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

65J22Inverse problems (numerical methods in abstract spaces)
47J25Iterative procedures (nonlinear operator equations)
47J06Nonlinear ill-posed problems
49N45Inverse problems in calculus of variations
15A29Inverse problems in matrix theory
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