Cross approximation in tensor electron density computations The paper studies approximations of a discrete electron density function (and its cubic root) related with the Hartree-Fock/Kohn-Sham equation. To be able to realize the computations for large scale problems, it is necessary to find a suitable structured approximation to such functions. First, it is assumed that a function is discretized on a tensor grid giving a tensor in the canonical format. Classical algorithms for computing Tucker and cross approximations of such tensors are compared with a new algorithm based on cross2D approximations. Then, approximations to a general form tensors are discussed. Storage and computational cost of all algorithms are compared. It is shown that the new algorithm is superior especially for tensors in the canonical or smooth elementwise canonical form.

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  1. Che, Maolin; Wei, Yimin; Yan, Hong: The computation of low multilinear rank approximations of tensors via power scheme and random projection (2020)
  2. Che, Maolin; Wei, Yimin: Randomized algorithms for the approximations of Tucker and the tensor train decompositions (2019)
  3. Canc├Ęs, Eric (ed.); Friesecke, Gero (ed.); Helgaker, Trygve Ulf (ed.); Lin, Lin (ed.): Mathematical methods in quantum chemistry. Abstracts from the workshop held March 18--24, 2018 (2018)
  4. Andreev, Roman; Tobler, Christine: Multilevel preconditioning and low-rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs. (2015)
  5. Dolgov, S. V.; Khoromskij, B. N.; Oseledets, I. V.; Savostyanov, D. V.: Computation of extreme eigenvalues in higher dimensions using block tensor train format (2014)
  6. Hackbusch, Wolfgang: Numerical tensor calculus (2014)
  7. Savostyanov, Dmitry V.: Quasioptimality of maximum-volume cross interpolation of tensors (2014)
  8. Grasedyck, Lars; Kressner, Daniel; Tobler, Christine: A literature survey of low-rank tensor approximation techniques (2013)
  9. Dolgov, Sergey; Khoromskij, Boris; Savostyanov, Dmitry: Superfast Fourier transform using QTT approximation (2012)
  10. Savostyanov, D. V.; Tyrtyshnikov, E. E.; Zamarashkin, N. L.: Fast truncation of mode ranks for bilinear tensor operations (2012)
  11. Khoromskij, B. N.; Khoromskaia, V.; Flad, H.-J.: Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format (2011)
  12. Oseledets, I. V.; Savostyanov, D. V.; Tyrtyshnikov, E. E.: Cross approximation in tensor electron density computations (2010)
  13. Savostyanov, D. V.; Tyrtyshnikov, E. E.: Approximate multiplication of tensor matrices based on the individual filtering of factors (2009)