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.
Highlights
Hamiltonian systems describe conservative dynamics and non-dissipative phenomena in, for example, classical mechanics, transport problems, fluids and kinetic models
For the model order reduction of (3) we propose an adaptive dynamical scheme based on approximating the full model solution in a lower-dimensional space that is evolving, and whose dimension may change over time
To overcome these issues in the numerical solution of the reduced dynamics (7), we introduce two approximations: first we replace the rank-deficient matrix S with an ε-regularization that preserves the skew-Hamiltonian structure of S and in finite precision arithmetic, we set as velocity field for the evolution of the reduced basis U an approximation of F in the space HU(t), for all t ∈ Tτ
Summary
To the best of our knowledge the only dynamical low-rank approximation methods able to preserve the geometric structure of Hamiltonian dynamics were proposed in [24] to deal with the spatial approximation of the stochastic wave equation and in [26] to deal with finite-dimensional Hamiltonian systems The gist of these methods is to approximate the full model solution in a low-dimensional manifold that evolves in time and possesses the symplectic structure of the full phase-space. Two major difficulties are associated with this approach: (i) to maintain the global Hamiltonian structure of the dynamics while modifying the reduced phase space; and (ii) to evolve the system on the updated space starting from a rank-deficient initial condition To address these problems, we devise a regularization of the velocity field of the reduced flow so that the resulting vector belongs to the tangent space of the updated reduced manifold, and the Hamiltonian structure is preserved
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