Low-rank TT/QTT Decomposition Of Operators And Vectors With Its Applications

When solving differential or linear equations, the complexity of the problem spurts once the number of variables increases. An effective method in the present research is to represent large-scale vectors and matrices by low-rank tensors. Among those low-rank decompositions, the tensor train (TT) decomposition and quantized tensor train (QTT) decomposition possess apparent advantages of stability and efficiency, as well as alleviate the "curse of dimensionality" in a large extent.In this dissertation, TT/QTT methods are used for low-rank explicit representation of the biharmonic operator, as well as the computation of the biharmonic equation and Schrodinger equation, respectively. For the latter, we take the following steps mainly: carry out the discretization of the original differential equation; represent the discretiza-tion result of linear equations in the TT/QTT format; transform the linear equations into an optimization problem; finally solve it by alternating-linear-scheme-type (ALS-type) methods. Among ALS-type methods, the density matrix renormalization group (DMRG) and alternating minimal energy (AMEn) method show improvement of the original ALS, broadening the scope of it.The main contributions of this dissertation are:· Demonstrate the existence of low-rank TT and QTT representation of the second-order equidistant finite difference discretization of the biharmonic operator in the matrix form, with its rank independent of the dimension and the grid points of each dimension;· For the evolution problem of the hydrogen atom in laser fields, with the use of space-time discretization, construct a global system Ax=b in the TT/QTT format of the Schrodinger equation;· Utilize the DMRG and AMEn method for the computation of the biharmonic equation and Schrodinger equation respectively. The numerical results show that these problems can be solved efficiently in the low-rank TT/QTT format.