Least Squares Solution of the Linear Operator Equation

2015 ◽  
Vol 170 (1) ◽  
pp. 205-219 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Linear Operator ◽  
Least Squares ◽  
Operator Equation ◽  
Linear Operator Equation ◽  
Least Squares Solution

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

A nonstandard approximation of pseudoinverse and a new stopping criterion for iterative regularization

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Anatoly Bakushinsky ◽  
Alexandra Smirnova ◽  
Hui Liu
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Stopping Criterion ◽  
Iterative Regularization ◽  
Linear Operator Equation ◽  
Non Linear ◽  
Non Linear Operator

AbstractA problem of solving a (non)linear operator equation,

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