Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates

1994 ◽  
Vol 83 (1) ◽  
pp. 217-222 ◽  
Author(s):  
S. George ◽  
M. T. Nair
Keyword(s):  
Convergence Rates ◽  
Parameter Choice ◽  
Optimal Convergence ◽  
Optimal Convergence Rates ◽  
Ill Posed

scholarly journals Multiscale Compression Algorithm for Solving Nonlinear Ill-Posed Integral Equations via Landweber Iteration

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 221
Author(s):  
Rong Zhang ◽  
Fanchun Li ◽  
Xingjun Luo
Keyword(s):  
Theoretical Analysis ◽  
Galerkin Method ◽  
Integral Equations ◽  
Convergence Rates ◽  
Numerical Results ◽  
Compression Algorithm ◽  
Optimal Convergence ◽  
Relaxation Factor ◽  
Optimal Convergence Rates ◽  
Ill Posed

In this paper, Landweber iteration with a relaxation factor is proposed to solve nonlinear ill-posed integral equations. A compression multiscale Galerkin method that retains the properties of the Landweber iteration is used to discretize the Landweber iteration. This method leads to the optimal convergence rates under certain conditions. As a consequence, we propose a multiscale compression algorithm to solve nonlinear ill-posed integral equations. Finally, the theoretical analysis is verified by numerical results.

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.

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