scholarly journals Nonlinear Operators as Concerns Convex Programming and Applied to Signal Processing

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 866
Author(s):  
Anantachai Padcharoen ◽  
Pakeeta Sukprasert
Keyword(s):  
Signal Processing ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonlinear Problems ◽  
Least Square ◽  
Accretive Operators ◽  
Convex Minimization Problem ◽  
Nonlinear Operators ◽  
Least Square Problem ◽  
Common Zeros

Splitting methods have received a lot of attention lately because many nonlinear problems that arise in the areas used, such as signal processing and image restoration, are modeled in mathematics as a nonlinear equation, and this operator is decomposed as the sum of two nonlinear operators. Most investigations about the methods of separation are carried out in the Hilbert spaces. This work develops an iterative scheme in Banach spaces. We prove the convergence theorem of our iterative scheme, applications in common zeros of accretive operators, convexly constrained least square problem, convex minimization problem and signal processing.

scholarly journals Forward-Backward Splitting Methods for Accretive Operators in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Genaro López ◽  
Victoria Martín-Márquez ◽  
Fenghui Wang ◽  
Hong-Kun Xu
Keyword(s):  
Banach Spaces ◽  
Nonlinear Problems ◽  
Split Feasibility Problem ◽  
Splitting Methods ◽  
Feasibility Problem ◽  
Nonlinear Operator Equation ◽  
Accretive Operators ◽  
Nonlinear Operators ◽  
Constrained Convex Minimization Problem ◽  
The Split Feasibility Problem

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce two iterative forward-backward splitting methods with relaxations and errors to find zeros of the sum of two accretive operators in the Banach spaces. We shall prove the weak and strong convergence of these methods under mild conditions. We also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem.

scholarly journals Design of Special Impacting Filter for Multicarrier ABPSK System

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhimin Chen ◽  
Lenan Wu
Keyword(s):  
Quadratic Programming ◽  
Cost Function ◽  
Frequency Response ◽  
Programming Problem ◽  
Iterative Scheme ◽  
Notch Filter ◽  
Least Square ◽  
Quadratic Programming Problem ◽  
Least Square Problem ◽  
The Cost

A rather intuitive technique known as pole-zero placement is introduced to illustrate the frequency response of the special impacting filters (SIFs) with a pair of conjugate zero-poles and deduce the equation of the pole radii. Based on that, the paper proposes an iterative scheme to derive the parameters of the cascade notch filter. The cost function is determined by the cascading notch filter’s influence on impacting filters, converting the cost function’s least square problem to a filter parameters’ standard quadratic programming problem. Finally, a cascading notch SIF (CNSIF) designed to demodulate the ABPSK signals is realized.

scholarly journals On ill-posedness concepts, stable solvability and saturation

2018 ◽  
Vol 26 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Bernd Hofmann ◽  
Robert Plato
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Solution Space ◽  
Nonlinear Problems ◽  
Special Role ◽  
Well Posedness ◽  
Operator Equations ◽  
Nonlinear Operators ◽  
Nonlinear Operator Equations ◽  
Linear Operator Equations

AbstractWe consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces. It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.

scholarly journals General Iterative Methods for System of Equilibrium Problems and Constrained Convex Minimization Problem in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Peichao Duan
Keyword(s):  
Iterative Methods ◽  
Hilbert Spaces ◽  
Minimization Problem ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Convex Minimization ◽  
Convex Minimization Problem ◽  
Constrained Convex Minimization ◽  
Constrained Convex Minimization Problem ◽  
System Of Equilibrium Problems

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of solutions of system of equilibrium problems and a constrained convex minimization problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by Tian and Liu (2012) and many others. Furthermore, we give numerical example to demonstrate the effectiveness of our iterative scheme.

scholarly journals Numerical method to a class of boundary value problems

2018 ◽  
Vol 22 (4) ◽  
pp. 1877-1883 ◽  
Author(s):  
Yu-Yang Qiu
Keyword(s):  
Boundary Value Problems ◽  
Numerical Method ◽  
Conjugate Gradient ◽  
Iteration Method ◽  
Boundary Value ◽  
Solution Process ◽  
Least Square ◽  
Least Square Problem ◽  
Matrix Iteration

A class of boundary value problems can be transformed uniformly to a least square problem with Toeplitz constraint. Conjugate gradient least square, a matrix iteration method, is adopted to solve this problem, and the solution process is elucidated step by step so that the example can be used as a paradigm for other applications.

scholarly journals Performance enhancement of MPM using SNR boosting techniques

2016 ◽  
Vol 5 (4) ◽  
pp. 115
Author(s):  
Shimaa Mamdouh ◽  
Amr Hussein ◽  
Hamdy Elmekaty
Keyword(s):  
Signal Processing ◽  
Performance Enhancement ◽  
Mimo Systems ◽  
Signal To Noise Ratio ◽  
Matrix Pencil ◽  
Doa Estimation ◽  
Least Square ◽  
Test Case ◽  
Recursive Least Square ◽  
Low Snr

Signal to noise ratio (SNR) boosting is one of the most important research areas in signal processing. The effectiveness of SNR boosting is not limited to a specific application rather, it is widely used in image processing, signal processing, cognitive radio, MIMO systems, digital beam forming, and direction of arrival (DOA) estimation …etc. In this paper, the recursive least square (RLS) and wavelet based de-noising filters are exploited for SNR boosting in DOA estimation techniques for further performance enhancement. The matrix pencil method (MPM) as an effortlessness and high resolution DOA estimation technique is taken as a test case. That is because it suffers from performance deterioration under low SNR regimes. The simulation results reveal that the MPM based RLS de-noising filter outperforms the MPM based wavelet de-noising filter and the traditional MPM in terms of mean square error (MSE) especially at low SNR regimes.

scholarly journals Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 156 ◽  
Author(s):  
Chanjuan Pan ◽  
Yuanheng Wang
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Minimization Problem ◽  
Numerical Experiments ◽  
Splitting Method ◽  
Splitting Methods ◽  
Strong Convergence Theorem ◽  
Accretive Operators ◽  
Convex Minimization Problem ◽  
Inertial Extrapolation

In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we extend our main results to solve the convex minimization problem. Moreover, the numerical experiments are presented to support the feasibility and efficiency of the proposed method.

scholarly journals Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 681-690
Author(s):  
Khairul Saleh ◽  
Hafiz Fukhar-ud-din
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces ◽  
Fixed Point Iterations

Abstract In this work, we propose an iterative scheme to approach common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in a CAT(0) space. We establish and prove convergence theorems for the algorithm. The results are new and interesting in the theory of $$CAT\left( 0\right) $$ C A T 0 spaces and are the analogues of corresponding ones in uniformly convex Banach spaces and Hilbert spaces.

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