Tensor Krylov subspace methods are popular technologies for solving Sylvester tensor equations, among which the GMRES method based on tensor format (GMRES_BTF) and the FOM method based on tensor format (FOM_BTF) are two commonly used ones. Both of them rely on the Arnoldi process based on tensor format (Arnoldi_BTF) to construct orthonormal bases for tensor Krylov subspace. However, the computational costs and storage requirements of the tensor Krylov subspace methods will increase tremendously as the Arnoldi_BTF process proceeds. Restarting is an efficient way to deal with this problem. To the best of our knowledge, there are few efficient restarting strategies for tensor Krylov subspace methods. In order to fill-in this gap, we apply the deflated restarting strategy to the GMRES_BTF and FOM_BTF methods, and propose two deflated restating methods for solving Sylvester tensor equations. The key is that the two proposed methods retain some useful information in the harmonic Ritz tensors or Ritz tensors obtained from the previous tensor Krylov subspace, respectively. Numerical experiments on both artificial and real data sets demonstrate the superiority of the proposed methods over many state-of-the-art methods for Sylvester tensor equations.

张量Krylov子空间方法是求解Sylvester张量方程的热门技术，其中基于张量格式的GMRES方法（GMRES_BTF）和基于张量格式的FOM方法（FOM_BTF）是两种常用的方法。两者都依赖基于张量格式的Arnoldi过程（Arnoldi_BTF）来构造张量Krylov子空间的正交基。然而，随着 Arnoldi_BTF 过程的进行，张量 Krylov 子空间方法的计算成本和存储要求将大幅增加。重新启动是解决此问题的有效方法。据我们所知，张量 Krylov 子空间方法几乎没有有效的重启策略。为了填补这个空白，我们将紧缩重启策略应用于 GMRES_BTF 和 FOM_BTF 方法，并提出了两种求解西尔维斯特张量方程的紧缩重述方法。关键是这两种方法分别在调和 Ritz 张量或从先前张量 Krylov 子空间获得的 Ritz 张量中保留了一些有用的信息。对人工和真实数据集的数值实验证明了所提出的方法相对于许多最先进的西尔维斯特张量方程方法的优越性。