## 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 $$\mathcal H_1$$ 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
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