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Accelerating the HS-type Richardson Iteration Method with Anderson Mixing

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Abstract

The Accelerated Hermitian/skew-Hermitian type Richardson (AHSR) iteration methods are presented for solving non-Hermitian positive definite linear systems with three schemes, by using Anderson mixing. The upper bounds of spectral radii of iteration matrices are studied, and then the convergence theories of the AHSR iteration methods are established. Furthermore, the optimal iteration parameters are provided, which can be computed exactly. In addition, the application to the model convection-diffusion equation is depicted and numerical experiments are conducted to exhibit the effectiveness and confirm the theoretical analysis of the AHSR iteration methods.

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References

  1. Bai, Z. Z., Golub, G. H., Ng, M. K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl., 24(3), 603–626 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z. Z., Golub, G. H., Pan, J. Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math., 98(1), 1–32 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartels, R. H., Golub, G. H.: The simplex method of linear programming using LU decomposition. Commun. ACM, 12(5), 266–268 (1969)

    Article  MATH  Google Scholar 

  4. Brezinski, C., Redivo-Zaglia, M., Saad, Y.: Shanks sequence transformations and Anderson acceleration. SIAM Rev., 60(3), 646–669 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Davis, T. A.: Direct methods for sparse linear systems. SIAM, Philadelphia, 2006

    Book  MATH  Google Scholar 

  6. Eiermann, M., Niethammer, W., Varga, R. S.: Acceleration of relaxation methods for non-Hermitian linear systems. SIAM J. Matrix Anal. Appl., 13(3), 979–991 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. El Guennouni, A., Jbilou, K., Sadok, H.: A block version of BiCGstab for linear systems with multiple right-hand sides. Electron. Trans. Numer. Anal., 16(2), 129–142 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Elman, H. C., Golub, G. H.: Iterative methods for cyclically reduced nonselfadjoint linear systems. Math. Comput., 54(190), 671–700 (1990)

    MathSciNet  MATH  Google Scholar 

  9. Elman, H. C., Golub, G. H.: Iterative methods for cyclically reduced nonselfadjoint linear systems. II. Math. Comput., 56(193), 215–242 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Fischer, B., Reichel, L.: A stable Richardson iteration method for complex linear systems. Numer. Math., 54(2), 225–242 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ganine, V., Hills, N., Lapworth, B.: Nonlinear acceleration of coupled fluid—structure transient thermal problems by Anderson mixing. Int. J. Numer. Meth. Fl., 71(8), 939–959 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Golub, G. H., Overton, M. L.: The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems. Numer. Math., 53(5), 571–593 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Golub, G. H., Vanderstraeten, D.: On the preconditioning of matrices with skew-symmetric splittings. Numer. Algorithms, 25(1), 223–239 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Golub, G. H., Varga, R. S.: Chebyshev semi-iterative methods, successive overrelaxation iterative methods and second order Richardson iterative methods. Numer. Math., 3(1), 147–156 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goossens, S., Tan, K., Roose, D.: An efficient FGMRES solver for the shallow water equations based on domain decomposition, Domain Decomposition Methods in Sciences and Engineering, 350–358, 1998

  16. Greif, C., Varah, J.: Iterative solution of cyclically reduced systems arising from discretization of the three-dimensional convection-diffusion equation. SIAM J. Sci. Comput., 19(6), 1918–1940 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Greif, C., Varah, J.: Block stationary methods for nonsymmetric cyclically reduced systems arising from three-dimensional elliptic equations. SIAM J. Matrix Anal. Appl., 20(4), 1038–1059 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ho, N., Olson, S. D., Walker, H. F.: Accelerating the Uzawa algorithm. SIAM J. Sci. Comput., 39(5), S461–S476 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Krukier, L., Chikina, L., Belokon, T.: Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real linear system of equations. Appl. Numer. Math., 41(1), 89–105 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, C. X., Wu, S. L.: A single-step HSS method for non-Hermitian positive definite linear systems. Appl. Math. Lett., 44, 26–29 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, L., Huang, T. Z., Liu, X. P.: Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems. Comput. Math. Appl., 54(1), 147–159 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, L., Huang, T. Z., Liu, X. P.: Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems. Numer. Linear Algebra Appl., 14(3), 217–235 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, Z. Z., Chu, R. S., Zhang, H.: Accelerating the shift-splitting iteration algorithm. Appl. Math. Comput., 361, 421–429 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Morgan, R. B.: GMRES with deflated restarting. SIAM J. Sci. Comput., 24(1), 20–37 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mouzouris, G. C., Mendel, J. M.: Designing fuzzy logic systems for uncertain environments using a singular-value-QR decomposition method. Proc. 5th IEEE Int. Fuzzy Systems, 1, 295–301 (1996)

    Article  Google Scholar 

  26. Ni, P.: Anderson acceleration of fixed-point iteration with applications to electronic structure computations. Ph.D. thesis, Worcester Polytechnic Institute, 2009

  27. Pour, H. N., Goughery, H. S.: New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems. Numer. Algorithms, 69(1), 207–225 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Saad, Y.: Iterative methods for sparse linear systems, SIAM, Philadelphia, 2003

    Book  MATH  Google Scholar 

  29. Saad, Y., Schultz, M. H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 7(3), 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Toth, A., Kelley, C.: Convergence analysis for Anderson acceleration. SIAM J. Numer. Anal., 53(2), 805–819 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Walker, H. F., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal., 49(4), 1715–1735 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, A. L., Cao, Y., Wu, Y. J.: Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems. BIT, 59(1), 1–21 (2018)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing, 100049, P. R. China

    Zhi Zhi Li & Huai Zhang

  2. Guangdong Key Laboratory of Intelligent Information Processing, Shenzhen Key Laboratory of Media Security, and Guangdong Laboratory of Artificial Intelligence and Digital Economy (SZ), College of Electronics and Information Engineering, Shenzhen University, Shenzhen, 518060, P. R. China

    Le Ou-Yang

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  1. Zhi Zhi Li
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  2. Huai Zhang
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  3. Le Ou-Yang
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Correspondence to Le Ou-Yang.

Additional information

Supported by National Science Foundation of China (Grant Nos. 41725017 and 42004085), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110184) and the National Key R & D Program of the Ministry of Science and Technology of China (Grant Nos. 2020YFA0713400 and 2020YFA0713401)

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Li, Z.Z., Zhang, H. & Ou-Yang, L. Accelerating the HS-type Richardson Iteration Method with Anderson Mixing. Acta. Math. Sin.-English Ser. 38, 2069–2089 (2022). https://doi.org/10.1007/s10114-022-0665-x

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  • DOI: https://doi.org/10.1007/s10114-022-0665-x

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