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Modified parameterized inexact Uzawa method for singular saddle-point problems

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Abstract

As well known, each of the consistent singular saddle-point (CSSP) problems has more than one solutions, and most of the iteration methods can only be proved to converge to one of the solutions of the CSSP problem. However, we do not know which solution it is and whether this solution depends on the initial iteration guesses. In this work, we introduce a new iteration method by slightly modifying the parameterized inexact Uzawa (PIU) iteration scheme. Theoretical analysis shows that, under suitable restrictions on the involved iteration parameters, the iteration sequence produced by the new method converges to the solution \(\mathcal {A}^{\dag }b\) for any initial guess, no matter the singular saddle-point system \(\mathcal {A}~x=b\) is consistent or inconsistent, where \(\mathcal {A}^{\dag }\) denotes the Moore-Penrose inverse of matrix \(\mathcal {A}\). In addition, the quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor are determined. Numerical examples are given to verify the correctness of the theoretical results and the effectiveness of our new method.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Yan Dou, Ai-Li Yang & Yu-Jiang Wu

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  1. Yan Dou
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  2. Ai-Li Yang
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  3. Yu-Jiang Wu
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Correspondence to Ai-Li Yang.

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Dou, Y., Yang, AL. & Wu, YJ. Modified parameterized inexact Uzawa method for singular saddle-point problems. Numer Algor 72, 325–339 (2016). https://doi.org/10.1007/s11075-015-0046-y

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  • DOI: https://doi.org/10.1007/s11075-015-0046-y

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