Time integration of quantized tensor trains using the interpolative dynamical low-rank approximation

TL;DR

This work investigates interpolative dynamical low-rank approximation (DLRA) for time integration on quantized tensor trains (QTTs), introducing two implementations: a purely interpolative approach (DLR-X) and an oblique-projection variant (DLR-P) that connects interpolative and orthonormal representations. It compares these with the traditional orthonormal DLRA (DLR-G) across three numerical tests—inviscid Burgers’ equation, Maxwell’s equations in a non-uniform dielectric, and linear 2-D advection with Boltzmann’s equation—to assess accuracy, stability, and rank behavior. The results show that interpolative DLRA can handle nonlinear, element-wise operations and enable higher-order time integration through basis expansion, though performance is problem-dependent and may require oversampling or projection to maintain stability. The study highlights the trade-offs between accuracy, stability, and computational cost, and demonstrates that oblique projections offer a robust bridge to standard DLRA methods, expanding the toolbox for high-resolution, multi-scale PDE simulations. It also discusses rank-adaptivity strategies (e.g., two-site variants) and conservation considerations as future directions.

Abstract

Quantized tensor trains (QTTs) are a low-rank and multiscale framework that allows for efficient approximation and manipulation of multi-dimensional, high resolution data. One area of active research is their use in numerical simulation of hyperbolic systems such as the Navier-Stokes equations and the Vlasov equations. One popular time integration scheme is the dynamical low-rank approximation (DLRA), in which the time integration is constrained to a low-rank manifold. However, until recently, DLRA has typically used orthogonal projectors to project the original dynamical system into a reduced space, which is only well-suited for linear systems. DLRA has also mostly been investigated in the context of non-quantized tensor trains. This work investigates interpolative DLRA schemes in which the low-rank manifold is constructed from aptly chosen interpolation points and interpolating polynomials, in the context of QTTs. Through various examples, its performance is compared to its orthogonal counterpart. This work demonstrates how interpolative DLRA is suitable for nonlinear systems and time integrators requiring nonlinear element-wise operations, such as upwind time integration schemes.

Figures (10)

• Figure 1: An overview of the three steps in a single iteration of the dynamical low-rank time integration algorithm (see Sec. \ref{['sec:final']}) with the interpolative TT construction (DLR-X), solving $\partial_t x + Az = b$. This figure explicitly shows the DLR procedure with the center of orthogonality at the fourth tensor core, with the sweeping procedure going from left to right. However, the procedure is iterated over all cores in the tensor train. In the diagrams, the TT with green cores depicts the vector $x$, the TT with blue cores depicts the operator $A$, and the TT with the purple cores depicts the vector $b$. The right/left-pointing triangles denote left/right canonical tensors in the interpolative construction, and the flat rectangles denote index selection.
• Figure 2: An overview of the three steps in a single iteration of the dynamical low-rank time integration algorithm (see Sec. \ref{['sec:final']}) with oblique projection (DLR-P), solving $\partial_t x + Ax = b$. This figure explicitly shows the DLR procedure with the center of orthogonality at the fourth tensor core, with the sweeping procedure going from left to right. However, the procedure is iterated over all cores in the tensor train. Here, the right/left-pointing triangles denote left/right canonical tensors in the orthonormal construction.
• Figure 3: DLR time evolution with rank truncation threshold $\varepsilon=10^{-4}$ on a uniform grid with $2^9$ grid points. Test problems include (a) shock formation, with initial condition $\sin(2\pi x)$, (b) shock propagation, with initial condition $1-H(x-0.5)$, where $H(x)$ is the Heaviside step function, and (c) rarefaction, with initial condition $H(x-0.5)$. The DLR-X calculations are performed with each of the specified truncation thresholds. (Top) the time-evolved velocity field at $T=0.5$ for (a) and (b), and $T=0.25$ for (c). (Second row) the error of field $u$ with respect to the $\varepsilon=10^{-14}$ result, denoted as the reference. (Third row) the error arising from the DLR approximation over the course of the simulation. (Bottom) the total number of evaluations of Eq. \ref{['eq:upwind_euler_burgers']}. For the dense calculation, the total number of evaluations is $2^9$, and is depicted by the thin dotted line.
• Figure 4: Results for the Gaussian packet in a dielectric cavity with $2^8$ grid points per dimension. Plots in the top row correspond to results obtained with AP DLR-X, while plots in the bottom row correspond to results obtained with step-and-truncate (SAT). (a-b) QTT ranks for DLRA and SAT before the final rank truncation procedure ($r_{in}$). The rank is smaller for DLR-X than for SAT, which is beneficial since the computational cost scales polynomially with $r_{in}$. (c-d) QTT ranks after the final rank truncation procedure. Notice that the rank for the DLR-X calculation is slightly larger than that of the SAT calculation after times $t=1$. This arises from the approximate nature of the tangent space constructed during the DLR procedure, and can be mitigated by using $\varepsilon_{in} < \varepsilon$ during the DLRA procedure. (e) The number of Euler evaluations in DLR-X compared to the dense grid calculation (the dotted line at $256^2$ evaluations per time step).
• Figure 5: Comparison of results for different AP DLR methods: the standard AP DLR time integration (AP-G), DLR time integration on an interpolative basis (AP-X), and DLR time integration with oblique projection (AP-P). Calculations are performed with $2^8$ grid points per dimension and an error threshold $\varepsilon=10^{-4}$. (a) Ranks of the QTT over simulation time. (b-d) Image of the $E_z$ field at time $t=1.2$ obtained using the three methods. The AP-G result is far from the true solution. The AP-P result shows more noise than the AP-X result.