, Volume 11, Issue 4-6, pp 259-271,
Open Access This content is freely available online to anyone, anywhere at any time.

On tensor approximation of Green iterations for Kohn-Sham equations

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.

Communicated by S. Sauter.
Dedicated to Wolfgang Hackbusch on the occasion of his 60th birthday.