We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge–Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost.

我们考虑描述非耗散现象的参数化哈密顿系统的模型降阶，如波浪型和输运主导问题。这种模型的简化基础方法的发展受到两个主要因素的挑战：编码动力学物理和稳定性特性的丰富几何结构及其局部低级性质。为了解决这些问题，我们提出了一种非线性结构保持模型缩减，其中缩减的相空间随时间演变。本着动力学低秩逼近的精神，简化的动力学是通过将哈密顿向量场辛投影到每个简化状态的逼近流形的切线空间来获得的。先验误差估计是根据完整模型解决方案在简化流形上的投影误差来建立的。对于减少动态的时间离散化，我们采用分裂技术。简化的基满足一维等于完整模型大小的辛和正交矩形矩阵的流形上的演化方程。我们在矩阵流形的切线空间上重新定义问题，并开发基于李群技术和显式 Runge-Kutta (RK) 方案的内在时间积分器。所得方法显示为收敛于 RK 积分器的阶数，并且它们的计算复杂度仅线性地取决于完整模型的维度，前提是对降低的流速的评估具有可比的成本。