New Gradient-Based Iterative-Like Algorithms for Solving a Class of Sylvester Tensor Equations

Abstract

In this paper, we consider deriving some new gradient-based iterative (GI)-like algorithms for solving a class of Sylvester tensor equations, which often arise from control systems and image processing. We first study the optimal parameter and the iteration matrix’s minimal spectral radius of the relaxed GI (RGI) algorithm proposed by Zhang and Wang (Taiwan J Math 26: 501–519, 2022) in terms of matricization of a tensor and straightening operator. Then based on the Jacobi method, by using the diagonal matrices to replace the system matrices in mode products contained in the RGI and the modified RGI (MRGI) algorithms (Taiwan J Math 26: 501–519, 2022), we design the Jacobi RGI (JRGI) and improved MRGI (IMRGI) algorithms for the Sylvester tensor equations, which require less computational load and are more efficient than the RGI and the MRGI ones, respectively. We deduce the sufficient convergence condition and quasi-optimal parameter of the JRGI algorithm, and sufficient and necessary conditions for the convergence of the IMRGI algorithm. Furthermore, we apply a new update strategy to the RGI algorithm and develop an updated RGI (URGI) algorithm for the Sylvester tensor equations, which is different from the MGI (Math Probl Eng 819479: 1–7, 2013) and the MRGI ones. The URGI algorithm takes full advantage of the latest computed results and returns better convergence behavior than RGI, MGI, and MRGI ones. Also, we prove that the proposed URGI algorithm is convergent under proper conditions. Finally, numerical experiments are performed to verify that the proposed algorithms are efficient and have advantages over some existing ones.

Keywords

Sylvester tensor equation / Improved modified relaxed gradient-based iterative (IMRGI) algorithm / Jacobi relaxed GI (JRGI) algorithm / Updated relaxed GI (URGI) algorithm / Convergence analysis / Optimal parameter

References

[1] BaiZ-Z. On Hermitian and skew-Hermitian splitting ietration methods for continuous Sylvester equations. J. Comput. Math., 2011, 29: 185-198

[2] BeikFPA, Ahmadi-AslS. Residual norm steepest descent based iterative algorithms for Sylvester tensor equations. J. Math. Model., 2015, 2: 115-131

[3] BeikFPA, MovahedFS, Ahmadi-AslS. On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations. Numer. Linear Algebra Appl., 2016, 23: 444-466

[4] BeikFPA, Najafi-KalyaniM, ReichelL. Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations. Appl. Numer. Math., 2020, 151: 425-447

[5] BentbibAH, KhouiaA, SadokH. The LSQR method for solving tensor least-squares problems. Electron. Trans. Numer. Anal., 2022, 55: 92-111

[6] BermanA, PlemmonsRJ Nonnegative Matrices in the Mathematical Sciences, 1994 Philadelphia SIAM

[7] ChenZ, LuL-Z. A projection method and Kronecker product preconditioner for solving Sylvester tensor equations. Sci. China Math., 2012, 55: 1281-1292

[8] ChenZ, LuL-Z. A gradient based iterative solutions for Sylvester tensor equations. Math. Probl. Eng., 2013, 819479: 1-7

[9] DehdeziEK. HOBi-CGSTAB and HOBi-CRSTAB methods for solving some tensor equations. Afrika Math., 2024, 35: 14

[10] DingF, ChenT-W. Gradient based iterative algorithms for solving a class of matrix equations. IEEE Trans. Automat. Control, 2005, 50: 1216-1221

[11] HajarianM. Matrix form of the Bi-CGSTAB method for solving the coupled Sylvester matrix equations. IET Control Theory Appl., 2013, 7: 1828-1833

[12] HajarianM. Matrix GPBiCG algorithms for solving the general coupled matrix equations. IET Control Theory Appl., 2015, 9: 74-81

[13] HuW-P, WangW-G. Improved gradient iteration algorithms for solving the coupled Sylvester matrix equation. J. Nanjing Univ. Math. Biq., 2016, 33: 177-192

[14] HuangB-H, MaC-F. The relaxed gradient-based iterative algorithms for a class of generalized coupled Sylvester-conjugate matrix equation. J. Franklin Inst., 2018, 355: 3168-3195

[15] HuangB-H, MaC-F. An iterative algorithm to solve the generalized Sylvester tensor equations. Linear Multilinear A., 2020, 68(6): 1175-1200

[16] HuangB-H, MaC-F. Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations. Appl. Math. Comput., 2020, 369 124892

[17] HuangG-X, ChenQ-X, YinF. Preconditioned TBiCOR and TCORS algorithms for solving the Sylvester tensor equation. Appl. Math. Comput., 2022, 422 126984

[18] HuangZ-G, CuiJ-J. Modified and accelerated relaxed gradient based iterative algorithms for the complex conjugate and transpose matrix equations. Numer. Algorithms, 2024, 97: 1955-2009

[19] KeY-F. Finite iterative algorithm for the complex generalized Sylvester tensor equations. J. Appl. Anal. Comput., 2020, 10: 972-985

[20] Kolda, T.G.: Multilinear operators for higher-order decompositions. In: Technical report, Sandia National Laboratories, Albuquerque, NM, and Livermore, CA (2006)

[21] KoldaTG, BaderBW. Tensor decompositions and applications. SIAM Rev., 2009, 51: 455-500

[22] LiB-W, SunY-S, YuY. Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer. Int. J. Heat Mass Transf., 2008, 51: 5887-5894

[23] LiB-W, TianS, SunY-S. Schur-decomposition for 3D matrix equations and its application in solvingradiative discrete ordinates equations discretized by Chebyshev collocation spectral method. J. Comput. Phys., 2010, 229: 1198-1212

[24] LiT, WangQ-W, DuanX-F. Numerical algorithms for solving discrete Lyapunov tensor equation. J. Comput. Appl. Math., 2020, 370 ArticleID: 112676

[25] LiT, WangQ-W, ZhangX-F. A modified conjugate residual method and nearest Kronecker product preconditioner for the generalized coupled Sylvester tensor equations. Mathematics, 2022, 10: 1730

[26] LvC-Q, MaC-F. A modified CG algorithm for solving generalized coupled Sylvester tensor equations. Appl. Math. Comput., 2020, 365 124699

[27] NiuQ, WangX, LuL-Z. A relaxed gradient based algorithm for solving Sylvester equations. Asian J. Control, 2011, 13: 461-464

[28] QiL-Q, ChenH-B, ChenY-N Tensor Eigenvalues and Their Applications, 2018 Singapore Springer 39

[29] QiL-Q, LuoZ-Y Tensor Analysis: Spectral Theory and Special Tensors, 2017 Philadelphia SIAM

[30] Saberi-MovahedF, TajaddiniA, HeyouniM, ElbouyahyaouiL. Some iterative approaches for Sylvester tensor equations, part i: a tensor format of truncated loose simpler GMRES. Appl. Numer. Math., 2022, 172: 428-445

[31] Sheng, X.-P., Sun, W.-W.: The relaxed gradient based iterative algorithm for solving matrix equations A i X B i = F i \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{i}XB_{i}=F_{i}$$\end{document}. Comput. Math. Appl. 74, 597–604 (2017)

[32] ShirilordA, DehghanM. Iterative algorithm for a generalized matrix equation with momentum acceleration approach and its convergence analysis. J. Franklin Inst., 2024, 361 ArticleID: 107021

[33] SoltaniS, KilmerME, HansenPC. A tensor-based dictionary learning approach to tomographic image reconstruction. BIT Numer. Math., 2016, 56: 1425-1454

[34] Tian, Z.-L., Duan, X.-F., Wu, N.-C., Liu, Z.-Y.: The parameterized accelerated iteration method for solving the matrix equation A X B = C \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$AXB=C$$\end{document}. Numer. Algorithms 97, 843–867 (2024)

[35] TianZ-L, TianM-Y, GuC-Q, HaoX-N. An accelerated Jacobi-gradient based iterative algorithm for solving Sylvester matrix equations. Filomat, 2017, 31: 2381-2390

[36] WangQ-W, XuX-J, DuanX-F. Least squares solution of the quaternion Sylvester tensor equation. Linear Multilinear A., 2021, 69: 104-130

[37] WangW-L, SongC-Q. Iterative algorithms for discrete-time periodic Sylvester matrix equations and its application in antilinear periodic system. Appl. Numer. Math., 2021, 168: 251-273

[38] WangW-L, SongC-Q, JiS-P. Iterative solution to a class of complex matrix equations and its application in time-varying linear system. J. Appl. Math. Comput., 2021, 67: 317-341

[39] WangX, DaiL, LiaoD. A modified gradient based algorithm for solving Sylvester equations. Appl. Math. Comput., 2012, 218: 5620-5628

[40] WuA-G, LvL-L, HouM-Z. Finite iterative algorithms for extended Sylvester-conjugate matrix equations. Math. Comput. Model., 2011, 54: 2363-2384

[41] XieY-J, MaC-F. The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation. Appl. Math. Comput., 2016, 273: 1257-1269

[42] XuX-J, WangQ-W. Extending BiCG and BiCR methods to solve the Stein tensor equation. Comput. Math. Appl., 2019, 77: 3117-3127

[43] ZhangH-M. A finite iterative algorithm for solving the complex generalized coupled Sylvester matrix equations by using the linear operators. J. Franklin Inst., 2017, 354: 1856-1874

[44] ZhangX-F, DingW, LiT. Tensor form of GPBiCG algorithm for solving the generalized Sylvester quaternion tensor equations. J. Franklin Inst., 2023, 360: 5929-5946

[45] ZhangX-F, WangQ-W. Developing iterative algorithms to solve Sylvester tensor equations. Appl. Math. Comput., 2021, 409 126403

[46] ZhangX-F, WangQ-W. On RGI algorithm for solving Sylvester tensor equations. Taiwan J. Math., 2022, 26: 501-519

Funding

National Natural Science Foundation of China(12361078)

Natural Science Foundation of Guangxi Province(2024GXNSFAA010498)

RIGHTS & PERMISSIONS

Shanghai University