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Bounded perturbations resilient iterative methods for linear systems and least squares problems: operator-based approaches, analysis, and performance evaluation

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Abstract

We examine some bounded perturbations resilient iterative methods for addressing (constrained) consistent linear systems of equations and (constrained) least squares problems. We introduce multiple frameworks rooted in the operator of the Landweber iteration, adapting the operators to facilitate the minimization of absolute errors or residuals. We demonstrate that our operator-based methods exhibit comparable speed to powerful methods like CGLS, and we establish that the computational cost of our methods is nearly equal to that of CGLS. Furthermore, our methods possess the capability to handle constraints (e.g. non-negativity) and control the semi-convergence phenomenon. In addition, we provide convergence analysis of the methods when the current iterations are perturbed by summable vectors. This allows us to utilize these iterative methods for the superiorization methodology. We showcase their performance using examples drawn from tomographic imaging and compare them with CGLS, superiorized conjugate gradient (S-CG), and the non-negative flexible CGLS (NN-FCGLS) methods.

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Acknowledgements

We thank anonymous reviewers for their detailed and constructive comments which much improved the paper. The second author wishes to thank the Isaac Newton Institute for Mathematical Sciences and the London Mathematical Society for the financial support and the School of Engineering at Newcastle University for their hospitality.

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Authors and Affiliations

  1. Department of Mathematics, University of Qom, Isfahan Old Road, Qom, 37161-46611, Iran

    Mokhtar Abbasi

  2. School of Mathematics and Computer Science, Iran University of Science and Technology, Tehran, 16846-13114, Iran

    Touraj Nikazad

  3. School of Engineering, Newcastle University, Newcastle Upon Tyne, Newcastle, NE1 7RU, UK

    Touraj Nikazad

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  1. Mokhtar Abbasi
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  2. Touraj Nikazad
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Correspondence to Mokhtar Abbasi.

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Communicated by Rosemary Anne Renaut.

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Abbasi, M., Nikazad, T. Bounded perturbations resilient iterative methods for linear systems and least squares problems: operator-based approaches, analysis, and performance evaluation. Bit Numer Math 64, 15 (2024). https://doi.org/10.1007/s10543-024-01015-y

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