scholarly journals Continuous Regularized Least Squares Polynomial Approximation on the Sphere

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Yang Zhou ◽  
Yanan Kong
Keyword(s):  
Least Squares ◽  
Regularization Parameter ◽  
Weighted Least Squares ◽  
Computation Time ◽  
Condition Numbers ◽  
Perturbation Bound ◽  
Smooth Functions ◽  
Approximation Quality ◽  
Point Set ◽  
Polynomial Reconstruction

In this paper, we consider the problem of polynomial reconstruction of smooth functions on the sphere from their noisy values at discrete nodes on the two-sphere. The method considered in this paper is a weighted least squares form with a continuous regularization. Preliminary error bounds in terms of regularization parameter, noise scale, and smoothness are proposed under two assumptions: the mesh norm of the data point set and the perturbation bound of the weight. Condition numbers of the linear systems derived by the problem are discussed. We also show that spherical tϵ-designs, which can be seen as a generalization of spherical t-designs, are well applied to this model. Numerical results show that the method has good performance in view of both the computation time and the approximation quality.

scholarly journals Mixed and component wise condition numbers for weighted Moore-Penrose inverse and weighted least squares problems

Filomat ◽  
2009 ◽  
Vol 23 (1) ◽  
pp. 43-59 ◽  
Author(s):  
Li Zhao ◽  
Jie Sun
Keyword(s):  
Numerical Analysis ◽  
Least Squares ◽  
Weighted Least Squares ◽  
Full Column Rank ◽  
Condition Numbers ◽  
Linear Least Squares ◽  
Least Squares Problems ◽  
Linear Least Squares Problem ◽  
Classical Condition ◽  
The Matrix

Condition numbers play an important role in numerical analysis. Classical condition numbers are norm-wise: they measure both input perturbations and output errors with norms. To take into account the relative scaling of data components or a possible sparseness, component-wise condition numbers have been increasingly considered. In this paper, we give explicit expressions for the mixed and component-wise condition numbers for the weighted Moore-Penrose inverse of a matrix A, as well as for the solution and residue of a weighted linear least squares problem ||W 1 2 (Ax-b) ||2 = minv2Rn ||W 1 2 (Av-b) ||2, where the matrix A with full column rank. .

Different inference approaches for the estimators of the sushila distribution

2021 ◽  
Vol 16 (4) ◽  
pp. 251-260
Author(s):  
Marcos Vinicius de Oliveira Peres ◽  
Ricardo Puziol de Oliveira ◽  
Edson Zangiacomi Martinez ◽  
Jorge Alberto Achcar
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Bayesian Method ◽  
Weighted Least Squares ◽  
Computational Cost ◽  
Estimation Method ◽  
Computation Time ◽  
Ordinary Least Squares ◽  
Likelihood Method ◽  
Estimation Methods

In this paper, we order to evaluate via Monte Carlo simulations the performance of sample properties of the estimates of the estimates for Sushila distribution, introduced by Shanker et al. (2013). We consider estimates obtained by six estimation methods, the known approaches of maximum likelihood, moments and Bayesian method, and other less traditional methods: L-moments, ordinary least-squares and weighted least-squares. As a comparison criterion, the biases and the roots of mean-squared errors were used through nine scenarios with samples ranging from 30 to 300 (every 30rd). In addition, we also considered a simulation and a real data application to illustrate the applicability of the proposed estimators as well as the computation time to get the estimates. In this case, the Bayesian method was also considered. The aim of the study was to find an estimation method to be considered as a better alternative or at least interchangeable with the traditional maximum likelihood method considering small or large sample sizes and with low computational cost.

scholarly journals Design of one‐way wavefield extrapolation operators, using smooth functions in WLSQ optimization

Geophysics ◽  
2004 ◽  
Vol 69 (4) ◽  
pp. 1037-1045 ◽  
Author(s):  
Jan W Thorbecke ◽  
Kees Wapenaar ◽  
Gerd Swinnen
Keyword(s):  
Least Squares ◽  
Recursive Algorithm ◽  
Weighted Least Squares ◽  
Convolution Operators ◽  
Smooth Functions ◽  
Exchange Method ◽  
Data Set ◽  
Depth Migration ◽  
Frequency Space ◽  
Small Depth

Many depth migration methods use one‐way frequency–space depth extrapolation methods. These methods are generally considered to be expensive, so it is important to find the most efficient way of implementing them. This usually means making spatial convolution operators that are as short as possible. Applying the extrapolation operators in a recursive way, using small depth steps, also demands that the operators do not amplify the wavefield at every depth step. Weighted least squares is an appropriate method to use for designing extrapolation operators that are accurate and efficient and that remain stable in a recursive algorithm. The extrapolated wavefields calculated with these operators are comparable with the extrapolation results obtained with other known operator design techniques as the Remez exchange method and nonlinear optimization. In this paper, the weighted least‐squares technique is refined by using different model functions. By smoothing the phase and amplitude transition at the evanescent cutoff, we can stabilize the resulting operators. The accuracy of the operators is shown in zero‐offset migration impulse responses in 2D and 3D media. The Sigsbee2A data set is used to illustrate the quality of the extrapolation operators in prestack depth migration in a complex medium.

A Coupled Finite Difference and Weighted Least Squares Simulation of Violent Breaking Wave Impact

2012 ◽  
Author(s):  
Ole Lindberg ◽  
Harry B. Bingham ◽  
Allan Peter Engsig Karup
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Least Squares ◽  
Flow Model ◽  
Weighted Least Squares ◽  
Surface Boundary ◽  
Wave Impact ◽  
Finite Point ◽  
Point Set ◽  
Vertical Breakwater

Two model for simulation of free surface flow are presented. The first model is a finite difference based potential flow model with non-linear kinematic and dynamic free surface boundary conditions. The second model is a weighted least squares based incompressible and inviscid flow model. A special feature of this model is a generalized finite point set method which is applied to the solution of the Poisson equation on an unstructured point distribution. The presented finite point set method is generalized to arbitrary order of approximation. The two models are applied to simulation of steep and overturning wave impacts on a vertical breakwater. Wave groups with five different wave heights are propagated from offshore to the vicinity of the breakwater, where the waves are steep, but still smooth and non-overturning. These waves are used as initial condition for the weighted least squares based incompressible and inviscid model and the wave impacts on the vertical breakwater are simulated in this model. The resulting maximum pressures and forces on the breakwater are relatively high when compared with other studies and this is due to the incompressible nature of the present model.

On Condition Numbers for the Weighted Moore-Penrose Inverse and the Weighted Least Squares Problem involving Kronecker Products

2014 ◽  
Vol 4 (1) ◽  
pp. 1-20
Author(s):  
T. T. Chen ◽  
W. Li
Keyword(s):  
Least Squares ◽  
Condition Number ◽  
Matrix Function ◽  
Kronecker Product ◽  
Weighted Least Squares ◽  
Condition Numbers ◽  
Kronecker Products ◽  
General Matrix ◽  
Least Squares Problem ◽  
Explicit Expressions

AbstractWe establish some explicit expressions for norm-wise, mixed and componentwise condition numbers for the weighted Moore-Penrose inverse of a matrix A ⊗ B and more general matrix function compositions involving Kronecker products. The condition number for the weighted least squares problem (WLS) involving a Kronecker product is also discussed.

Application of New Least-Squares Methods for the Quantitative Infrared Analysis of Multicomponent Samples

1982 ◽  
Vol 36 (6) ◽  
pp. 665-673 ◽  
Author(s):  
David M. Haaland ◽  
Robert G. Easterling
Keyword(s):  
Least Squares ◽  
Data Storage ◽  
Quantitative Estimation ◽  
Weighted Least Squares ◽  
Computation Time ◽  
Threshold Value ◽  
Infrared Analysis ◽  
Least Squares Regression ◽  
Beer's Law ◽  
Beer’S Law

Improvements have been made in previous least-squares regression analyses of infrared spectra for the quantitative estimation of concentrations of multicomponent mixtures. Spectral baselines are fitted by least-squares methods, and overlapping spectral features are accounted for in the fitting procedure. Selection of peaks above a threshold value reduces computation time and data storage requirements. Four weighted least-squares methods incorporating different baseline assumptions were investigated using FT-IR spectra of the three pure xylene isomers and their mixtures. By fitting only regions of the spectra that follow Beer's Law, accurate results can be obtained using three of the fitting methods even when baselines are not corrected to zero. Accurate results can also be obtained using one of the fits even in the presence of Beer's Law deviations. This is a consequence of pooling the weighted results for each spectral peak such that the greatest weighting is automatically given to those peaks that adhere to Beer's Law. It has been shown with the xylene spectra that semiquantitative results can be obtained even when all the major components are not known or when expected components are not present. This improvement over previous methods greatly expands the utility of quantitative least-squares analyses.

Sign in / Sign up

Export Citation Format

Share Document