Modified Tseng’s Method with Inertial Viscosity Type for Solving Inclusion Problems and Its Application to Image Restoration Problems

Abstract

In this paper, we study a monotone inclusion problem in the framework of Hilbert spaces. (1) We introduce a new modified Tseng’s method that combines inertial and viscosity techniques. Our aim is to obtain an algorithm with better performance that can be applied to a broader class of mappings. (2) We prove a strong convergence theorem to approximate a solution to the monotone inclusion problem under some mild conditions. (3) We present a modified version of the proposed iterative scheme for solving convex minimization problems. (4) We present numerical examples that satisfy the image restoration problem and illustrate our proposed algorithm’s computational performance.

Fast hybrid iterative schemes for solving variational inclusion problems

“…In summary, We propose two modifications of Tseng's forward‐backward‐forward splitting method for solving variational inclusion problems and prove a strong convergence theorem for each of them. Our methods combine the inertial technique and viscosity approximation method with Tseng's forward‐backward‐forward splitting method. The unique feature of our methods compared to the existing inertial‐viscosity hybrid techniques in the literature (see, for example, [18–22]) is that we compute the inertial extrapolation and the viscosity approximation simultaneously and at the initial step of each iteration. We aim to introduce simple and accelerated strongly convergent modified forward‐backward splitting methods for solving variational inclusion problems in the framework of real Hilbert spaces. …”

“…Our methods combine the inertial technique and viscosity approximation method with Tseng's forward‐backward‐forward splitting method. The unique feature of our methods compared to the existing inertial‐viscosity hybrid techniques in the literature (see, for example, [18–22]) is that we compute the inertial extrapolation and the viscosity approximation simultaneously and at the initial step of each iteration.…”