Building Z-Permuted Matrices in the QTT Format

Abstract

The paper presents a method for building matrices in the QTT format, the columns and rows of which are reordered in a special way, by z-permutation. To obtain a matrix in this permutation, a new operation in the QTT (Quantized Tensor Train) format, z-kron, is introduced. This reordering allows one to reduce the QTT ranks of the approximation of the stiffness matrix, which makes it possible to accelerate the convergence of the numerical solution of the system. For example, when solving the Dirichlet problem for Poisson’s equation by the finite element method (FEM), where the QTT format is used to store the coefficient matrix, reordering the rows and columns in a coefficients matrix with dimensions $$n \times n$$ , where $$n = {{4}^{d}}$$ , makes it possible to prevent the exponential in $$d$$ growth of ranks.