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Zbl 1213.65085
Xu, Hong-Kun
Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. (English)
[J] Inverse Probl. 26, No. 10, Article ID 105018, 17 p. (2010). ISSN 0266-5611

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 $\cal H_1$ such that $x^* \in C$ and $Ax^*\in Q$. Here $C$ and $Q$ are nonempty closed convex subsets of $\cal H_1$, and $A$ is a bounded linear operator from $\cal H_1$ into another infinite-dimensional Hilbert space $\cal H_2$. 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.
[Zhen Mei (Toronto)]

MSC 2000:: *65J22 Inverse problems
47J25 Methods for solving nonlinear operator equations (general)
47J06 Nonlinear ill-posed problems
49N45 Inverse problems in calculus of variations
15A29 Inverse problems in matrix theory

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

Cited in: Zbl 1253.65093

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