求解大规模线性问题的张量GMRES算法
Tensor GMRES Algorithm for Solving Large-Scale Linear Problems
DOI: 10.12677/aam.2025.144223, PDF,    科研立项经费支持
作者: 王仕伟*, 杨 志, 冷震北：重庆对外经贸学院数学与计算机学院，重庆
关键词: 大规模线性问题；Krylov子空间方法；t-积；GMRES；Large-Scale Linear Problems； Krylov Subspace Methods； t-Product； GMRES
摘要: 彩色图像和视频通常可以被描述为高阶张量。本文基于三阶张量t-积，讨论了Krylov子空间方法用以解决图像恢复中的大规模线性问题。本文通过张量GMRES算法构建Krylov子空间，将大规模线性问题转换为小规模问题，且构建的子空间始终保持张量的空间结构。数值例子和彩色图像修复的应用说明了算法的有效性。
Abstract: Color images and video sequences can typically be characterized as higher-order tensors. This paper investigates Krylov subspace methods based on the third-order tensor t-product for solving large-scale linear systems arising in image restoration. This paper employs the tensor GMRES algorithm to construct the Krylov subspace, effectively reducing large-scale linear problems to manageable small-scale formulations, while consistently preserving the spatial architecture of tensors within the constructed subspace. Numerical experiments and applications in color image inpainting demonstrate the efficacy of the proposed methodology.
文章引用：王仕伟, 杨志, 冷震北. 求解大规模线性问题的张量GMRES算法[J]. 应用数学进展, 2025, 14(4): 1007-1018. https://doi.org/10.12677/aam.2025.144223

参考文献

[1] Kolda, T.G. and Bader, B.W. (2009) Tensor Decompositions and Applications. SIAM Review, 51, 455-500. [Google Scholar] [CrossRef] 
[2] Tucker, L.R. (1966) Some Mathematical Notes on Three-Mode Factor Analysis. Psychometrika, 31, 279-311. [Google Scholar] [CrossRef] [PubMed]
[3] De Lathauwer, L., De Moor, B. and Vandewalle, J. (2000) A Multilinear Singular Value Decomposition. SIAM Journal on Matrix Analysis and Applications, 21, 1253-1278. [Google Scholar] [CrossRef] 
[4] Goodfellow, I., Bengio, Y. and Courville, A. (2016) Deep Learning. MIT Press.
[5] He, K., Zhang, X., Ren, S. and Sun, J. (2015) Delving Deep into Rectifiers: Surpassing Human-Level Performance on Imagenet Classification. 2015 IEEE International Conference on Computer Vision (ICCV), Santiago, 7-13 December 2015, 1026-1034. [Google Scholar] [CrossRef] 
[6] Rasmussen, C.E. and Williams, C.K.I. (2005) Gaussian Processes for Machine Learning. The MIT Press. [Google Scholar] [CrossRef] 
[7] Paszke, A., Gross, S., Massa, F., et al. (2019) PyTorch: An Imperative Style, High-Performance Deep Learning Library. Advances in Neural Information Processing Systems (NeurIPS), Vancouver, 8-14 December 2019.
[8] Kilmer, M.E. and Martin, C.D. (2011) Factorization Strategies for Third-Order Tensors. Linear Algebra and Its Applications, 435, 641-658. [Google Scholar] [CrossRef] 
[9] Braman, K. (2010) Third-Order Tensors as Linear Operators on a Space of Matrices. Linear Algebra and Its Applications, 433, 1241-1253. [Google Scholar] [CrossRef] 
[10] Cichocki, A. and Amari, S.I. (2010) Tensor Decompositions for Signal Processing Applications: From Two-Way to Multi-Way Component Analysis. IEEE Transactions on Signal Processing, 58, 1226-1241.
[11] Vasilenko, D. and Savich, A. (2016) Multidimensional Signal Processing Using Tensor Decomposition: A Survey. IEEE Transactions on Signal Processing, 64, 1263-1275.
[12] Reichel, L. and Ugwu, U.O. (2021) The Tensor Golub-Kahan-Tikhonov Method Applied to the Solution of Ill‐Posed Problems with a T‐Product Structure. Numerical Linear Algebra with Applications, 29, e2412. [Google Scholar] [CrossRef] 
[13] Ugwu, U.O. and Reichel, L. (2021) Tensor Regularization by Truncated Iteration: A Comparison of Some Solution Methods for Large-Scale Linear Discrete Ill-Posed Problems with a T-Product. arXiv preprint arXiv:2110.02485
[14] Zheng, M. and Ni, G. (2023) Approximation Strategy Based on the T-Product for Third-Order Quaternion Tensors with Application to Color Video Compression. Applied Mathematics Letters, 140, Article 108587. [Google Scholar] [CrossRef] 
[15] Khaleel, H.S., Mohd Sagheer, S.V., Baburaj, M. and George, S.N. (2018) Denoising of Rician Corrupted 3D Magnetic Resonance Images Using Tensor-SVD. Biomedical Signal Processing and Control, 44, 82-95. [Google Scholar] [CrossRef] 
[16] Cichocki, A., Mandic, D., De Lathauwer, L., Zhou, G., Zhao, Q., Caiafa, C., et al. (2015) Tensor Decompositions for Signal Processing Applications: From Two-Way to Multiway Component Analysis. IEEE Signal Processing Magazine, 32, 145-163. [Google Scholar] [CrossRef] 
[17] Zhang, J., Saibaba, A.K., Kilmer, M.E. and Aeron, S. (2018) A Randomized Tensor Singular Value Decomposition Based on the T-Product. Numerical Linear Algebra with Applications, 25, e2179. [Google Scholar] [CrossRef] 
[18] Hao, N., Kilmer, M.E., Braman, K. and Hoover, R.C. (2013) Facial Recognition Using Tensor-Tensor Decompositions. SIAM Journal on Imaging Sciences, 6, 437-463. [Google Scholar] [CrossRef] 
[19] Kilmer, M.E., Braman, K., Hao, N. and Hoover, R.C. (2013) Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging. SIAM Journal on Matrix Analysis and Applications, 34, 148-172. [Google Scholar] [CrossRef] 
[20] El Guide, M., El Ichi, A., Jbilou, K. and Sadaka, R. (2021) On Tensor GMRES and Golub-Kahan Methods via the T-Product for Color Image Processing. The Electronic Journal of Linear Algebra, 37, 524-543. [Google Scholar] [CrossRef] 
[21] Song, H., Wang, S. and Huang, G. (2023) Tensor Conjugate-Gradient Methods for Tensor Linear Discrete Ill-Posed Problems. AIMS Mathematics, 8, 26782-26800. [Google Scholar] [CrossRef] 
[22] Wang, S., Huang, G. and Yin, F. (2024) Tensor Conjugate Gradient Methods with Automatically Determination of Regularization Parameters for Ill-Posed Problems with T-Product. Mathematics, 12, Article 159. [Google Scholar] [CrossRef] 
[23] Kernfeld, E., Kilmer, M. and Aeron, S. (2015) Tensor-Tensor Products with Invertible Linear Transforms. Linear Algebra and Its Applications, 485, 545-570. [Google Scholar] [CrossRef] 
[24] Hansen, P.C. (1998) Rank-Deficient and Discrete Ill-Posed Problems. Society for Industrial and Applied Mathematics. [Google Scholar] [CrossRef] 

为你推荐