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Tensor Arnoldi–Tikhonov and GMRES-Type Methods for Ill-Posed Problems with a t-Product Structure

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Abstract

This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in (Linear Algebra Appl 435:641–658, 2011). A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The discrepancy principle is used to determine the regularization parameter and the number of steps of the t-Arnoldi process. Numerical examples compare results for several solution methods, and illustrate the potential superiority of solution methods that tensorize over solution methods that matricize linear discrete ill-posed problems for third order tensors.

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Notes

  1. https://www.github.com/canyilu/tproduct.

  2. https://github.com/jnagy1/IRtools/blob/master/Extra/test_data/HSTgray.jpg.

  3. http://www.hlevkin.com/TestImages.

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Acknowledgements

The authors would like to thank the referees for comments that led to improvements of the presentation. Research by LR was supported in part by NSF Grant DMS-1720259.

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  1. Department of Mathematical Sciences, Kent State University, Kent, OH, 44240, USA

    Lothar Reichel & Ugochukwu O. Ugwu

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  1. Lothar Reichel
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  2. Ugochukwu O. Ugwu
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Correspondence to Ugochukwu O. Ugwu.

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Reichel, L., Ugwu, U.O. Tensor Arnoldi–Tikhonov and GMRES-Type Methods for Ill-Posed Problems with a t-Product Structure. J Sci Comput 90, 59 (2022). https://doi.org/10.1007/s10915-021-01719-1

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  • DOI: https://doi.org/10.1007/s10915-021-01719-1

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