Somyot Plubtieng

The split convex feasibility problem serves as a unified mathematical framework for solving inverse problems, medical image reconstruction, and signal recovery across multiple Hilbert spaces. Traditional iterative schemes, such as hybrid and shrinking projection algorithms, typically enforce strong convergence by projecting the initial point onto the intersection of successive half-spaces. However, this strategy introduces a severe computational bottleneck due to the necessity of solving constrained quadratic subproblems at every iteration. In this paper, we propose a novel Parallel Moving Point Projection (PMPP) algorithm designed to overcome this limitation. The proposed framework integrates a parallel projection architecture across multiple bounded linear operators with a dynamic moving point projection mechanism, where projections are performed onto the intersection of two adaptively updated half-spaces, which can be computed in explicit closed form. Under mild and standard assumptions on the control parameters, we establish a strong convergence theorem for the generated sequence in real Hilbert spaces without requiring any compact embedding. Furthermore, to validate the theoretical framework and computational efficiency of our approach, we perform extensive numerical experiments on both synthetic benchmark problems in finite-dimensional Euclidean spaces and color image restoration tasks corrupted by motion blur, average blur, and additive Gaussian noise. The experimental results demonstrate that the proposed PMPP algorithm achieves competitive execution times, lower iteration counts in some cases, and higher restoration quality in terms of signal-to-noise ratio (SNR), improvement in signal-to-noise ratio (ISNR), and structural similarity index measure (SSIM) compared to prominent existing parallel hybrid projection methods.

This paper introduces a novel inertial forward–backward–forward algorithm driven by a newly conceptualized moving point projection technique for solving monotone inclusion problems in real Hilbert spaces. By leveraging the properties of a Lipschitz continuous, monotone operator and a maximally monotone operator alongside this innovative projection strategy, we dynamically construct a sequence of nonempty, closed, and convex sets that contain the zeros of the sum of the two operators. This geometric construction ensures that the resulting sequence is well defined and guarantees its weak convergence to a solution. Furthermore, to validate the practical efficacy of the proposed theoretical framework, we evaluate our method on image restoration problems. Numerical experiments measuring the improvement in signal-to-noise ratio (ISNR) and the structural similarity index measure (SSIM) confirm that the proposed algorithm is highly efficient and significantly outperforms existing state-of-the-art methods.

In this work, we propose and study an inertial Krasnosel’ski˘ı-Mann iterative algorithm with step-size parameters involving nonexpansive mapping to find a solution of a fixed point problem of a nonex- pansive mapping in the frame work of Hilbert spaces. Strong convergence of the new proposed algorithm is proved under some useful properties of a nonexpansive mapping and inequalities on real Hilbert spaces to- gether with the appropriate conditions of scalar controls without relying on the concept of viscosity methods. For the applications, we employ the obtained results to find a zero point of some monotone inclusion problems and then apply to solve image restoration problems. For representing the advantage of the new algorithm, the signal to noise ratio (SNR) with various types of blurring operators and some numerical experiments are pre- sented to compare and illustrate the behavior of the new algorithm with numerical results of some previous existing algorithms.

In this research paper, we propose a novel approach termed the inertial subgradient extragradient algorithm to solve bilevel system equilibrium problems within the realm of real Hilbert spaces. Our algorithm is capable of circumventing the necessity for prior knowledge about the Lipschitz constant of the involving bifunction and only computes the minimization of strong bifunctions onto the feasible set that is required. Under appropriate conditions, we establish strong convergence theorems for our proposed algorithms. To validate our algorithms, we illustrate a series of numerical examples. Through these examples, we demonstrate the performance of the algorithms we have put forth in this paper.

In this paper, we introduce and study a modified Mann-type algorithm that combines inertial terms for solving common fixed point problems of two countable families of nonexpansive mappings in Hilbert spaces. Under appropriate assumptions on the sequences of parameters, we establish a strong convergence result for the sequence generated by the proposed method in finding a common fixed point of two countable families of nonexpansive mappings. This method can be applied to solve the monotone inclusion problem. Additionally, we employ a modified Mann-type iterative algorithm to address image restoration problems. Furthermore, we present numerical results across different scenarios to demonstrate the superior efficiency of our algorithm compared to existing algorithms.