This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches.

翻译：这项工作为模拟非分裂现象的有限维度的汉密尔顿系统提出了一种适应性结构保留命令减少模式示范方法,用于模拟非分裂现象。为了克服运输问题典型的逐渐衰落的科尔莫戈罗夫宽度,整个模型大约用于使用动态低级近似技术及时调整的当地缩小的空间。通过将汉密尔顿矢量场在近距离空间的同步投影与当地缩小的空间相近,可以确定较低的动力。这确保了汉密尔密尔顿动态的峡谷静脉冲结构在减少期间得以保持。此外,通过允许缩小的空间在时间演变期间发生变化而获得低位解决方案的精确近似值。如果通过误差指标评估的减少的解决方案的质量不令人满意,则降低的基点在参数方向上得到增强,而目前的基点则最差。涉及波相互作用、非线性运输问题和Vlasov方程式的大规模数字测试显示了拟议方法相对于全球和传统缩小基点方法的高度稳定性和大量运行速度。