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

This is the authors' abstract. We don't add key points for this paper.

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 the 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 typically uses orthogonal projectors to project the original dynamical system into a reduced space, which is best-suited for linear partial differential equations (PDEs). For nonlinear PDEs requiring element-wise evaluations, one can instead consider interpolative DLRA, in which the low-rank manifold is constructed from aptly chosen interpolation points and interpolating polynomials. Thus far, interpolative DLRA has only been investigated in the context of non-quantized tensor trains. This work extends the method to QTTs, and combines the interpolative representation with various basis expansion schemes to achieve higher order time integration. Through various examples, its performance is compared to its orthogonal counterpart, demonstrating that interpolative DLRA is indeed suitable for nonlinear systems and time integrators requiring nonlinear element-wise operations, such as upwind time integration schemes.