Abstract
In the present paper we discuss efficient rank-structured tensor approximation methods for 3D integral transforms representing the Green iterations for the Kohn-Sham equation. We analyse the local convergence of the Newton iteration to solve the Green’s function integral formulation of the Kohn-Sham model in electronic structure calculations. We prove the low-separation rank approximations for the arising discrete convolving kernels given by the Coulomb and Yukawa potentials 1/|x|, and e −λ|x|/|x|, respectively, with \(x \in {\mathbb{R}}^{d} \). Complexity analysis of the nonlinear iteration with truncation to the fixed Kronecker tensor-product format is presented. Our method has linear scaling in the univariate problem size. Numerical illustrations demostrate uniform exponential convergence of tensor approximations in the orthogonal Tucker and canonical formats.
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Communicated by S. Sauter.
Dedicated to Wolfgang Hackbusch on the occasion of his 60th birthday.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Khoromskij, B.N. On tensor approximation of Green iterations for Kohn-Sham equations. Comput. Visual Sci. 11, 259–271 (2008). https://doi.org/10.1007/s00791-008-0097-x
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DOI: https://doi.org/10.1007/s00791-008-0097-x