatial dependence for the spatial autoregressive model (SAR) with latent variables (LVs). It was arranged by the standard SAR, where the independent variables were replaced by factor scores of the exogenous latent variables from a measurement model (in structural equation modeling) as well as their dependent variables. As a result, an error distribution of the SAR-LVs should have a different distribution from the standard SAR. Therefore, this LM test for the SAR-LVs is based on the new distribution. The estimation of the latent variables used a weighted least squares (WLS) method. The estimation of the SAR-LVs parameter used a two-stage least squares (2SLS) method. The SAR-LVs model was applied to the model with a positive and negative spatial autoregressive coefficient to illustrate how it was interpreted.

Sign in / Sign up

Export Citation Format

Share Document