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

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.

💡 Research Summary

The paper introduces a novel time‑integration framework that combines Quantized Tensor Trains (QTTs) with an interpolative version of the Dynamical Low‑Rank Approximation (DLRA). Traditional DLRA projects the governing equations onto a low‑rank manifold using orthogonal projectors. While this works well for linear problems, it becomes cumbersome for nonlinear PDEs that require element‑wise operations such as upwind fluxes, nonlinear source terms, or pointwise multiplications. The authors replace the orthogonal construction with an interpolative (CUR‑based) construction, where each tensor core is represented by a set of interpolation points and the corresponding values of the original tensor at those points. The interpolation indices are selected by algorithms such as maxvol or q‑DEIM, yielding matrices C, R, and a small core (\hat M) that satisfy relaxed orthogonality conditions.

QTTs are obtained by folding a one‑dimensional grid of size (N=2^L) into an (L)-dimensional binary tensor, thus reducing the effective dimensionality from (N) to (\log_2 N). This multiscale representation allows the computational complexity of many operations to scale as (\mathcal{O}(r^3 K \log N)) rather than (\mathcal{O}(N K)), where (r) is the QTT rank and (K) the number of physical dimensions. By embedding the interpolative DLRA within the QTT format, the authors obtain a low‑rank manifold that supports direct element‑wise evaluation, making it naturally suited for nonlinear time‑integration schemes.

The paper details two ways to realize interpolative DLRA for QTTs: (i) directly using the interpolative TT cores within the projector‑splitting integrator, and (ii) converting between interpolative and orthogonal representations via oblique projections, thereby reusing existing orthogonal DLRA solvers when convenient. Both approaches preserve the low‑rank structure while allowing the nonlinear terms to be evaluated without reconstructing the full tensor.

Three numerical experiments illustrate the advantages. First, the inviscid Burgers equation is solved with an upwind scheme; the interpolative DLRA accurately captures the nonlinear flux without the excessive rank growth observed in the orthogonal DLRA. Second, an electromagnetic pulse propagating in a dielectric cavity is simulated; the interpolative method yields lower errors and smaller ranks compared to both orthogonal DLRA and a global step‑and‑truncate approach, even when the underlying operator (curl) is not separable across the quantized dimensions. Third, a simple advection problem is integrated with higher‑order Runge‑Kutta methods, demonstrating that the interpolative framework can accommodate sophisticated time integrators while adaptively adjusting the rank. Across all tests, the QTT‑interpolative DLRA combination reduces memory consumption and CPU time dramatically, especially at high resolutions (e.g., (N = 2^{20})).

The authors also discuss the theoretical aspects of projecting dynamics onto an interpolative manifold. Although CUR decompositions are near‑optimal for matrices, their compatibility with the projector‑splitting DLRA for tensors is non‑trivial. The paper shows that the projection error introduced by the interpolative manifold is bounded and does not dominate the overall discretization error, provided the selected interpolation indices capture the dominant subspace of each core. Adaptive re‑selection of indices can further control rank growth and maintain accuracy.

In conclusion, this work demonstrates that interpolative DLRA, when coupled with the multiscale QTT representation, offers a powerful tool for simulating nonlinear PDEs that require element‑wise operations. It overcomes the limitations of orthogonal DLRA, delivers substantial computational savings, and opens avenues for future research, including adaptive index selection, hybrid global‑local update schemes, and implementation on modern hardware accelerators.