A Localized Symplectic Model Reduction Technique for Parameterized Hamiltonian systems

In this article, a localized symplectic model reduction technique, locally weighted proper symplectic decomposition (LWPSD), is proposed to simplify parameterized Hamiltonian systems. Our aim is two-fold. First, to achieve computational savings for large-scale Hamiltonian systems with parameter variation. Second, to preserve the symplectic structure of the original system. As an analogy to the proper orthogonal decomposition, the proper symplectic decomposition (PSD) can be used to construct a symplectic subspace to fit empirical data, and yield a low-order Hamiltonian system on the subspace. Instead of using a global basis to construct a global reduced model, the locally weighted approach approximates the original system by multiple lower-dimensional subspaces. Each local reduced basis is generated by the PSD of a weighted snapshot ensemble. Compared with the standard PSD, the LWPSD could yield a more accurate solution with a fixed subspace dimension. The stability, accuracy, and efficiency of the proposed technique are illustrated through the numerical simulation of the wave equation.