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.
翻译:我们用GMRES算法来考虑使用高压产品结构的线性系统的解决办法。为了在浮动点操作和内存要求这两个大方面处理计算的复杂性,我们的算法以低压强表示法为基础,即Tensor列车格式。在后向错误分析框架内,我们展示了高压近离子如何影响计算解决方案的准确性。用Bacwkwkward的视角,我们调查了美元(d+1)美元(美元)维度问题需要通过对美元维度问题(如参数线性操作员或参数右侧问题)进行排列的结果解决的情况。我们提供了后向误差界限,将美元(d+1)美元-维计算的解决方案的准确性与可以提取的美元维度解决方案序列的数值质量联系起来。这样就可以在解决美元(d+1)美元-维度问题时规定趋同的门槛,以确保从美元(d+1)美元-维度计算法解决方案中提取的数值质量,一旦解算出,则从$(d+1)美元-米计算法的答案汇合起来,则用不同层次的特征显示。