A DEIM Tucker Tensor Cross Algorithm and its Application to Dynamical Low-Rank Approximation

TL;DR

This work tackles the computational bottleneck of Tucker-based DLRA in high-dimensional tensor differential equations by introducing DEIM-FS, a cross Tucker tensor algorithm that constructs near-optimal low-rank models from sparsely sampled fibers. The method uses DEIM to select fibers and achieves accuracy close to HOSVD while reducing data access and memory to scalable levels, enabling efficient DLRA even for nonlinear, high-order RHS tensors. A black-box variant (DEIM-FS iterative) removes the need for exact singular vectors, and rank adaptivity further tunes the model to a user-defined tolerance. The authors demonstrate substantial memory and time savings in DLRA applications, including 4D Fokker–Planck and 4D nonlinear advection, illustrating practical impact for solving high-dimensional, nonlinear tensor differential equations.

Abstract

We introduce a Tucker tensor cross approximation method that constructs a low-rank representation of a $d$-dimensional tensor by sparsely sampling its fibers. These fibers are selected using the discrete empirical interpolation method (DEIM). Our proposed algorithm is referred to as DEIM fiber sampling (DEIM-FS). For a rank-$r$ approximation of an $\mathcal{O}(N^d)$ tensor, DEIM-FS requires access to only $dNr^{d-1}$ tensor entries, a requirement that scales linearly with the tensor size along each mode. We demonstrate that DEIM-FS achieves an approximation accuracy close to the Tucker-tensor approximation obtained via higher-order singular value decomposition at a significantly reduced cost. We also present DEIM-FS (iterative) that does not require access to singular vectors of the target tensor unfolding and can be viewed as a black-box Tucker tensor algorithm. We employ DEIM-FS to reduce the computational cost associated with solving nonlinear tensor differential equations (TDEs) using dynamical low-rank approximation (DLRA). The computational cost of solving DLRA equations can become prohibitive when the exact rank of the right-hand side tensor is large. This issue arises in many TDEs, especially in cases involving non-polynomial nonlinearities, where the right-hand side tensor has full rank. This necessitates the storage and computation of tensors of size $\mathcal{O}(N^d)$. We show that DEIM-FS results in significant computational savings for DLRA by constructing a low-rank Tucker approximation of the right-hand side tensor on the fly. Another advantage of using DEIM-FS is to significantly simplify the implementation of DLRA equations, irrespective of the type of TDEs. We demonstrate the efficiency of the algorithm through several examples including solving high-dimensional partial differential equations.

Figures (7)

• Figure 1: Schematic of the DEIM-FS cross algorithm for a 3D tensor. For simplicity, we assume that all selected fibers are adjacent to each other.
• Figure 2: Toy Example: Comparison of the FSTD, DEIM-FS, DEIM-FS (iterative), and HOSVD: (a) approximation errors of $\mathcal{F}_1$ and $\mathcal{F}_2$ versus $r'_{\mathcal{F}}$ for a fixed rank of $r_{\mathcal{F}}$; (b) approximation error of $\mathcal{F}_1$ versus rank; (c) approximation error of $\mathcal{F}_2, (b=3,5)$ versus rank; (d) approximation error of $\mathcal{F}_2, (b=5)$ versus rank; (e) effect of fiber initialization on the approximation error of $\mathcal{F}_2, (b=5)$ for a fixed rank of $r_{\mathcal{F}}=20$ for 100 different initializations; (f) convergence of the singular values of the core tensor ($\mathcal{S}_{\mathcal{F}}$) versus iterations for DEIM-FS (iterative) algorithm.
• Figure 3: Fokker Planck Equation: (a) first 5 singular values of analytical solution and DLRA-DEIM-FS; (b) relative error evolution for different $r$ and $r^{}_{\mathcal{F}}$.
• Figure 4: Four-dimensional nonlinear advection equation (Example 3): (a) Relative error evolution with fixed $r$ for different $\epsilon_l$ and $\epsilon_u$; (b) Evolution of $r^{}_{\mathcal{F}}$ associated with the plots of Figure 4a; (c) Relative error evolution for different $r$ with fixed $\epsilon_l = 10^{-5}$ and $\epsilon_u = 10^{-4}$; (d) Relative error evolution for different $\Delta t$ with fixed $r$, $\epsilon_l$, and $\epsilon_u$.
• Figure 5: Four-dimensional nonlinear advection equation (Example 3): (a) First 11 singular values of FOM and TDB-DEIM-FS; (b) Evolution of $r^{}_{\mathcal{F}}$ associated with the solved system in Figure 5a.

Theorems & Definitions (1)

• Remark 1