论文标题
张量火车格式的强大gmres算法
A robust GMRES algorithm in Tensor Train format
论文作者
论文摘要
我们考虑使用GMRE算法使用张量产品结构的线性系统解决方案。为了在浮点操作和内存要求方面应对大维的计算复杂性，我们的算法基于低级数张量表示，即张量列车格式。 在向后的误差分析框架中，我们展示了张量近似如何影响计算解决方案的准确性。从Bacwkward的角度来看，我们调查了以下情况，在这种情况下，要解决的$（d+1）$ - 维度问题是由于$ d $维问题的串联的串联而导致的结果（例如，参数线性运算符或参数右手侧面问题），我们提供了向后误差，以相关联级的序列与$（D+1）$（D+1）计算的序列相关联（ $ D $ - 二维解决方案，可以提取它。当解决$（d+1）$ - 尺寸问题时，可以规定融合阈值，以确保将从$（d+1）$计算的解决方案中提取的$ d $维解决方案的数值质量。上述功能在一组不同的维度和大小的学术示例中进行了说明。
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.
