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On the spectral problem for trivariate functions

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Abstract

Using a variational approach applied to generalized Rayleigh functionals, we extend the concepts of singular values and singular functions to trivariate functions defined on a rectangular parallelepiped. We also consider eigenvalues and eigenfunctions for trivariate functions whose domain is a cube. For a general finite-rank trivariate function, we describe an algorithm for computing the canonical polyadic (CP) decomposition, provided that the CP factors are linearly independent in two variables. All these notions are computed using Chebfun3; a part of Chebfun for numerical computing with 3D functions. Application in finding the best rank-1 approximation of trivariate functions is investigated. We also prove that if the function is analytic and two-way orthogonally decomposable (odeco), then the CP values decay geometrically, and optimal finite-rank approximants converge at the same rate.

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Notes

  1. Unless in the exceptional case that the initial vectors correspond to a saddle point of (2.1) [25, Thm. 2].

  2. Relative to a tolerance to which (2.10) holds; machine epsilon by default.

  3. See e.g. [52] for an analogous discussion regarding eigenvalues of symmetric discrete tensors.

  4. The singular quadruplets for odeco tensors is fully studied in [44], which shows that the number of singular values grows rapidly with both the dimension and size, and that the singular vector tuples form a positive-dimensional variety. Extending such results to the continuous case is an interesting open problem.

  5. This is the function doublehelix from the Chebfun gallery.

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Acknowledgements

The authors would like to thank Alex Townsend, André Uschmajew and Nick Vannieuwenhoven for helpful comments on the manuscript. We thank the referees for their comments and constructive suggestions. The work of the first author was in part supported by a grant from IPM (No. 96150051). The second author was supported by JSPS as an Overseas Research Fellow.

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Authors and Affiliations

  1. Department of Mathematics, Shiraz University of Technology, Modarres BLVD., Shiraz, 71555-313, Iran

    Behnam Hashemi

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran

    Behnam Hashemi

  3. National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Yuji Nakatsukasa

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  1. Behnam Hashemi
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Correspondence to Behnam Hashemi.

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Communicated by Daniel Kressner.

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Hashemi, B., Nakatsukasa, Y. On the spectral problem for trivariate functions. Bit Numer Math 58, 981–1008 (2018). https://doi.org/10.1007/s10543-018-0710-4

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