求解具有张量积结构线性系统的共轭梯度法

The Conjugate Gradient Method for Linear Systems with Tensor Product Structure

本文考虑具有张量积结构线性系统的数值解法.该线性系统常常来源于高维立方体上线性偏微分方程的有限差分离散化.利用张量-矩阵乘法,给出了基于张量格式的求解这类线性系统的共轭梯度法.与求解标准线性系统的共轭梯度法比较,新的算法能够节约大量的计算量及存储空间.

The numerical solution of linear systems with tensor product structures is considered. Such structures arise from the finite difference diseretization of linear partial differential equations on a high dimensional hypercube. By ex-ploiting the tensor - matrix multiplication, we present a conjugate method based on tensor format for linear systems with tensor product structure. Comparing with the standard conjugate gradient method, the algorithm we proposed can reduce much computational cost and memory.