In this work we generalize the Jacobi-Davidson method to the case when eigenvector can be reshaped into a low-rank matrix. In this setting the proposed method inherits advantages of the original Jacobi-Davidson method, has lower complexity and requires less storage. We also introduce low-rank version of the Rayleigh quotient iteration which naturally arises in the Jacobi-Davidson method.
In @cite_13 authors proposed a version of inverse iteration based on the alternating linear scheme ALS procedure, which is similar to . By contrast, the present work considers inverse iteration on the whole tangent space. We also provide an interpretation of the method as an inexact Newton method.
@cite_13
2404706163
We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (locally optimal block preconditioned conjugate gradient method, inverse iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH3CN on a laptop in one hour using only 100 MB of memory to represent all computed eigenfunctions.