An improved fast iterative shrinkage thresholding algorithm with an error for image deblurring problem

Pattanapong Tianchai

doi:10.1186/s13663-021-00703-6

Abstract

In this paper, we introduce a new iterative forward-backward splitting method with an error for solving the variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters such that another strong convergence theorem for these problem is obtained. We also apply our main result to improve the fast iterative shrinkage thresholding algorithm (IFISTA) with an error for solving the image deblurring problem. Finally, we provide numerical experiments to illustrate the convergence behavior and show the effectiveness of the sequence constructed by the inertial technique to the fast processing with high performance and the fast convergence with good performance of IFISTA.

Highlights

Let C be a nonempty closed convex subset of a real Hilbert space H
We focus attention on the inertial parameter θn which controls the momentum of xn – xn–1 in the fast iterative shrinkage thresholding algorithm (FISTA)
The outline of our research is as follows: in Sect. 2, we give some well-known definitions and lemmas which are used in Sect. 3 to prove the strong convergence theorem of improve the fast iterative shrinkage thresholding algorithm (IFISTA) for solving the variational inclusion problem (1.1), and we apply its result in Sect. 4 for solving the image deblurring problem, which is a special case of convex minimization problem (1.2); and in Sect. 5, we provide numerical experiments to illustrate the fast processing with high performance and the fast convergence with good performance of IFISTA by the inertial technique

Summary

Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H. Many researchers have proposed and analyzed the iterative shrinkage thresholding algorithms for solving the convex minimization problem (1.2) under a few specific conditions as follows. We introduce a new iterative forward-backward splitting method with an error for solving the variational inclusion problem (1.1) as follows:. It can be applied to improve the fast iterative shrinkage thresholding algorithm (IFISTA) with an error for solving the convex minimization problem (1.2) by letting A = ∇F and B = ∂G as follows:. Lemma 2.5 ([29] (Demiclosedness principle)) Let C be a nonempty closed convex subset of a real Hilbert space H, and let S : C → C be a nonexpansive mapping with Fix(S) = ∅.

Main result

IFISTA

Conclusion