A parallel Basis Update and Galerkin Integrator for Tree Tensor Networks

TL;DR

This work develops a robust, fully parallel dynamical low-rank integrator for high-dimensional tensor differential equations by extending the parallel Basis-Update & Galerkin (BUG) approach from matrices to Tucker tensors and general tree tensor networks (TTNs). By updating all bases and connecting tensors in parallel within each time step and incorporating augmentation and rank truncation, the method achieves parallel scalability while remaining robust to small singular values, with a proven error bound that is independent of those singular values. Theoretical results are complemented by numerical experiments on quantum spin systems and radiative transfer, demonstrating accuracy comparable to rank-adaptive approaches but with notable speedups. The framework offers a scalable tool for DLRA in high-dimensional settings, enabling efficient simulations in quantum physics and uncertainty quantification. Overall, the parallel TTN BUG integrator advances robust, parallel time integration for hierarchical tensor representations, providing concrete benefits in both efficiency and stability for complex DLRA problems.

Abstract

Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update & Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer.

Figures (8)

• Figure 1: Augmentation of an order three tensor. Left: Illustration of the augmentation of a connecting tensor $\bar{C}_\tau^1$ (light grey left top) with $\widetilde{C}_{\tau_1}^1$ in the first dimension (light grey left down) and with $\widetilde{C}_{\tau_2}^1$ in the second dimension (light grey right top). Right: Illustration of the augmentation of the connecting tensor from the left in the $0$-dimension (dark grey block). All remaining blocks are set to zero.
• Figure 1: Unspecified parameters are $\Omega=\Delta=V=\alpha=1$, and $\vartheta=10^{-8}$. Left: Errors for $d=8$ particles at time $T=1$. Right: Maximal ranks over time for $d=16$ particles and $r_{\text{max}}=30$.
• Figure 2: Expected scalar flux at time $t = 2$, using $200$ spatial cells, $100$ moments in direction, and $100$ points in both random variables. The parallel integrator takes 12 seconds, while the rank-adaptive BUG integrator takes 30 seconds.
• Figure 3: Variance of the scalar flux at time $t = 2$, using $200$ spatial cells, $100$ moments in direction, and $100$ points in both random variables. The parallel integrator takes 12 seconds, while the rank-adaptive BUG integrator takes 30 seconds.
• Figure 4: Ranks chosen by the integrator during simulation. The root tensor always has rank $1$. The connecting tensor of the tree $\tau_1$ connects spatial and angular dimensions, whereas the connecting tensor of the tree $\tau_2$ connects the uncertain dimensions.

Theorems & Definitions (7)

• Theorem 3.1
• Proof 1
• Definition 4.1: CLW2021 Ordered trees with unequal leaves
• Definition 4.2: CLW2021 Tree tensor network
• Theorem 4.3: CLS2023 Rank truncation error
• Remark 4.4
• Theorem 4.5