Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. (English) Zbl 1213.65085

Xu, Hong-Kun

Inverse Probl. 26, No. 10, Article ID 105018, 17 p. (2010).

The author discusses iterative methods for solving the split feasibility problem (SFP), i.e., to find a point \(x^*\) in a infinite-dimensional Hilbert space \(\mathcal H_1\) such that \(x^* \in C\) and \(Ax^*\in Q\). Here \(C\) and \(Q\) are nonempty closed convex subsets of \(\mathcal H_1\), and \(A\) is a bounded linear operator from \(\mathcal H_1\) into another infinite-dimensional Hilbert space \(\mathcal H_2\). Such problems have been considered as models in phase retrievals, medical image reconstruction and recently also in the intensity-modulated radiation therapy.
In this paper the SFP problem is reformulated as a minimization problem to enable approximations via gradient-projection methods. Equivalently, the SFP problem is also considered as a fixed point equation so that iterative methods with adaptive relaxation factors can be used to approximate the solution as a fixed point. Both weak convergence to the original solution and strong convergence to the regularized solution are established.

Reviewer: Zhen Mei (Toronto)

MSC:

65J22	Numerical solution to inverse problems in abstract spaces
47J25	Iterative procedures involving nonlinear operators
47J06	Nonlinear ill-posed problems
49N45	Inverse problems in optimal control
15A29	Inverse problems in linear algebra

Keywords:

inverse problems; iteration methods; Hilbert space; fixed point iterations; gradient projection methods; split feasibility problem; bounded linear operator; weak convergence; strong convergence