[Paper Review] Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse

Boris N. Khoromskij|arXiv (Cornell University)|Dec 6, 2018

Tensor decomposition and applications19 references4 citations

TL;DR

This paper introduces an operator-dependent range-separated (RS) tensor approximation for the discretized multidimensional Dirac delta and elliptic operator inverse, using low-rank tensor decompositions of the Green kernel. By splitting the Green kernel into short- and long-range components and applying the elliptic operator, the method achieves $O(n)$ complexity scaling and enables efficient, accurate solution of potential equations with many singular charges—critical for modeling biomolecular electrostatics via the Poisson-Boltzmann equation.

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 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.

Motivation & Objective

• To address the numerical instability and high computational cost of traditional grid-based methods when solving elliptic PDEs with singular Dirac delta sources.
• To develop a tensor-based, operator-dependent splitting of the Dirac delta into short- and long-range components for improved conditioning and accuracy.
• To extend the approach to the inverse of elliptic operators and the solution of potential equations in non-homogeneous media with many point charges.
• To enable scalable simulations of large biomolecular systems governed by the Poisson-Boltzmann equation using low-rank tensor formats.
• To demonstrate that the long-range part of the collective Dirac delta scales logarithmically with particle count, enabling linear-logarithmic complexity in large N-particle systems.

Proposed method

• Decompose the discretized Green kernel of an elliptic operator into short- and long-range components via range-separated (RS) tensor formats.
• Apply the elliptic operator $\mathcal{L}$ to each component to generate the corresponding RS splitting of the Dirac delta: $\delta = \mathcal{L}p_s + \mathcal{L}p_l = \delta_s + \delta_l$.
• Represent both $\delta_s$ and $\delta_l$ in low-rank tensor formats, ensuring the total approximation error is controlled by $\varepsilon$-truncation ranks and grid size $n$.
• Use the canonical RS tensor representation of the Newton kernel ($1/\|x\|$) on a $d$-fold $n \times \cdots \times n$ Cartesian grid as the foundation for the decomposition.
• Leverage the fact that radial functions like $1/\|x\|$ and $1/\|x\|^3$ admit low-rank RS tensor decompositions, enabling efficient computation of hydrodynamic and elastic potentials.
• Apply the method to the Poisson-Boltzmann equation by regularizing the singular charge distribution via the long-range smoothed component, reducing conditioning issues.

Experimental results

Research questions

• RQ1 Can the discretized Dirac delta in $\mathbb{R}^d$ be approximated efficiently using operator-dependent range-separated tensor formats?
• RQ2 Does the RS splitting of the Green kernel lead to a stable, low-rank representation of the Dirac delta that separates singular and smooth components?
• RQ3 Can the long-range part of the collective Dirac delta for $N$ point charges be represented with rank growing only logarithmically in $N$?
• RQ4 How does the tensor-based approach improve the conditioning and scalability of solving potential equations with many singular sources?
• RQ5 Can the method be extended to the inverse of elliptic operators with constant coefficients, such as the biharmonic or Stokes operators?

Key findings

• The operator-dependent RS splitting of the Dirac delta is constructed by applying the elliptic operator $\mathcal{L}$ to the RS decomposition of the Green kernel, yielding a sum of short- and long-range tensor components.
• The long-range component $\delta_l$ is smooth and amenable to low-rank tensor approximation, while the short-range component $\delta_s$ captures the singularity with localized correction.
• The method achieves $O(n)$ complexity scaling in grid size $n$ due to the tensor format, enabling efficient computation on large 3D grids.
• The rank of the long-range part of the collective Dirac delta grows only logarithmically with the number of particles $N$, ensuring scalable treatment of large $N$-particle systems.
• Numerical tests confirm the localization and accuracy of the approximation, with controlled error dependent on $\varepsilon$-truncation and grid size.
• The approach is successfully applied to the Poisson-Boltzmann equation for biomolecular electrostatics, providing a regularization scheme that avoids numerical instabilities from point-charge singularities.