Structure-Preserving Reduced Basis Methods for Hamiltonian Systems with a Nonlinear Poisson Structure

We develop structure-preserving reduced basis methods for a large class of problems by resorting to their semi-discrete formulation as Hamiltonian dynamical systems. In this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries and conservation laws of the dynamics. We design reduced basis methods for the general case of nonlinear state-dependent degenerate Poisson structures based on a two-step approach. First, via a local approximation of the Poisson tensor we split the Hamiltonian dynamics into an “almost symplectic” part and the trivial evolution of the Casimir invariants. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, whereas the local Poisson tensor kernel is preserved exactly. The global Poisson structure and the conservation properties of the phase flow are retained by the reduced model in the constant-valued case and up to errors in the Poisson tensor approximation in the state-dependent case. The proposed reduction scheme is combined with a discrete empirical interpolation method (DEIM) to deal with nonlinear Hamiltonian functionals and ensure a computationally competitive reduced model. A priori error estimates for the solution of the reduced system are established. A set of numerical simulations is presented to corroborate the theoretical findings.