Perturbed fixed point iterative methods based on pattern structure

Abstract

We introduce a new idea of algorithmic structure, called assigning algorithm, using a finite collection of a subclass of strictly quasi‐nonexpansive operators. This new algorithm allows the iteration vectors to take steps on a pattern which is based on a connected directed acyclic graph. The sequential, simultaneous, and string‐averaging methods for solving convex feasibility problems are the special cases of the new algorithm which may be used to reduce idle time of processors in parallel implementations. We give a convergence analysis for such algorithmic structure with perturbation. Also, we extend some existence results of the split common fixed point problem based on the new algorithm. The performance of the new algorithm is illustrated with numerical examples from computed tomography.

Superiorization of block accelerated cyclic subgradient methods

“…Φ-class is closed under composition and convex combination of operators. This property is very important because it allows to use Φ-class operators in different algorithmic structure such as (block) sequential, (block) simultaneous iterative methods, string averaging and recently introduced pattern structures [18].…”