UNCECOMP 2025
Date:
In recent years, dynamical low-rank (DLR) methods have been employed in various filtering algorithms (e.g., Kalman-like filters, particle filters) to significantly reduce the computational cost associated with the forecast step. The main idea is to approximate the solution by a low-rank matrix (i.e., a SVD-like decomposition) to speedup computations. Compared to projection-based reduced-order models, the difference is that both the reduced basis and the coefficients (i.e., all the factors of the SVD-like decomposition) evolve over time. Several methods have been proposed in the literature to integrate in time the factors of the SVD-like decomposition. In particular, we focus here on the Basis Update & Galerkin (BUG) integrator, which is based on the explicit Euler method and which has recently been extended to second order using the midpoint rule. In this work, we further extend the BUG integrator to high-order Runge-Kutta methods. Notably, any Runge-Kutta method can be used. The performance of the resulting integrators is assessed on two-dimensional, time-dependent, nonlinear problems.