AI 中文总结
本文针对t-积下的三阶不相容张量线性系统，提出TGDBEK方法，通过贪心策略动态调整活动块大小，理论上线性收敛，数值实验显示其性能优于现有同类张量Kaczmarz求解器。
AI 中文摘要
随机扩展Kaczmarz方法是求解大规模不相容线性系统的有效迭代框架。本文将该框架扩展到t-积下的三阶不相容张量线性系统，提出张量贪心双块扩展Kaczmarz（Tensor Greedy Double Block Extended Kaczmarz，TGDBEK）方法。每次迭代中，TGDBEK通过基于残差的贪心选择策略动态构造行切片和列切片的活动块，优先选择与最大残差范数相关联的切片。不同于现有依赖静态预定义划分的张量块Kaczmarz变体——张量随机扩展块Kaczmarz（tensor randomized extended block Kaczmarz，TREBK）、其贪心对应版本（TREGBK）以及张量随机扩展平均块Kaczmarz（TREABK）——TGDBEK在每一步使用单一直观阈值参数η动态调整活动块大小。我们证明了TGDBEK线性收敛到唯一的最小范数最小二乘解$\boldsymbol{\rm A}^\boldsymbol{\rm \text{\textdagger}} * \boldsymbol{\rm B}$。在合成稠密与稀疏张量系统，以及多维多通道彩色图像、三维体积MRI图像去模糊问题上的大量数值基准测试表明，TGDBEK在迭代次数和CPU运行时间两方面均显著优于最先进的张量Kaczmarz求解器。
英文摘要
The randomized extended Kaczmarz method is an effective iterative framework for solving large-scale inconsistent linear systems. In this paper, we extend this framework to third-order inconsistent tensor linear systems under the t-product and propose the Tensor Greedy Double Block Extended Kaczmarz (TGDBEK) method. At each iteration, TGDBEK dynamically constructs active blocks of row and column slices via a residual-based greedy selection strategy, prioritizing the slices associated with the largest residual norms. Unlike existing tensor block Kaczmarz variants that rely on static, predefined partitions --the tensor randomized extended block Kaczmarz (TREBK) method, its greedy counterpart (TREGBK), and the tensor randomized extended average block Kaczmarz (TREABK) method -- TGDBEK adapts the active block sizes dynamically at each step using a single intuitive threshold parameter $η$. We establish the theoretical linear convergence of TGDBEK to the unique minimum-norm least-squares solution $\mathcal{A}^\dagger * \mathcal{B}$. Extensive numerical benchmarks on synthetic dense and sparse tensor systems, as well as multidimensional multichannel color and 3D volumetric MRI image deblurring problems, demonstrate that TGDBEK substantially outperforms state-of-the-art tensor Kaczmarz solvers in both iteration count and CPU running time.
