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

TL;DR

This paper explores interpolative dynamical low-rank approximation (DLRA) methods for quantized tensor trains (QTTs), demonstrating their effectiveness for nonlinear systems and complex time integration schemes in high-dimensional data simulations.

Contribution

It introduces and analyzes interpolative DLRA schemes for QTTs, extending their applicability to nonlinear systems and advanced time integrators.

Findings

Interpolative DLRA performs well on nonlinear systems.

It is suitable for time integrators with nonlinear element-wise operations.

Compared to orthogonal DLRA, it offers advantages in certain complex scenarios.

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.