In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor-train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix $A$, generated by the first column of the form $(a_0,\dots,a_{m-1},0,\dots,0,a_{-n},\dots, a_{-1})^\top$ admits a QTT representation with the QTT ranks bounded by $(m+n)$. Under certain assumptions on the entries of $A$, we also derive an explicit QTT representation of $A^{-1}$. The latter can be used, for instance, to overcome stability issues arising when numerically solving differential equations with periodic boundary conditions in the QTT format.
翻译：在本文中,我们关注循环剂基体及其量化的抗拉力结构的倒置,特别是,我们表明,以美元(a_0,\dots,a ⁇ m-1},0,\dots,0,a ⁇ -n},\docks,a ⁇ -}},a ⁇ -}}}} ⁇ top$为QTT在QT排中的代表机构,按美元(m+n)的顺序排列。在美元条目的某些假设下,我们还得出了明确的QTT代表单位为$A}-1美元。例如,后者可以用来克服在QTT格式中以数字方式解决带有定期边界条件的差别方程式时产生的稳定性问题。
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