一种鲁棒的张量训练格式GMRES算法
A robust GMRES algorithm in Tensor Train format
摘要：
我们考虑用GMRES算法求解张量积结构的线性方程组。为了解决大维数下浮点运算和内存需求的计算复杂性问题，我们的算法基于低秩张量表示，即张量训练格式.在向后误差分析框架中，我们展示了张量近似如何影响计算解的准确性。从后向的角度，我们研究了待解的$(d+1)$维问题是由一系列$d$维问题串联而成的情况（如参数线性算子或参数右侧问题），我们提供了向后误差界，以将$(d+1)$维计算解的精度与$d$序列的数值质量相关联。这使得能够在求解$(d+1)$维问题时规定收敛阈值，该收敛阈值确保一旦求解器收敛就将从$(d+1)$维计算解提取的$(d+1)$维解的数值质量。上述特征在一组不同尺寸和大小的学术示例上进行了说明。
英文摘要：
We consider the solution of linear systems with tensor product structure using a GMRES algorithm. In order to cope with the computational complexity in large dimension both in terms of floating point operations and memory requirement, our algorithm is based on low-rank tensor representation, namely the Tensor Train format. In a backward error analysis framework, we show how the tensor approximation affects the accuracy of the computed solution. With the bacwkward perspective, we investigate the situations where the $(d+1)$-dimensional problem to be solved results from the concatenation of a sequence of $d$-dimensional problems (like parametric linear operator or parametric right-hand side problems), we provide backward error bounds to relate the accuracy of the $(d+1)$-dimensional computed solution with the numerical quality of the sequence of $d$-dimensional solutions that can be extracted form it. This enables to prescribe convergence threshold when solving the $(d+1)$-dimensional problem that ensures the numerical quality of the $d$-dimensional solutions that will be extracted from the $(d+1)$-dimensional computed solution once the solver has converged. The above mentioned features are illustrated on a set of academic examples of varying dimensions and sizes.
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