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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 (www.arxiv.org) 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
104/2018

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

Boris N. Khoromskij

Abstract

In this paper, we introduce the operator dependent range-separated tensor approximation of the discretized Dirac delta in $\mathbb{R}^d$. 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 $\mathbb{R}^d$. 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.

Received:
Dec 19, 2018
Published:
Dec 19, 2018
MSC Codes:
65F30, 65F50, 65N35, 65F10
Keywords:
electrostatic potential, Green function, Dirac delta, low-rank tensor decomposition, range-separated tensor formats, Coulomb potential

Related publications

inJournal
2020 Repository Open Access
Boris N. Khoromskij

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

In: Journal of computational physics, 401 (2020), p. 108998