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Levenberg–Marquardt method in Banach spaces with general convex regularization terms

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Abstract

We propose a Levenberg–Marquardt method with general uniformly convex regularization terms to solve nonlinear inverse problems in Banach spaces, which is an extension of the scheme proposed by Hanke in (Inverse Probl 13:79–95, 1997) in Hilbert space setting. The method is so designed that it can be used to detect the features of the sought solutions such as sparsity or piecewise constancy. It can also be used to deal with the situation that the data is contaminated by noise containing outliers. By using tools from convex analysis in Banach spaces, we establish the convergence of the method. Numerical simulations are reported to test the performance of the method.

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Acknowledgments

Q Jin is partly supported by the ARC discovery project grant DP150102345 and H Yang is partly supported by the Natural Science Foundation of China under grant 11071264 and 11571386.

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Authors and Affiliations

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

    Qinian Jin

  2. Guangdong Provincial Key Lab of Computational Science, School of Data and Computer Science, Sun Yat-sen University, Guangzhou, 510275, China

    Hongqi Yang

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  1. Qinian Jin
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  2. Hongqi Yang
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Correspondence to Qinian Jin.

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Jin, Q., Yang, H. Levenberg–Marquardt method in Banach spaces with general convex regularization terms. Numer. Math. 133, 655–684 (2016). https://doi.org/10.1007/s00211-015-0764-z

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  • DOI: https://doi.org/10.1007/s00211-015-0764-z

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