The Dynamical Low-rank Approximation Of Tensors Based On ST-HOSVD

The dynamic low-rank approximation of tensors plays a very important role in the approximation problem of tensors.Recently,there are much research about the low-rank approximation of tensors.In this paper,for the given time-varying tensorsA(t)?R^{n₁×n₂×···×n_d},we consider the dynamical low-rank approximation based on sequential truncation higher-order singular value decomposition of rank r=(r₁,r₂,r₃,...,r_d).First,the derivatives of a series of tensorsY^k(t)with ST-HOSVD approximation are orthogonally projected onto the tangent space T_YkM_rkof the tensor manifoldM_rk,and for the decomposition facors of the ST-HOSVD format,this yields nonlinear differential equations that are well suited for numerical solution.During the process,we apply the increments of the decomposition factors of ST-HOSVD,instead of the matrix decomposition of the k-mode unfolding of tensors.Second,we analyze the approximation properties of this method,and show that this method is also applicable to the approximation of solutions of tensor differential equations.Finally,we present some numerical experiments to verify the feasibility and effectiveness of the method.