Uncertainty quantification for random Hamiltonian systems by using polynomial expansions and geometric integrators

Recent advances in the field of uncertainty quantification are based on achieving suitable functional representations of the solutions to random systems. This aims at improving the performance of Monte Carlo simulation, at least for low-dimensional problems and moderately large independent variable. One of these functional representations are the so-called generalized polynomial chaos (gPC) expansions, based upon orthogonal polynomial decompositions. When the input random parameters are independent (a germ), a Galerkin projection technique applied to the truncated gPC expansion is usually employed. This approach exhibits fast mean-square convergence for smooth dynamics, whenever applicable. However, the main difficulty arises when solving the Galerkin system for the gPC coefficients, which may rely on different solvers (algorithms and codes) than those for the original system. A recent contribution noticed that, for random Hamiltonian systems, the Galerkin system is Hamiltonian too. Thus, the well-known symplectic integrators can be applied. The present paper investigates random Hamiltonian systems in general, when the input random parameters may be non-independent. In such a case, polynomial expansions based on the canonical basis and an imitation of the Galerkin projection technique are proposed. The Hamiltonian structure of the original system is unfortunately not conserved, but volume preservation is. Hence volume-preserving integrators are of use. Numerical experiments suggest that the proposed polynomial expansions may be useful for fast and accurate uncertainty quantification.