Parallel Hybrid Algorithms for a Finite Family of G-Nonexpansive Mappings and Its Application in a Novel Signal Recovery

Abstract

This article considers a parallel monotone hybrid algorithm for a finite family of G-nonexpansive mapping in Hilbert spaces endowed with graphs and suggests iterative schemes for finding a common fixed point by the two different hybrid projection methods. Moreover, we show the computational performance of our algorithm in comparison to some methods. Strong convergence theorems are proved under suitable conditions. Finally, we give some numerical experiments of our algorithms to show the efficiency and implementation of the LASSO problems in signal recovery with different types of blurred matrices and noise.

A novel Noor iterative method of operators with property (E) as concerns convex programming applicable in signal recovery and polynomiography

“…where 𝜁 > 0. As a result, various techniques and iterative schemes have been developed over the years to solve the LASSO problem, see the literature [44][45][46][47]. In this case, we set…”

“…It is well known that the problem () can be solved by the LASSO problem: $$$$ \underset{x\in {\mathrm{\mathbb{R}}}^N}{\min}\frac{1}{2}{\left\Vert y- Ax\right\Vert}_2^2+\zeta {\left\Vert x\right\Vert}_1, $$$$ where $$$ \zeta >0 $$$. As a result, various techniques and iterative schemes have been developed over the years to solve the LASSO problem, see the literature [44–47]. In this case, we set $$$$ T{x}_n= pro{x}_{\zeta g}\left({x}_n-\zeta \nabla f\left({x}_n\right)\right), $$$$ where $$$ f(x)=\frac{1}{2}{\left\Vert y- Ax\right\Vert}_2^2,g(x)=\zeta {\left\Vert x\right\Vert}_1 $$$ and $$$ \zeta \in \left(0,\frac{2}{{\left\Vert A\right\Vert}_2^2}\right) $$$.…”

A Modified Inertial Parallel Viscosity-Type Algorithm for a Finite Family of Nonexpansive Mappings and Its Applications

“…T l×1 with l = 50, 100. For simplicity, the proposed method (15) with M ≤ 3 and the nonexpansive mapping T are chosen from T WJ , T SOR , and T GS , and f (u) = u. The results of the WJ, GS, SOR, and proposed methods are given for the following cases: These are demonstrated and discussed for solving the linear system (16).…”

“…where λ > 0. As a result, various techniques and iterative schemes have been developed over the years to solve the LASSO problem; see [15,16]. In this case, we set…”

“…Recently, there has been some research involving the parallel method for solving many problems. It is shown that the method can be applied in real-world problems such as image deblurring and related applications [15,16].…”