TY - JOUR
T1 - Rank-adaptive structure-preserving model order reduction of Hamiltonian systems
AU - Hesthaven, Jan S.
AU - Pagliantini, Cecilia
AU - Ripamonti, Nicolò
PY - 2022/3/8
Y1 - 2022/3/8
N2 - 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.
AB - 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.
KW - math.NA
KW - cs.NA
KW - Adaptive algorithms
KW - Dynamical low-rank approximation
KW - Hamiltonian dynamics
KW - Reduced basis methods (RBM)
KW - Symplectic manifolds
UR - http://www.scopus.com/inward/record.url?scp=85126980532&partnerID=8YFLogxK
U2 - 10.1051/m2an/2022013
DO - 10.1051/m2an/2022013
M3 - Article
SN - 2822-7840
VL - 56
SP - 617
EP - 650
JO - Mathematical Modelling and Numerical Analysis
JF - Mathematical Modelling and Numerical Analysis
IS - 2
ER -