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An efficient Gauss–Newton algorithm for solving regularized total least squares problems

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Abstract

The total least squares (TLS) method is a well-known technique for solving an overdetermined linear system of equations Ax ≈ b, that is appropriate when both the coefficient matrix A and the right-hand side vector b are contaminated by some noise. For ill-posed TLS poblems, regularization techniques are necessary to stabilize the computed solution; otherwise, TLS produces a noise-dominant output. In this paper, we show that the regularized total least squares (RTLS) problem can be reformulated as a nonlinear least squares problem and can be solved by the Gauss–Newton method. Due to the nature of the RTLS problem, we present an appropriate method to choose a good regularization parameter and also a good initial guess. Finally, the efficiency of the proposed method is examined by some test problems.

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Notes

  1. Global convergence means convergence to a local minimum from any initial point [22].

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Acknowledgements

The authors would like to express their heartfelt thanks to the editor and anonymous referees for their useful comments and constructive suggestions which substantially improved the quality and presentation of this paper.

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Authors and Affiliations

  1. Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran

    Hossein Zare

  2. Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

    Masoud Hajarian

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  1. Hossein Zare
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  2. Masoud Hajarian
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Correspondence to Masoud Hajarian.

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Zare, H., Hajarian, M. An efficient Gauss–Newton algorithm for solving regularized total least squares problems. Numer Algor 89, 1049–1073 (2022). https://doi.org/10.1007/s11075-021-01145-2

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  • DOI: https://doi.org/10.1007/s11075-021-01145-2

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