MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

On Tensor Approximation of Green Iterations for Kohn-Sham Equations

Boris N. Khoromskij


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. %with linear scaling in the univariate problem size.

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^{-\lambda|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.

MSC Codes:
65F30, 65F50, 65N35
Hartree-Fock method, tensor-product approximation, Density functional theory

Related publications

2008 Journal Open Access
Boris N. Khoromskij

On tensor approximation of Green iterations for Kohn-Sham equations

In: Computing and visualization in science, 11 (2008) 4/6, pp. 259-271