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Novel Multi-level Projected Iteration to Solve Inverse Problems with Nearly Optimal Accuracy

Author

Listed:
  • Gaurav Mittal

    (Indian Institute of Technology Roorkee)

  • Ankik Kumar Giri

    (Indian Institute of Technology Roorkee)

Abstract

In this paper, we introduce a novel nonlinear projected iterative method to solve the ill-posed inverse problems in Banach spaces. This method is motivated by the well-known iteratively regularized Landweber iteration method. We analyze the convergence of our novel method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, we consider a nested family of convex and compact sets on which stability holds, and based on this family, we develop a multi-level algorithm with nearly optimal accuracy. To enhance the accuracy between neighboring levels, we couple the increase in accuracy with the growth of stability constants. This ensures that the algorithm terminates within a finite number of iterations after achieving a certain discrepancy criterion. Moreover, we discuss example of an ill-posed problem on which our both the methods are applicable and deduce various constants appearing in our work.

Suggested Citation

  • Gaurav Mittal & Ankik Kumar Giri, 2022. "Novel Multi-level Projected Iteration to Solve Inverse Problems with Nearly Optimal Accuracy," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 643-680, August.
    • Handle: RePEc:spr:joptap:v:194:y:2022:i:2:d:10.1007_s10957-022-02044-9
    DOI: 10.1007/s10957-022-02044-9
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    References listed on IDEAS

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    1. Mittal, Gaurav & Giri, Ankik Kumar, 2021. "Iteratively regularized Landweber iteration method: Convergence analysis via Hölder stability," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    2. Y. Alber & D. Butnariu, 1997. "Convergence of Bregman Projection Methods for Solving Consistent Convex Feasibility Problems in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 33-61, January.
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    Cited by:

    1. Mittal, Gaurav, 2024. "Nonstationary iterated frozen Tikhonov regularization with uniformly convex penalty terms for solving inverse problems," Applied Mathematics and Computation, Elsevier, vol. 468(C).

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