A practical investigation on time integration in the quantized tensor train format

TL;DR

This paper explores how different time integration strategies, numerical dissipation, and problem representations impact the efficiency and accuracy of quantized tensor train methods in simulating advection-dominated problems.

Contribution

It provides a practical investigation into improving QTT-based simulations by analyzing the effects of various numerical choices on long-time dynamical computations.

Findings

Choice of time integrator affects low-rank approximation stability.

Adding numerical dissipation can reduce noise and rank growth.

Problem representation influences the efficiency of QTT calculations.

Abstract

Quantized tensor trains (QTTs) are a multiscale computational framework that can potentially reduce the computational cost of solving partial differential equations and initial value problems by making low-rank approximations. However, its use is somewhat limited in practice, in part due to the challenges that arise when making low-rank approximations of the quantized data. For example, when performing long-time dynamical numerical simulations, it has been observed that the accumulation of numerical errors arising from both the discretization of the partial differential equation itself and the low-rank approximation can lead to increased rank and noise-dominated results. Focusing on a set of advection-dominated test problems relevant to electromagnetic plasmas and electromagnetic fields, this work investigates how the choice in time integrator, the addition of numerical dissipation, and the choice in problem representation can affect the efficiency and success of the QTT calculations.