Efficient HSS-based preconditioners for generalized saddle point problems

Abstract

A fast iteration method based on HSS is proposed for solving the nonsymmetric generalized saddle point problem. It converges to the unique solution of the generalized saddle point problem unconditionally. We devise a new preconditioner induced by the new iteration method. We analyze the spectrum of the preconditioned coefficient matrix, and reveal the relation between the theoretically required number of iteration steps and the dimension of the preconditioned Krylov subspace. Furthermore, some practical inexact variants of the new preconditioner have been developed to reduce the computational overhead. Numerical experiments validate the effectiveness of the proposed preconditioners.

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Notes

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    In this case, EVHSS-preconditioned GMRES uses two full restarts (each of 10 inner steps) and a restart with just 1 inner step. Thus, the total number of inner steps is \((3-1)\times 10+1=21\).

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 11601323,11801362) and the Key Discipline Fund (Multivariate Statistical Analysis 2018-2019) of College of Arts and Sciences in Shanghai Maritime University. The authors would like to thank Dr. Ju-Li Zhang of Shanghai University of Engineering Science for helpful discussions. The first author also would like to thank Professor Jun-Feng Yin for gracious invitation and hospitality during his visit to Tongji University.

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Correspondence to Ke Zhang.

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Zhang, K., Wang, L. Efficient HSS-based preconditioners for generalized saddle point problems. Comp. Appl. Math. 39, 154 (2020). https://doi.org/10.1007/s40314-020-01180-0

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Keywords

  • Generalized saddle point problem
  • Preconditioner
  • Hermitian and skew-Hermitian splitting
  • Krylov subspace method

Mathematics Subject Classification

  • 65F10
  • 65N22