Structure-Preserving Galerkin POD Reduced-Order Modeling of Hamiltonian Systems

摘要

The proper orthogonal decomposition reduced-order models (POD-ROMs) have beenwidely used as a computationally efficient surrogate models in large-scalenumerical simulations of complex systems. However, when it is applied to aHamiltonian system, a naive application of the POD method can destroy itsHamiltonian structure in the reduced-order model. In this paper, we develop anew reduce-order modeling approach for the Hamiltonian system, which uses thetraditional framework of Galerkin projection-based model reduction but modifiesthe ROM so that the appropriate Hamiltonian structure is preserved. Since thePOD truncation can degrade the approximation of the Hamiltonian function, wepropose to use the POD basis from shifted snapshots to improve the Hamiltonianfunction approximation. We further derive a rigorous a priori error estimate ofthe structure-preserving ROM and demonstrate its effectiveness in severalnumerical examples. This approach can be readily extended to dissipativeHamiltonian systems, port-Hamiltonian systems etc.