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Regularization of inverse problems by two-point gradient methods in Banach spaces

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Abstract

In this paper, we propose and analyze a two-point gradient method for solving inverse problems in Banach spaces which is based on the Landweber iteration and an extrapolation strategy. The method allows to use non-smooth penalty terms, including the \(L^1\)-like and the total variation-like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy in practical applications. The design of the method involves the choices of the step sizes and the combination parameters which are carefully discussed. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

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Acknowledgements

The work of M. Zhong is supported by the National Natural Science Foundation of China (Nos. 11871149, 11671082) and and supported by Zhishan Youth Scholar Program of SEU. The work of W. Wang is partially supported by the National Natural Science Foundation of China (No. 11871180) and Natural Science Foundation of Zhejiang Province (No. LY19A010009). The work of Q Jin is partially supported by the Future Fellowship of the Australian Research Council (FT170100231).

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, People’s Republic of China

    Min Zhong

  2. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, Zhejiang, People’s Republic of China

    Wei Wang

  3. Mathematical Sciences Institute, Australian National University, Canberra, ACT, 2601, Australia

    Qinian Jin

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  1. Min Zhong
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  2. Wei Wang
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  3. Qinian Jin
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Correspondence to Wei Wang.

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Zhong, M., Wang, W. & Jin, Q. Regularization of inverse problems by two-point gradient methods in Banach spaces. Numer. Math. 143, 713–747 (2019). https://doi.org/10.1007/s00211-019-01068-0

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