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Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces

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  • Yixuan Yang

    (Department of Mathematics, NanChang University, Nanchang 330031, China)

  • Yuchao Tang

    (Department of Mathematics, NanChang University, Nanchang 330031, China)

  • Chuanxi Zhu

    (Department of Mathematics, NanChang University, Nanchang 330031, China)

Abstract

The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator. Second, we propose a fixed point approach for computing this resolvent operator in a general case. Based on the Krasnoselskii–Mann algorithm for finding fixed points of non-expansive operators, we prove the strong convergence of the sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator, which has wide applications in image restoration and image reconstruction problems. Furthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper, lower semi-continuous convex functions.

Suggested Citation

  • Yixuan Yang & Yuchao Tang & Chuanxi Zhu, 2019. "Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces," Mathematics, MDPI, vol. 7(2), pages 1-16, February.
  • Handle: RePEc:gam:jmathe:v:7:y:2019:i:2:p:131-:d:202437
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    References listed on IDEAS

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    3. Abdellatif Moudafi, 2014. "Computing the resolvent of composite operators," Documents de Travail 2014-02, CEREGMIA, Université des Antilles et de la Guyane.
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