Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems

Abstract

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

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows

“…More recently, physical constraints have started to be used in standard (ie, without ROM) LES closure modeling (see, eg, the works of Duraisamy et al and Wang et al). Finally, physical constraints have also been used in standard ROM (ie, without closure modeling) . The CDDC‐ROM proposed in this paper uses physical constraints to improve the physical accuracy of the ROM closure model (ie, the Correction term in the DDC‐ROM).…”

“…Finally, physical constraints have also been used in standard ROM (ie, without closure modeling). [45][46][47][48][49][50][51][52] The CDDC-ROM proposed in this paper uses physical constraints to improve the physical accuracy of the ROM closure model (ie, the Correction term in the DDC-ROM).…”

Structure-preserving model reduction for dynamical systems with a first integral

“…Therefore, model reduction while preserving such properties would be preferred: for example, if the reduced-order system inherits the mathematical structures, one could easily choose an appropriate numerical integrator for the reduced-order system. Structure-preserving model reduction methods have received attention in recent years (see [1,5,12,22] and references therein).…”

“…Peng and Mohseni [22] proposed model reduction techniques that find a lower order Hamiltonian system, to which any structure-preserving integrators developed for Hamiltonian systems can be applicable. For the case S(y y y) is a constant skew-symmetric matrix but is not necessarily of the form (2), Gong et al [12] proposed a model reduction approach that yields a lower-order skew-gradient system with a constant skew-symmetric matrix. These structure-reserving model reduction methods are briefly reviewed in Section 2.…”

“…This situation often arises, for example, as a Hamiltonian system with some constraints and from discretizing a Hamiltonian partial differential equation (PDE). The simple application of the approach [12] gives a lower order skew-gradient system; however, the computational complexity for evaluating the vector field may still depend on n (the size of the original problem). In this paper, we study structure-preserving model reduction techniques so that the vector field of the reduced-order system can be evaluated efficiently.…”