Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse

Boris N. Khoromskij

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Submission date: 19. Dec. 2018
published in: Journal of computational physics, 401 (2020), art-no. 108998 
DOI number (of the published article): 10.1016/j.jcp.2019.108998
Bibtex
with the following different title: Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: electrostatic potential, Green function, Dirac delta, low-rank tensor decomposition, range-separated tensor formats, Coulomb potential
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
n this paper, we introduce the operator dependent range-separated tensor approximation of the discretized Dirac delta in ℝ^d. It is constructed by application of the discrete elliptic operator to the range-separated decomposition of the associated Green kernel discretized on the Cartesian grid in ℝ^d. The presented operator dependent local-global splitting of the Dirac delta can be applied for solving the potential equations in non-homogeneous medium when the density in the right-hand side is given by the large sum of pointwise singular charges. We show how the idea of the operator dependent RS splitting of the Dirac delta can be extended to the closely related problem on the range separated tensor representation of the elliptic resolvent. The numerical tests confirm the expected localization properties of the obtained operator dependent approximation of the Dirac delta represented on a tensor grid. As an example of application, we consider the regularization scheme for solving the Poisson-Boltzmann equation for modeling the electrostatics in bio-molecules.