We propose a novel framework of generalised Petrov–Galerkin Dynamical Low Rank (DLR) Approximations in the context of random PDEs. It builds on the standard Dynamical Low Rank Approximations in their Dynamically Orthogonal formulation. It allows to seamlessly build-in many standard and well-studied stabilisation techniques that can be framed as either generalised Galerkin methods, or Petrov–Galerkin methods. The framework is subsequently applied to the case of Streamline Upwind/Petrov–Galerkin (SUPG) stabilisation of advection-dominated problems with small stochastic perturbations of the transport field. The norm-stability properties of two time discretisations are analysed. Numerical experiments confirm that the stabilising properties of the SUPG method naturally carry over to the DLR framework.

中文翻译：

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Petrov-Galerkin 动力学低秩近似：平流主导问题的 SUPG 稳定


我们在随机偏微分方程的背景下提出了一种新的广义 Petrov-Galerkin 动态低秩 （DLR） 近似框架。它建立在标准的 Dynamical Low Rank Approximations 的 Dynamical Orthogonal 公式之上。它允许无缝地内置许多标准的和经过充分研究的稳定技术，这些技术可以被构建为广义的 Galerkin 方法或 Petrov-Galerkin 方法。该框架随后应用于流线逆风/彼得罗夫-加辽金 （SUPG） 稳定对流主导的问题，以及输运场的小随机扰动。分析了两个时间离散的范数稳定性特性。数值实验证实，SUPG 方法的稳定特性自然而然地延续到 DLR 框架中。