A mesh-free hybrid Chebyshev-Tucker tensor format with applications to multi-particle modelling

TL;DR

The paper addresses scalable, accurate approximation of high-dimensional multivariate functions, especially long-range multi-particle potentials, by introducing a mesh-free two-level Chebyshev-Tucker tensor format (ChebTuck) that couples global Chebyshev interpolation with ALS-based Tucker compression on the coefficient tensor. The approach yields near-optimal Tucker ranks, supports CP and RS input, and provides explicit error and complexity bounds, including exponential decay for long-range components and log-scale Tucker-rank growth in particle number. Key contributions include formal definitions of the hybrid Chebyshev-Tucker format, transformation to standard Tucker format, detailed numerical schemes for functional and algebraic inputs, and demonstrated efficiency on Newton kernels, biomolecular potentials, and lattice structures. The method offers significant practical impact by enabling fast, accurate, mesh-free tensor representations for large-scale multi-particle simulations, with reproducible results and potential extensions to higher dimensions and other Green functions.

Abstract

We introduce and analyze a mesh-free two-level hybrid Tucker tensor format for approximating multivariate functions, which combines the product Chebyshev interpolation with the alternating least-squares (ALS) based Tucker decomposition of the tensor of Chebyshev coefficients. This construction allows to avoid the expensive rank-structured grid-based approximation of function-related tensors on large spatial grids, while benefiting from the Tucker decomposition of the rather small core tensor of Chebyshev coefficients. Thus, we can compute the nearly optimal Tucker decomposition of the 3D function with controllable accuracy $\varepsilon >0$ without discretizing the function on the full grid in the domain, but only using its values at small set of Chebyshev nodes. Finally, we can represent the function in the algebraic Tucker format with optimal $\varepsilon$-rank on an arbitrarily large 3D tensor grid in the computational domain by discretizing the Chebyshev polynomials on that grid. The rank parameters of the Tucker-ALS decomposition of the coefficient tensor can be much smaller than the polynomial degrees of the initial Chebyshev interpolant obtained via a function independent polynomial basis set. It is shown that our techniques can be gainfully applied to the long-range part of the singular electrostatic potential of multi-particle systems approximated in the range-separated tensor format. We provide error and complexity estimates and demonstrate the computational efficiency of the proposed techniques on challenging examples, including the multi-particle electrostatic potential for large bio-molecular systems and lattice-type compounds.

Figures (9)

• Figure 1: Left: the canonical vectors ${\bf p}_k^{(1)}$ for $k = 1, \ldots, R$, $R=26$, along the $x$-axis for the rank-$R$ CP tensor approximation of collocation tensor of size ${\bf P}_R \in \mathbb{R}^{n \times n \times n}$ with $n = 256$. Right: canonical vectors ${\bf a}_k^{(1)}$ of the long-range part in the total potential ${\bf A}_{R_l} \in \mathbb{R}^{n\times n \times n}$ with $N=500$ particles and grid size $n=256$ for $k = 1, \ldots, 8$.
• Figure 2: Relative $\ell_\infty$ error \ref{['eq:err_single_newton_1d']} of the Chebyshev interpolation of the canonical vectors vs. the degree of the Chebyshev interpolant. Each line represents the error of the Chebyshev interpolant of a canonical vector ${\bf p}_k$ for $k = 1, \ldots, 15$ ( left) and $k = 16, \ldots, 39$ ( middle). Right: Relative $\ell_\infty$ error of the degree $m=129$ Chebyshev interpolation of the canonical vectors vs. the index $k$ of the univariate function.
• Figure 3: Absolute values of the Chebyshev coefficients of the degree-129 Chebyshev interpolant of ${\bf p}_{10}$ ( left) and ${\bf p}_{30}$ ( right). Only coefficients of even degree are shown, i.e., ${\bf c}_k(0), {\bf c}_k(2), \ldots, {\bf c}_k(128)$ since $p(\|x\|)$ is even and ${\bf c}_k(1), \ldots, {\bf c}_k(129)= 0$
• Figure 4: Left: Middle slice ${\bf A}_{R_l}(:,:,n/2)$ of the (compressed) long-range part of the RS tensor for the electrostatic potential of the biomolecule with $N=500$ particles computed on the 3D $n\times n\times n$ Cartesian grid with $n=1024$ and rank $R_l=506$. Middle: the error of the canonical-to-Tucker transform, with $\varepsilon=10^{-6}$. Right: the total error of the canonical-to-canonical tensor transform with the $\varepsilon=10^{-5}$.
• Figure 5: Left: the middle slice $\hat{f}_{\bf m}(:,:,t_{n/2}) \approx {\bf A}_{R_l}(:,:,n/2)$ of the degree $m=129$ ChebTuck format of the biomolecule with $N=500$ and $n=256$. Middle: the error during RHOSVD compression, i.e. $\hat{f}_{\bf m}(:,:,t_{n/2}) - \tilde{f}_{\bf m}(:,:,t_{n/2})$. Right: the total error $\hat{f}_{\bf m}(:,:,t_{n/2}) - {\bf A}_{R_l}(:,:,n/2)$.

Theorems & Definitions (27)

• Definition 2.1: Functional Tucker format
• Definition 2.2: Two-level hybrid Tucker formats
• Definition 2.3: ChebTuck format
• Proposition 2.4
• Proof 1
• Proposition 2.5
• Proof 2
• Remark 2.6
• Remark 3.1
• Proposition 3.2