Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices
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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,…,a_m-1,0,…,0,a_-n,…, a_-1)^⊤ 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.
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