Robust high-order low-rank BUG integrators based on explicit Runge--Kutta methods

2025/02/10 by Fabio Nobile, Sébastien Riffaud, Nobile, Fabio +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical methods for differential equations #cs.NA #math.DS #math.NA

paper · pdf · doi:10.48550/arxiv.2502.07040

openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work, we introduce high-order Basis-Update & Galerkin (BUG) integrators based on explicit Runge-Kutta methods for large-scale matrix differential equations. These dynamical low-rank integrators extend the BUG integrator to arbitrary explicit Runge-Kutta schemes by performing a BUG step at each stage of the method. The resulting Runge-Kutta BUG (RK-BUG) integrators are robust with respect to small singular values, fully forward in time, and high-order accurate, while enabling conservation and rank adaptivity. We prove that RK-BUG integrators retain the order of convergence of the underlying Runge-Kutta method until the error reaches a plateau corresponding to the low-rank truncation error, which vanishes as the rank becomes full. This theoretical analysis is supported by several numerical experiments. The results demonstrate the high-order convergence of the RK-BUG integrator and its superior accuracy compared to other existing dynamical low-rank integrators.

Citations

• The Approximation of One Matrix by Another of Lower Rank

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• Dynamical Low-Rank Filters for Data Assimilation
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