Condition numbers with their condition numbers for the weighted moore-penrose inverse and the weighted least squares solution

2006 ◽  
Vol 22 (1-2) ◽  
pp. 95-112 ◽  
Author(s):  
Wenhua Kang ◽  
Hua Xiang
Keyword(s):  
Least Squares ◽  
Weighted Least Squares ◽  
Condition Numbers ◽  
Least Squares Solution

Convolutional quelling in seismic tomography

Geophysics ◽  
1989 ◽  
Vol 54 (5) ◽  
pp. 570-580 ◽  
Author(s):  
Keith A. Meyerholtz ◽  
Gary L. Pavlis ◽  
Sally A. Szpakowski
Keyword(s):  
Least Squares ◽  
Seismic Tomography ◽  
Nearest Neighbor ◽  
Sparse Matrix ◽  
Weighted Least Squares ◽  
Synthetic Data ◽  
Least Squares Solution ◽  
Smoothing Filter ◽  
Imaging Artifacts ◽  
Tomography Data

This paper introduces convolutional quelling as a technique to improve imaging of seismic tomography data. We show the result amounts to a special type of damped, weighted, least‐squares solution. This insight allows us to implement the technique in a practical manner using a sparse matrix, conjugate gradient equation solver. We applied the algorithm to synthetic data using an eight nearest‐neighbor smoothing filter for the quelling. The results were found to be superior to a simple, least‐squares solution because convolutional quelling suppresses side bands in the resolving function that lead to imaging artifacts.

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. .

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.

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