Global and extended global Hessenberg processes for solving Sylvester tensor equation with low-rank right-hand side

Iranian Journal of Numerical Analysis and Optimization

مقاله 9، دوره 12، Issue 3 (Special Issue) - On the occasion of the 75th birthday of Professor A. Vahidian and Professor F. Toutounian - شماره پیاپی 23، بهمن 2022، صفحه 658-679   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2022.78966.1186

نویسندگان

T. Cheraghzadeh¹؛ F. Toutounian^{* 2}؛ R. Khoshsiar Ghaziani¹

¹Department of Applied Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord, Iran.

²Department of Applied Mathematics, Faculty of Mathematical Science, The Center of Excellence on Modeling and Control Systems, Ferdowsi University of Mashhad, Iran.

چکیده

In this paper, we introduce two new schemes based on the global Hessen-berg processes for computing approximate solutions to low-rank Sylvester tensor equations. We first construct bases for the matrix and extended matrix Krylov subspaces by applying the global and extended global Hes-senberg processes. Then the initial problem is projected into the matrix or extended matrix Krylov subspaces with small dimensions. The reduced Sylvester tensor equation obtained by the projection methods can be solved by using a recursive blocked algorithm. Furthermore, we present the upper bounds for the residual tensors without requiring the computation of the approximate solutions in any iteration. Finally, we illustrate the perfor-mance of the proposed methods with some numerical examples.
 

کلیدواژه‌ها

Low-rank Sylvester tensor equation؛ Global Hessenberg process؛ Extended Global Hessenberg process؛ CP decomposition

مراجع

1. Addam, M., Elbouyahyaoui, L. and Heyouni, M. On Hessenberg type methods for low-rank Lyapunov matrix equations, Applicationes Mathematicae 45 (2018) 255–273.
2. Bader, B.W. and Kolda, T.G. MATLAB tensor toolbox version 3.2. http://www.sandia.gov/tgkolda/TensorToolbox/.
3. Ballani, J. and Grasedyck, L. A projection method to solve linear systems in tensor format, Numer. Linear Algebra Appl. 20 (1) (2013) 27–43.
4. Beik, F.P.A., Saberi Movahed, F. and Ahmadi-Asl, S. On the krylov subspace methods based on tensor format for positive definite Sylvester tensor equations, Numer. Linear Algebra Appl. 23 (3) (2016) 444–466.
5. Bentbib, A.H., El-Halouy, S. and Sadek, El M. Krylov subspace projection method for Sylvester tensor equation with low rank right-hand side, Numer. Algorithms 84 (4) (2020) 1411–1430.
6. Bentbib, A.H., El-Halouy, S. and Sadek, El M. Extended Krylov subspace methods for solving Sylvester and Stein tensor equations, Discrete Continuous Dyn. Syst.-s 15 (1) (2022) 41–56.
7. Bouyouli, R., Jbilou, K., Sadaka, R. and Sadok, H. Convergence properties of some block Krylov subspace methods for multiple linear systems, J. Comp. Appl. Math. 196 (2) (2006) 498–511.
8. Chang, K.C., Pearson, K. and Zhang, T. Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci. 6 (2) (2008) 507–520.
9. Chen, Z. and Lu, L.Z. A projection method and Kronecker product preconditioner for solving Sylvester tensor equations, Sci. China Math. 55 (6) (2012) 1281–1292.

10. Chen, Z. and Lu, L.Z. A gradient based iterative solutions for Sylvester tensor equations, Math. Probl. Eng. (2013) Article ID 819479, 7 pp.
11. Chen, M. and Kressner, D. Recursive blocked algorithms for linear systems with Kronecker product structure, Numer. Algorithms 84 (3) (2020) 1199–1216.
12. Cheraghzadeh, T., Khoshsiar Ghaziani, R. and Toutounian, F. Projection schemes based on Hessenberg process for Sylvester tensor equation with low-rank right-hand side, Comput. Appl. Math. 41 (2022) Article number 311.
13. Colda, T.G. and Bader, B.W. Tensor decompositions and applications, SIAM Rev. 51 (3) (2009) 455–500.
14. Cichocki, A., Zdunek, R., Phan, A.H. and Amari, S.-I. Nonnegative matrix and tensor factorizations: Applications to exploratory multi-way data analysis and blind source separation, John Wiley and Sons, 2009.
15. Dehdezi, E.K. and Karimi, S. Extended conjugate gradient squared and conjugate residual squared methods for solving the generalized coupled Sylvester tensor equations, T. I. Meas. Control. 9(4)(2021) 645–664.
16. Dehdezi, E.K. and Karimi, S. A gradient based iterative method and associated preconditioning technique for solving the large multilinear systems, Calcolo. 58(4) (2021) 1–19.
17. Heyouni, M. The global Hessenberg and global CMRH methods for linear systems with multiple right-hand sides, Numer. Algorithms 26 (4) (2001) 317–332.
18. Heyouni, M. Extended Arnoldi methods for large low-rank Sylvester matrix equations, Appl. Numer. Math. 60 (11) (2010) 1171–1182.
19. Heyouni, M. and Essai, A. Matrix Krylov subspace methods for linear systems with multiple right-hand sides, Numer. Algorithms 40 (2) (2005) 137–156.
20. Heyouni M. and Jbilou, K. An extended block Arnoldi algorithm for largescale solution of the continuous-time algebraic Riccati equation, Electron. Trans. Numer. Anal. 33 (2009) 53–62.
21. Heyouni, M., Saberi-Movahed, F. and Tajaddini, A. A tensor format for the generalized Hessenberg method for solving Sylvester tensor equations, J. Comput. Appl. Math. 377 (2020) 112878.
22. Huang, B., Xie, Y. and Ma, C. Krylov subspace methods to solve a class of tensor equations via the Einstein product, Numer. Linear Algebra Appl. 26 (2019) e2254.

23. Jbilou, K., Messaoudi, A. and Sadok, H. Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math. 31 (1) (1999) 49–63.
24. Karimi, S. and Dehghan, M. Global least squares method based on tensor form to solve linear systems in Kronecker format, Trans. Inst. Measur. Contr. 40 (7) (2018) 2378–2386.
25. Kressner, D. and Tobler, C. Low-rank tensor Krylov subspace methods for parametrized linear systems, SIAM J. Matrix Anal. Appl. 32 (4) (2011) 1288–1316.
26. Lee, N. and Cichocki, A. Fundamental tensor operations for large-scale data analysis using tensor network formats, Multidim. Syst. Sign. Process. 29 (3) (2018) 921–960.
27. Li, B.W., Tian, S., Sun, Y.S. and Hu, Z.M. Schur decomposition for 3d matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method, J. Comput. Phys. 229 (4) (2010) 1198–1212 .
28. Malek, A. and Momeni-Masuleh, S.H. A mixed collocation finite difference method for 3d microscopic heat transport problems, J. Comput. Appl. Math. 217 (1) (2008) 137–147.
29. Penzl, T. et al., A Matlab toolbox for large Lyapunov and Riccati equations, model reduction problems, and linear quadratic optimal control problems, https://www.tu-chemnitz.de/ sfb393/lyapack/(2000).
30. Mele, G. and Jarlebring, E. On restarting for the tensor infinite Arnoldi method, BIT 58 (1) (2018) 133–162.
31. Qi, L. Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (6) (2005) 1302–1324.
32. Ramezani, Z. and Toutounian, F. Extended and rational Hessenberg methods for the evaluation of matrix functions, BIT Numer. Math. 59, (2019) 523–545.
33. Saberi-Movahed, F., Tajaddini, A., Heyouni, M. and Elbouyahyaoui, L. Some iterative approaches for Sylvester tensor equations, PartI: A tensor format of truncated Loose Simpler GMRES, Appl. Numer. Math. 172 (2022) 428–445.
34. Saberi-Movahed, F., Tajaddini, A., Heyouni, M. and Elbouyahyaoui, L. Some iterative approaches for Sylvester tensor equations, Part II: A tensor format of truncated Loose Simpler GMRES, Appl. Numer. Math. 172 (2022) 428–445.

35. Sadok, H. CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm, Numer. Algorithms 20 (4) (1999) 303–321.
36. Simoncini, V. A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comp. 29 (3) (2007) 1268–1288.
37. Sun, Y., Ma, J. and Li, B.W. Chebyshev collocation spectral method for three-dimensional transient coupled radiative conductive heat transfer, J. Heat Transfer 134 (9) (2012):092701.
38. Xu, X. and Wang, Q.-W. Extending Bi-CG and Bi-CR methods to solve the Stein tensor equation, Comput. Math. Appl. 77 (12) (2019) 3117–3127.
39. Zhang, X.-F.and Wang, Q.-W. Developing iterative algorithms to solve Sylvester tensor equations, Appl. Math. Comput. 409 (2021) 126403.