AI 中文总结
针对Tucker格式大规模张量线性系统求解的GMRES迭代瓶颈，提出两种随机化sketched GMRES算法，在对称/非对称场景及逆问题上均优于标准低秩Tucker求解器。
AI 中文摘要
我们研究求解Tucker格式的大规模张量结构化线性系统的问题。在该场景下，GMRES等标准迭代求解器面临一个根本瓶颈：Krylov基向量的多线性秩随迭代次数增长，导致张量运算成本和内存需求快速增加。为克服这些挑战，我们在sketched GMRES框架内提出两种随机算法，用短递推替代完整的Arnoldi正交化。第一种是RHOSVD-Tucker sGMRES，采用带每迭代秩选择的随机HOSVD，对广泛问题具有鲁棒性；第二种是MLN-Tucker sGMRES，利用固定秩的多线性Nyström近似，支持流式计算，该近似的可流性还能以无额外成本从Krylov基的紧凑草图表示中实现内存高效的解重构。两种方法在对称和非对称场景下均优于标准低秩Tucker求解器。将其应用于逆问题时，低秩Tucker约束作为隐式正则化项，结合自适应投影Tikhonov惩罚与自动正则化参数选择，可得到稳定的重构结果。
英文摘要
We address the problem of solving large-scale tensor-structured linear systems in the Tucker format. In this setting, standard iterative solvers such as GMRES face a fundamental bottleneck: the multilinear ranks of the Krylov basis vectors grow with the iteration count, leading to rapidly increasing tensor operation costs and memory requirements. To overcome these challenges, we propose two randomized algorithms within the sketched GMRES framework that replace full Arnoldi orthogonalization with short recurrences. The first, RHOSVD-Tucker sGMRES, uses randomized HOSVD with per-iteration rank selection, providing robustness across a wide range of problems. The second method, MLN-Tucker sGMRES, leverages the multilinear Nyström approximation with a fixed rank, enabling streaming computations; the streamability of the approximation further allows, at no additional cost, a memory-efficient reconstruction of the solution from a compact sketched representation of the Krylov basis. Both methods outperform standard low-rank Tucker solvers in symmetric and non-symmetric settings. Applied to inverse problems, the low-rank Tucker constraint acts as an implicit regularizer; combined with adaptive projected Tikhonov penalization and automatic regularization parameter selection, the methods yield stable reconstructions.
Comments29 pages, 6 figures, 2 tables
