In the context of Uncertainty Quantification, simulating high-dimensional stochastic dynamical systems remains a challenging task owing to the high computational cost they incur. The Dynamical Low-Rank (DLR) method, which has gained increasing attention in recent years, has characteristics that are naturally adapted to that setting. The method approximates a random field u(t, x, w) by Sum_{i=1}^R U_i(t, x) Y_i(t, w), where both the physical bases {U_i}{i=1}^R and stochastic bases {Y_i}{i=1}^R evolve in time. This allows the bases to dynamically accommodate the onset of new features, thereby evolving at all times a low-rank approximation of the true solution. This thesis extends the state-of-the-art of DLR approximations in two settings where the method has the potential to provide substantial benefits: random Partial Differential Equations (PDEs) and Data Assimilation
In the setting of random PDEs, our first contribution is the development of a generalised Petrov-Galerkin Dynamical Low-Rank (PG-DLR) framework, which allows to embed various numerical stabilisation techniques within the DLR framework. To showcase the uses of the framework, we specialise it to the case of the Streamline Upwind Petrov-Galerkin (SUPG) stabilisation method, yielding the SUPG-DLR method. We establish L^2 stability results and error estimates analogous to those available for SUPG stabilisations of the full order model. Our second contribution is the analysis of a modified Basis Update & Galerkin (BUG) scheme for random parabolic PDEs. The algorithm combines explicit basis updates with a non-standard retraction and either explicit or implicit Galerkin steps, resulting in two practical variants for which we establish norm-stability bounds. We generalise those norm-stability properties to extensions of the BUG algorithm for tensors, specifically for the BUG-TF (for Tucker format) and BUG-TT (for Tensor Trains).
In the setting of Data Assimilation, we propose a framework merging Kalman-Bucy filters with the Dynamical Low-Rank approximation technique. We establish a DLR-based reduction of the Kalman-Bucy process, named DLR-KBP. The approximation, which is valid in the case of small model noise, evolves the filtering distribution on a low-dimensional, time-varying subspace, thereby reducing computational cost.
We prove several defining properties of the DLR-KBP, including reduced Kalman-Bucy equations characterising their mean and covariance. We then propose an ensemble Kalman filter (ENKF) extension of the DLR-KBP, the DLR-ENKF, and investigate its theoretical properties, establishing its well-posedness and a propagation-of-chaos property linking it to its mean-field limit, the DLR-KBP. In our final contribution, we apply the DLR-ENKF framework to joint state/parameter estimation problems. We formulate a suitable DLR-ENKF system for nonlinear dynamics and propose an efficient forecast/analysis-based time-integration scheme, which is combined with a novel DEIM (Discrete
