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Golub–Kahan bidiagonalization for ill-conditioned tensor equations with applications

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Abstract

This paper is concerned with the solution of severely ill-conditioned linear tensor equations. These kinds of equations may arise when discretizing partial differential equations in many space-dimensions by finite difference or spectral methods. The deblurring of color images is another application. We describe the tensor Golub–Kahan bidiagonalization (GKB) algorithm and apply it in conjunction with Tikhonov regularization. The conditioning of the Stein tensor equation is examined. These results suggest how the tensor GKB process can be used to solve general linear tensor equations. Computed examples illustrate the feasibility of the proposed algorithm.

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Notes

  1. All computations for this section were carried out on a 64-bit 2.50-GHz core i5 processor with 8.00-GB RAM using MATLAB version 9.4 (R2018a).

  2. http://personalpages.manchester.ac.uk/staff/d.h.foster

  3. The image is available at https://www.hlevkin.com/TestImages/Boats.ppm

References

  1. Bader, B.W, Kolda, T.G.: MATLAB tensor toolbox version 2.5. http://www.sandia.gov/tgkolda/TensorToolbox

  2. Ballani, J., Grasedyck, L.: A projection method to solve linear systems in tensor format. Numer. Linear Algebra Appl. 20, 27–43 (2013)

    Article  MathSciNet  Google Scholar 

  3. Beik, F.P.A., Movahed, F.S., Ahmadi-Asl, S.: On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations. Numer. Linear Algebra Appl. 23, 444–466 (2016)

    Article  MathSciNet  Google Scholar 

  4. Bentbib, A.H., El Guide, M., Jbilou, K., Reichel, L.: A global Lanczos method for image restoration. J. Comput. Appl. Math. 300, 233–244 (2016)

    Article  MathSciNet  Google Scholar 

  5. Bentbib, A.H., El Guide, M., Jbilou, K, Reichel, L.: Global Golub–Kahan bidiagonalization applied to large discrete ill-posed problems. J. Comput. Appl. Math. 322, 46–56 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bouhamidi, A., Jbilou, K., Reichel, L., Sadok, H.: A generalized global Arnoldi method for ill-posed matrix equations. J. Comput. Appl. Math. 236, 2078–2089 (2012)

    Article  MathSciNet  Google Scholar 

  7. Buccini, A., Pasha, M., Reichel, L.: Generalized singular value decomposition with iterated Tikhonov regularization. J. Comput. Appl. Math. 373, 112276 (2020)

    Article  MathSciNet  Google Scholar 

  8. Calvetti, D., Reichel, L.: Tikhonov regularization of large linear problems. BIT 43, 263–283 (2003)

    Article  MathSciNet  Google Scholar 

  9. Chen, Z., Lu, L.Z.: A projection method and Kronecker product preconditioner for solving Sylvester tensor equations. Sci. China Math. 55, 1281–1292 (2012)

    Article  MathSciNet  Google Scholar 

  10. Cichocki, A., Zdunek, R., Phan, A. H., Amari, S. I.: Nonnegative matrix and tensor factorizations: Applications to exploratory multi-way data analysis and blind source separation. John Wiley & Sons, Chichester (2009)

    Book  Google Scholar 

  11. Eldén, L.: Algorithms for the regularization of ill-conditioned least squares problems. BIT 17, 134–145 (1977)

    Article  MathSciNet  Google Scholar 

  12. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Kluwer, Dordrecht (1996)

    Book  Google Scholar 

  13. Golub, G.H., Van Loan, C.F.: Matrix computations. The Johns Hopkins University Press, Batimore (1996)

    MATH  Google Scholar 

  14. Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  15. Huang, G., Reichel, L., Yin, F.: On the choice of subspace for large-scale Tikhonov regularization problems in general form. Numer. Algorithms 81, 33–55 (2019)

    Article  MathSciNet  Google Scholar 

  16. Huang, B., Xie, Y., Ma, C.: Krylov subspace methods to solve a class of tensor equations via the Einstein product. Numer. Linear Algebra Appl. 26, e2254 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Kindermann, S.: Convergence analysis of minimization-based noise level-freeparameter choice rules for linear ill-posed problems. Electron. Trans. Numer. Anal. 38, 233–257 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Kindermann, S., Raik, K.: A simplified L-curve method as error estimator. Electron. Trans. Numer. Anal. 53, 217–238 (2020)

    Article  Google Scholar 

  19. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)

    Article  MathSciNet  Google Scholar 

  20. Liang, L., Zheng, B.: Sensitivity analysis of the Lyapunov tensor equation. Linear Multilinear Algebra 67, 555–572 (2019)

    Article  MathSciNet  Google Scholar 

  21. Malek, A., Bojdi, Z.K., Golbarg, P.N.N.: Solving fully three-dimensional microscale dual phase lag problem using mixed-collocation finite difference discretization. J. Heat Transf. 134, 094504 (2012)

    Article  Google Scholar 

  22. Malek, A., Masuleh, S.H.M.: Mixed collocation-finite difference method for 3D microscopic heat transport problems. J. Comput. Appl. Math. 217, 137–147 (2008)

    Article  MathSciNet  Google Scholar 

  23. Masuleh, S.H.M., Phillips, T.N.: Viscoelastic flow in an undulating tube using spectral methods. Comput. Fluids 33, 1075–1095 (2004)

    Article  Google Scholar 

  24. Najafi-Kalyani, M., Beik, F.P.A., Jbilou, K.: On global iterative schemes based on Hessenberg process for (ill-posed) Sylvester tensor equations. J. Comput. Appl. Math. 373, 112216 (2020)

    Article  MathSciNet  Google Scholar 

  25. Paige, C.C., Saunders, M.A.: LSQR: An algorithm for sparse linear equations and sparse least squares problems. ACM Trans. Math. Softw. 8, 43–71 (1982)

    Article  Google Scholar 

  26. Reichel, L., Shyshkov, A.: A new zero-finder for Tikhonov regularization. BIT 48, 627–643 (2008)

    Article  MathSciNet  Google Scholar 

  27. Sun, Y.S., Jing, M., Li, B.W.: Chebyshev collocation spectral method for three-dimensional transient coupled radiative-conductive heat transfer. J. Heat Transf. 134, 092701–092707 (2012)

    Article  Google Scholar 

  28. Xu, X., Wang, Q.-W.: Extending BiCG and BiCR methods to solve the Stein tensor equation. Comput. Math. Appl. 77, 3117–3127 (2019)

    Article  MathSciNet  Google Scholar 

  29. Zak, M.K., Toutounian, F.: Nested splitting conjugate gradient method for matrix equation AXB = C and preconditioning. Comput. Math. Appl. 66, 269–278 (2013)

    Article  MathSciNet  Google Scholar 

  30. Zak, M.K., Toutounian, F.: Nested splitting CG-like iterative method for solving the continuous Sylvester equation and preconditioning. Adv. Comput. Math. 40, 865–880 (2014)

    Article  MathSciNet  Google Scholar 

  31. Zak, M.K., Toutounian, F.: An iterative method for solving the continuous Sylvester equation by emphasizing on the skew-Hermitian parts of the coefficient matrices. Internat. J. Comput. Math. 94, 633–649 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank the anonymous referees for their suggestions and comments.

Funding

Research by LR was supported in part by NSF grants DMS-1720259 and DMS-1729509.

Author information

Authors and Affiliations

  1. Department of Mathematics, Vali-e-Asr University of Rafsanjan, PO Box 518, Rafsanjan, Iran

    Fatemeh P. A. Beik & Mehdi Najafi-Kalyani

  2. Laboratory LMPA, 50 Rue F. Buisson, ULCO calais cedex, Lyon, France

    Khalide Jbilou

  3. Laboratory CSEHS, University UM6P, Benguérir, Morocco

    Khalide Jbilou

  4. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    Lothar Reichel

Authors
  1. Fatemeh P. A. Beik
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  2. Khalide Jbilou
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  3. Mehdi Najafi-Kalyani
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  4. Lothar Reichel
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Correspondence to Lothar Reichel.

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Dedicated to Gérard Meurant on the occasion of his 70th birthday.

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Beik, F.P.A., Jbilou, K., Najafi-Kalyani, M. et al. Golub–Kahan bidiagonalization for ill-conditioned tensor equations with applications. Numer Algor 84, 1535–1563 (2020). https://doi.org/10.1007/s11075-020-00911-y

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