Generalized Iteration

2004 ◽  
Vol 3 (1) ◽  
pp. 201-252 ◽  
Author(s):  
Rainer Brück ◽  
Matthias Büger
Keyword(s):  
Generalized Iteration

scholarly journals On nonlinear boundary value problems with deviating arguments and discontinuous right hand side

1993 ◽  
Vol 6 (1) ◽  
pp. 83-91
Author(s):  
B. C. Dhage ◽  
S. Heikkilä
Keyword(s):  
Differential Equation ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problems ◽  
Deviating Arguments ◽  
Deviating Argument ◽  
Right Hand ◽  
Nonlinear Boundary Value ◽  
The Right ◽  
Generalized Iteration

In this paper we shall study the existence of the extremal solutions of a nonlinear boundary value problem of a second order differential equation with general Dirichlet/Neumann form boundary conditions. The right hand side of the differential equation is assumed to contain a deviating argument, and it is allowed to possess discontinuities in all the variables. The proof is based on a generalized iteration method.

scholarly journals Generalized Iteration Method via Green's Function for the Solution of the Fourth-order Boundary Value Problems

2021 ◽  
Vol 14 (3) ◽  
pp. 969-979
Author(s):  
Fatma Aydın Akgün ◽  
Zaur Rasulov
Keyword(s):  
Fixed Point ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Fourth Order ◽  
Iteration Method ◽  
Boundary Value ◽  
Fixed Point Iteration ◽  
Linear And Nonlinear ◽  
Generalized Iteration

The aim of this paper is to extend and generalize Picard-Green’s fixed point iteration method for the solution of fourth-order Boundary Value Problems. Several numerical applications to linear and nonlinear fourth-order Boundary Value Problems are discussed to illustrate the main results.

scholarly journals Simultaneous Patch-Group Sparse Coding with Dual-Weighted ℓp Minimization for Image Restoration

Micromachines ◽  
2021 ◽  
Vol 12 (10) ◽  
pp. 1205
Author(s):  
Jiachao Zhang ◽  
Ying Tong ◽  
Liangbao Jiao
Keyword(s):  
Image Restoration ◽  
Sparse Representation ◽  
Sparse Coding ◽  
Image Inpainting ◽  
Perceptual Quality ◽  
Novel Approach ◽  
Smooth Effects ◽  
Convex Regularization ◽  
Generalized Iteration ◽  
Patch Group

Sparse coding (SC) models have been proven as powerful tools applied in image restoration tasks, such as patch sparse coding (PSC) and group sparse coding (GSC). However, these two kinds of SC models have their respective drawbacks. PSC tends to generate visually annoying blocking artifacts, while GSC models usually produce over-smooth effects. Moreover, conventional ℓ1 minimization-based convex regularization was usually employed as a standard scheme for estimating sparse signals, but it cannot achieve an accurate sparse solution under many realistic situations. In this paper, we propose a novel approach for image restoration via simultaneous patch-group sparse coding (SPG-SC) with dual-weighted ℓp minimization. Specifically, in contrast to existing SC-based methods, the proposed SPG-SC conducts the local sparsity and nonlocal sparse representation simultaneously. A dual-weighted ℓp minimization-based non-convex regularization is proposed to improve the sparse representation capability of the proposed SPG-SC. To make the optimization tractable, a non-convex generalized iteration shrinkage algorithm based on the alternating direction method of multipliers (ADMM) framework is developed to solve the proposed SPG-SC model. Extensive experimental results on two image restoration tasks, including image inpainting and image deblurring, demonstrate that the proposed SPG-SC outperforms many state-of-the-art algorithms in terms of both objective and perceptual quality.

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