[Paper Review] Tucker tensor method for fast grid-based summation of long-range potentials on 3D lattices with defects

Venera Khoromskaia, Boris N. Khoromskij|arXiv (Cornell University)|Nov 7, 2014

Tensor decomposition and applications36 references3 citations

TL;DR

This paper presents a Tucker tensor method for fast, grid-based summation of long-range potentials on 3D lattices with defects, using low-rank canonical and Tucker tensor formats to achieve O(L) complexity instead of O(L³). The approach enables efficient computation of millions of interactions in seconds via rank-reduced tensor arithmetic, with stable error bounds and linear scaling in grid size N.

ABSTRACT

In this paper, we present a method for fast summation of long-range potentials on 3D lattices with multiple defects and having non-rectangular geometries, based on rank-structured tensor representations. This is a significant generalization of our recent technique for the grid-based summation of electrostatic potentials on the rectangular $L imes L imes L$ lattices by using the canonical tensor decompositions and yielding the $O(L)$ computational complexity instead of $O(L^3)$ by traditional approaches. The resulting lattice sum is calculated as a Tucker or canonical representation whose directional vectors are assembled by the 1D summation of the generating vectors for the shifted reference tensor, once precomputed on large $N imes N imes N$ representation grid in a 3D bounding box. The tensor numerical treatment of defects is performed in an algebraic way by simple summation of tensors in the canonical or Tucker formats. To diminish the considerable increase in the tensor rank of the resulting potential sum the $\varepsilon$-rank reduction procedure is applied based on the generalized reduced higher-order SVD scheme. For the reduced higher-order SVD approximation to a sum of canonical/Tucker tensors, we prove the stable error bounds in the relative norm in terms of discarded singular values of the side matrices. The required storage scales linearly in the 1D grid-size, $O(N)$, while the numerical cost is estimated by $O(N L)$. Numerical tests confirm the efficiency of the presented tensor summation method: we demonstrate that a sum of millions of Newton kernels on a 3D lattice with defects/impurities can be computed in seconds in Matlab implementation.

Motivation & Objective

• To develop a fast, scalable method for computing long-range potentials on 3D lattices with defects, such as vacancies and impurities.
• To extend previous tensor-based lattice summation techniques to non-rectangular and defective lattices while preserving computational efficiency.
• To enable algebraic summation of tensor representations of individual potentials using low-rank formats, minimizing storage and computational cost.
• To provide stable, error-controlled rank reduction via reduced higher-order SVD (RHOSVD) for canonical and Tucker tensors.
• To support further functional calculus (e.g., integration, differentiation) on the resulting potential fields using tensor arithmetic.

Proposed method

• Represent the fundamental potential kernel (e.g., Newton, Yukawa) on a large N×N×N grid using low-rank Tucker or canonical tensor formats via Laplace transform and sinc-quadrature approximation.
• Generate shifted tensor copies of the reference potential for each lattice site by translating the core tensor and skeleton vectors along lattice vectors.
• Assemble the full lattice sum by 1D summation of the directional vectors (skeletons) of the shifted tensors, preserving low separation ranks.
• Apply ε-rank reduction via reduced higher-order SVD (RHOSVD) to control the tensor rank of the sum, especially after algebraic summation over defective sites.
• Use the canonical-to-Tucker transformation to convert high-rank canonical sums into low-rank Tucker formats with stable error bounds in terms of discarded singular values.
• Leverage tensor arithmetic to enable efficient downstream operations like integration and differentiation with 1D complexity scaling.

Experimental results

Research questions

• RQ1 Can low-rank tensor formats be used to represent and sum long-range potentials on 3D lattices with defects while maintaining linear complexity?
• RQ2 How can tensor rank be controlled after algebraic summation of multiple defective lattice contributions?
• RQ3 What are the stability and error bounds of the reduced higher-order SVD approximation for sum of canonical or Tucker tensors?
• RQ4 Can the method achieve O(L) computational complexity for lattice sums on non-rectangular or defective lattices, compared to O(L³) in standard methods?
• RQ5 To what extent can the tensor representation support further numerical operations like differentiation or integration with minimal additional cost?

Key findings

• The method achieves computational complexity of O(NL) and storage of O(N), scaling linearly with the 1D grid size N, instead of O(L³) in classical approaches.
• The tensor rank of the resulting sum remains bounded by the rank of the reference potential, even after summation over millions of lattice sites with defects.
• Numerical experiments confirm that millions of Newton kernels on a 3D lattice with defects can be summed in seconds using a Matlab implementation.
• The reduced HOSVD approximation provides stable error bounds in the relative norm, proportional to the sum of discarded singular values from side matrices.
• The approach is applicable to a broad class of interaction kernels, including Newton, Yukawa, Lennard-Jones, and dipole-dipole potentials.
• The Tucker and canonical tensor formats allow efficient functional calculus (e.g., integration, differentiation) on the potential field using 1D-complexity tensor arithmetic.