Should We Search for a Global Minimizer of Least Squares Regularized with an ℓ0 Penalty to Get the Exact Solution of an under Determined Linear System?

2012 ◽  
pp. 508-519
Author(s):  
Mila Nikolova
Keyword(s):  
Exact Solution ◽  
Linear System ◽  
Least Squares ◽  
Global Minimizer

scholarly journals Splitting least squares and collocation procedures for two-point boundary value problems

1985 ◽  
Vol 27 (2) ◽  
pp. 167-193
Author(s):  
John Locker ◽  
P. M. Prenter
Keyword(s):  
Linear System ◽  
Boundary Value Problem ◽  
Numerical Solution ◽  
Least Squares ◽  
Boundary Value ◽  
Linear Operators ◽  
Point Boundary ◽  
System Error ◽  
Densely Defined ◽  
Split System

AbstractLet L, T, S, and R be closed densely defined linear operators from a Hubert space X into X where L can be factored as L = TS + R. The equation Lu = f is equivalent to the linear system Tv + Ru = f and Su = v. If Lu = f is a two-point boundary value problem, numerical solution of the split system admits cruder approximations than the unsplit equations. This paper develops the theory of such splittings together with the theory of the Methods of Least Squares and of Collocation for the split system. Error estimates in both L2 and L∞ norms are obtained for both methods.

Optimal Synthesis of a Robomech: Procedure and Application

1999 ◽  
Author(s):  
Stephen Canfield ◽  
Giridhar Kolanupaka ◽  
Ahmad Smaili
Keyword(s):  
Linear System ◽  
Least Squares ◽  
Lagrange Multipliers ◽  
Direct Application ◽  
Optimal Synthesis ◽  
Nonlinear Constraints ◽  
New Class ◽  
Least Squares Optimization ◽  
Synthesis Procedure ◽  
Synthesis Approach

Abstract This paper presents and compares two optimal synthesis techniques for direct application in creating a robomech-II, the second manipulator presented in a new class of linkage arms called Robomcchs. The first optimal synthesis approach will solve the problem as a nonlinear optimization, with a subset of the device parameters described in a linear system and solved directly in a least squares sense. The second approach will employ a least squares optimization using Lagrange Multipliers to contend with nonlinear constraints. In this paper, each optimal synthesis procedure is developed for the general case and then applied to robomcch-II through an example.

A Least-Squares Solution of a Linear System

SIAM Review ◽  
1962 ◽  
Vol 4 (2) ◽  
pp. 150-150
Author(s):  
Franklin W. Diederich
Keyword(s):  
Linear System ◽  
Least Squares ◽  
Least Squares Solution

A Least-Squares Solution of a Linear System (Franklin W. Dtederich)

SIAM Review ◽  
1964 ◽  
Vol 6 (2) ◽  
pp. 182-183
Author(s):  
Victor Chew ◽  
M. J. Synge
Keyword(s):  
Linear System ◽  
Least Squares ◽  
Least Squares Solution

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 Projection Methods for Ill-Posed Problems Revisited

2016 ◽  
Vol 16 (2) ◽  
pp. 257-276 ◽  
Author(s):  
Stefan Kindermann
Keyword(s):  
Exact Solution ◽  
Global Convergence ◽  
Least Squares ◽  
Local Convergence ◽  
Sufficient Conditions ◽  
Projection Methods ◽  
Bounded Sequence ◽  
Oblique Projection ◽  
Ill Posed

AbstractWe consider the discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solutions) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent conditions when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as an oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence, we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.

Sign in / Sign up

Export Citation Format

Share Document