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.

中文翻译：

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以QTT格式构建Z置换矩阵

摘要

本文提出了一种以QTT格式构建矩阵的方法，该矩阵的列和行通过z置换以特殊方式重新排序。为了获得此排列的矩阵，引入了QTT（量化张量列）格式的新操作z-kron。这种重新排序使人们可以减少刚度矩阵近似值的QTT秩，从而可以加快系统数值解的收敛速度。例如，当通过有限元方法（FEM）求解Poisson方程的Dirichlet问题时，使用QTT格式存储系数矩阵，并对维度为\ {n \ timesn \的系数矩阵中的行和列进行重新排序），其中\（n = {{4} ^ {d}} \）使得可以防止\（d \）排名增长。