Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces
We study the influence of computational errors on the convergence to compact sets of orbits of nonexpansive mappings in Banach and metric spaces. We first establish a convergence theorem assuming that the computational errors are summable and then provide examples which show that the summability of errors is necessary for convergence.
XCT image reconstruction by a modified superiorized iteration and theoretical analysis
ABSTRACT In this paper, we propose an improved iteration superiorization method for X-ray computed tomography image reconstruction. We simplify the classic superiorized iteration by removing two constraints imposed on the perturbation. A novel method is proposed to determine the perturbation amount and direction for the superiorized iteration simultaneously. Some theoretical properties (convergence for instance) of the superiorized iteration sequence with the proposed perturbation are analysed. We present a general proof for the convergence of ART-like iterations with summable perturbations. In addition, we prove the convergence of simultaneous iterations without the summable perturbation assumption. Experiments on simulated and real data not only verify the theoretical result but also show that the proposed algorithm is superior to the classic superiorized iteration and can reconstruct desirable images.
Transience and Solvability of a Non-Linear Diffusion Equation
This paper is concerned with the existence of bounded solutions to an operator inequality which is a non-linear version of a discrete time diffusion equation. Here, the solvability of the inequality will be closely related to the transience of a corresponding random walk. In particular, the inequality will generally be solvable in three or more dimensions, but not in one or two dimensions if appropriate moment conditions hold. 1. The basic equation and initialization. Let Q be a probability measure