Symplectic Dynamical Low Rank approximation of wave equations with random parameters

In this paper we propose a dynamical low-rank strategy for the approximation of second order wave equations with random parameters. The governing equation is rewritten in Hamiltonian form and the approximate solution is expanded over a set of $2S$ dynamical symplectic-orthogonal deterministic basis functions with timedependent stochastic coefficients. The reduced (low rank) dynamics is obtained by a symplectic projection of the governing Hamiltonian system onto the tangent space to the approximation manifold along the approximate trajectory. The proposed formulation is equivalent to recasting the governing Hamiltonian system in complex setting and looking for a dynamical low rank approximation in the low dimensional manifold of all complex-valued random fields with rank equal to $S$. Thanks to this equivalence, we are able to properly define the approximation manifold in the real setting, endow it with a differential structure and obtain a proper parametrization of its tangent space, in terms of orthogonal constraints on the dynamics of the deterministic modes. Finally, we recover the Symplectic Dynamically Orthogonal reduced order system for the evolution of both the stochastic coefficients and the deterministic basis of the approximate solution. This consists of a system of $S$ deterministic PDEs coupled to a reduced Hamiltonian system of dimension $2S$. As a result, the approximate solution preserves the mean Hamiltonian energy over the flow.