A new block preconditioner for weighted Toeplitz regularized least-squares problems

References

• N. Aghazadeh, M. Bastani, and D.K. Salkuyeh, Generalized Hermitian and skew-Hermitian splitting iterative method for image restoration, Appl. Math. Model. 39 (2015), pp. 6126–6138.
   Web of Science ®Google Scholar
• H. Aminikhah and M. Yousefi, A special generalized HSS method for discrete ill-posed problems, Comput. Appl. Math. 37 (2018), pp. 1507–1523.
   Web of Science ®Google Scholar
• H. Andrews and B. Hunt, Digital Image Restoration, Prentice-Hall, Englewood Cliffs, NJ, 1977.
   Google Scholar
• Z.-Z. Bai, G.H. Golub, and M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003), pp. 603–626.
   Web of Science ®Google Scholar
• Z.-Z. Bai, G.H. Golub, and C.-K. Li, Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput. 28 (2006), pp. 583–603.
   Web of Science ®Google Scholar
• Z.-Z. Bai, M.K. Ng, and Z.Q. Wang, Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl. 31 (2009), pp. 410–433.
   Web of Science ®Google Scholar
• Z.-Z. Bai and G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27 (2007), pp. 1–23.
   Web of Science ®Google Scholar
• F.P.A. Beik, S. Ahmadi-Asl, and A. Ameri, On the iterative refinement of the solution of ill-conditioned linear system of equations, Int. J. Comput. Math. 95 (2018), pp. 427–443.
   Web of Science ®Google Scholar
• M. Benzi, G.H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer. 14 (2005), pp. 1–137.
   Google Scholar
• M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl. 31 (2009), pp. 360–374.
   Web of Science ®Google Scholar
• M. Benzi and C. Faccio, Solving linear systems of the form (A+γUUT)x=b by preconditioned iterative methods, SIAM J. Sci. Comput. (2023), pp. S51–S70.
   Google Scholar
• M. Benzi and G.H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl. 26 (2004), pp. 20–41.
   Web of Science ®Google Scholar
• M. Benzi and M.K. Ng, Preconditioned iterative methods for weighted Toeplitz least squares problems, SIAM J. Matrix Anal. Appl. 27 (2006), pp. 1106–1124.
   Web of Science ®Google Scholar
• M. Benzi and M.A. Olshanskii, An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28 (2006), pp. 2095–2113.
   Web of Science ®Google Scholar
• Z.-H. Cao, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math. 57 (2007), pp. 889–990.
   Web of Science ®Google Scholar
• Y. Cao, M.-Q. Jiang, and Y.-L. Zheng, A splitting preconditioner for saddle point problems, Numer. Linear Algebra Appl. 18 (2011), pp. 875–895.
   Web of Science ®Google Scholar
• Y. Cao, J. Du, and Q. Niu, Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math. 272 (2014), pp. 239–250.
   Web of Science ®Google Scholar
• C. Chen and C. Ma, A generalized shift-splitting preconditioners for nonsymmetric saddle point problems, Appl. Math. Lett. 43 (2015), pp. 49–55.
   Web of Science ®Google Scholar
• G.-H. Cheng, X. Rao, and X.-G. Lv, The comparisons of two special Hermitian and skew-Hermitian splitting methods for image restoration, Appl. Math. Model. 39 (2015), pp. 1275–1280.
   Web of Science ®Google Scholar
• J.-J. Cui, G.-H. Peng, L. Quan, and Z.-G. Huang, Accelerated GNHSS iterative method for weighted Toeplitz regularized least-squares problems from image restoration, Comput. Appl. Math. 37 (2018), pp. 6152–6175.
   Web of Science ®Google Scholar
• J.-J. Cui, G.-H. Peng, L. Quan, and Z.-G. Huang, Modified special HSS method for discrete ill-posed problems and image restoration, Int. J. Comput. Math. 97 (2020), pp. 739–758.
   Web of Science ®Google Scholar
• E. de Sturler and J. Liesen, Block-diagonal and constraint preconditioners for nonsymmetric indefinite linear systems, SIAM J. Sci. Comput. 26 (2005), pp. 1598–1619.
   Web of Science ®Google Scholar
• H.S. Dollar, Constraint-style preconditioners for regularized saddle point problems, SIAM J. Matrix Anal. Appl. 29 (2007), pp. 672–684.
   Web of Science ®Google Scholar
• H.-T. Fan, M. Bastani, B. Zheng, and X.-Y. Zhu, A class of upper and lower triangular splitting iteration methods for image restoration, Appl. Math. Model. 54 (2018), pp. 82–111.
   Web of Science ®Google Scholar
• J.A. Fessler and S.D. Booth, Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction, IEEE Trans. Image Process. 8 (1999), pp. 688–699.
   PubMed Web of Science ®Google Scholar
• G.H. Golub and C. Greif, On solving block-structured indefinite linear systems, SIAM J. Sci. Comput. 24 (2003),pp. 2076–2092.
   Web of Science ®Google Scholar
• P.C. Hansen, J.G. Nagy, and D.P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM, Philadelphia, 2006.
   Google Scholar
• A. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989.
   Google Scholar
• A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. Comput. 19 (1998), pp. 172–184.
   Web of Science ®Google Scholar
• L.-D. Liao and G.-F. Zhang, New variant of the HSS iteration method for weighted Toeplitz regularized least-squares problems from image restoration, Comput. Math. Appl. 73 (2017), pp. 2482–2499.
   Web of Science ®Google Scholar
• X.-G. Lv, T.-Z. Huang, Z.-B. Xu, and X.-L. Zhao, A special Hermitian and skew-Hermitian splittingmethod for image restoration, Appl. Math. Model. 37 (2013), pp. 1069–1082.
   Web of Science ®Google Scholar
• M.K. Ng, Iterative methods for Toeplitz systems, in Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004.
   Google Scholar
• M.K. Ng and J.-Y. Pan, Weighted Toeplitz regularized least squares computation for image restoration, SIAM J. Sci. Comput. 36 (2014), pp. B94–B121.
   Web of Science ®Google Scholar
• J. Nocedal and S.J. Wright, Numerical Optimization, 2nd ed., Springer, New York, 2006.
   Google Scholar
• J. Pestana, On the eigenvalues and eigenvectors of block triangular preconditioned block matrices, SIAM J. Matrix Anal. Appl. 35 (2014), pp. 517–525.
   Web of Science ®Google Scholar
• J. Scott and M. Tůma, Sparse stretching for solving sparse-dense linear least-squares problems, SIAM J. Sci. Comput. 41 (2019), pp. A1604–A1625.
   Web of Science ®Google Scholar
• V. Simoncini, Block triangular preconditioners for symmetric saddle-point problems, Appl. Numer. Math. 49 (2004), pp. 63–80.
   Web of Science ®Google Scholar
• M.K. Zak and F. Toutounian, A shifted nested splitting iterative method with applications to ill-posed problems and image restoration, Comput. Math. Appl. 71 (2016), pp. 213–223.
   Web of Science ®Google Scholar
• M.-L. Zeng, A circulant-matrix-based new accelerated GSOR preconditioned method for block two-by-two linear systems from image restoration problems, Appl. Numer. Math. 164 (2021), pp. 245–257.
   Web of Science ®Google Scholar