Abstract
In this paper, we introduce tensors with Toeplitz structure. These structured tensors occur in different kinds of applications such as discretization of multidimensional PDE’s or Fredholm integral equations with an invariant kernel. We investigate the main properties of the new structured tensor and show the tensor contractive product with such tenors can be carried out with the fast Fourier transform. Also, we show that approximation of Toeplitz tensors with a specially structured tensor (that will be named \(\circledast \)-product tensors) can be reduced to the rank-1 approximation of a smaller tensor. Tensor equations with such \(\circledast \)-product coefficient tenors can be solved by a direct method. So, this approximation of a Toeplitz tensor can be used to find an approximate solution of the original tensor equation or can be used as a preconditioner. Our main goal is to show the ability of the tensor framework to handle structured multidimensional problems in their original format.
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Notes
Higher order singular value decomposition.
Block Toeplitz with Toeplitz Blocks.
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Acknowledgements
This research was in part supported by a Grant from IPM (No. 90650024).
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Communicated by Jinyun Yuan.
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Rezghi, M., Amirmazlaghani, M. Even-order Toeplitz tensor: framework for multidimensional structured linear systems. Comp. Appl. Math. 38, 143 (2019). https://doi.org/10.1007/s40314-019-0919-0
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DOI: https://doi.org/10.1007/s40314-019-0919-0