Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse
This work provides a novel numerical technique for efficiently solving potential equations with singular sources, benefiting computational physics and chemistry applications like biomolecular electrostatics.
The paper introduces an operator-dependent range-separated tensor approximation of the discretized Dirac delta in ℝ^d, constructed by applying the discrete elliptic operator to a range-separated decomposition of the Green kernel. Numerical tests confirm localization properties, and the method is applied to regularize the Poisson-Boltzmann equation for biomolecular electrostatics.
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 media 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.