Rank-adaptive structure-preserving model order reduction of Hamiltonian systems

TL;DR

This paper introduces an adaptive, structure-preserving model order reduction technique for Hamiltonian systems that dynamically adjusts the reduced space to maintain accuracy and symplectic structure, enabling efficient simulations of complex non-dissipative phenomena.

Contribution

It develops a novel adaptive reduction method that preserves symplectic structure and adjusts the reduced space dimension in real-time for Hamiltonian systems.

Findings

Ensures preservation of symplectic structure during reduction.

Achieves significant runtime speedups over traditional methods.

Demonstrates stability and accuracy on complex physical models.

Abstract

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches.