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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 ${ℋ}_{1}$ such that ${x}^{*}\in C$ and $A{x}^{*}\in Q$. Here $C$ and $Q$ are nonempty closed convex subsets of ${ℋ}_{1}$, and $A$ is a bounded linear operator from ${ℋ}_{1}$ into another infinite-dimensional Hilbert space ${ℋ}_{2}$. 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.

##### MSC:
 65J22 Inverse problems (numerical methods in abstract spaces) 47J25 Iterative procedures (nonlinear operator equations) 47J06 Nonlinear ill-posed problems 49N45 Inverse problems in calculus of variations 15A29 Inverse problems in matrix theory