Error estimates for SUPG-stabilised Dynamical Low Rank Approximations

TL;DR

This work provides an a priori error analysis for a SUPG-stabilised Dynamical Low Rank (DLR) method applied to random, time-dependent advection-dominated problems. By embedding SUPG stabilization into the DLR manifold (with Dual DO tangents) and using a collocation discretisation in probability along with a splitting-like time integration, the authors derive an error bound that scales with $h^{k+1}$, $\triangle t$, $\delta^{1/2} h^{k}$, and $\delta^{-1/2} h^{k+1}$, plus a model-error term $\nu$ and an initial-projection error; they also prove norm-stability of the scheme. The analysis relies on an oblique projection onto the tangent space, a stability estimate for the SUPG bilinear form, and a Local Basis Inverse inequality for the spatial basis. Numerically, the results validate the predicted convergence rates and demonstrate the method's efficiency and robustness for low-rank representations in stochastic advection-diffusion problems. This provides a practical pathway to stable, efficient reduced-order simulations in uncertain, transport-dominated regimes.

Abstract

We perform an error analysis of a fully discretised Streamline Upwind Petrov Galerkin Dynamical Low Rank (SUPG-DLR) method for random time-dependent advection-dominated problems. The time integration scheme has a splitting-like nature, allowing for potentially efficient computations of the factors characterising the discretised random field. The method allows to efficiently compute a low-rank approximation of the true solution, while naturally "inbuilding" the SUPG stabilisation. Standard error rates in the L2 and SUPG-norms are recovered. Numerical experiments validate the predicted rates.

Figures (1)

• Figure 1: SUPG error for $k = 1, 2$ and small approximation rank $R$.

Theorems & Definitions (10)

• Lemma 2.1
• proof
• Proposition 2.1
• Proposition 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• proof