Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

Abstract

AbstractIn 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 , generated by the first column of the form admits a QTT representation with the QTT ranks bounded by . Under certain assumptions on the entries of , we also derive an explicit QTT representation of . 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.