Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n-widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities, and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e., the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.

由于所需投影基数较大，参数化问题的经典模型降阶 (MOR) 可能会变得计算效率低下，特别是对于缓慢衰减 Kolmogorov n宽度的问题。此外，动力系统的哈密顿结构可能是可用的，并且应该在还原过程中保留。在当前的演示中，我们通过提出相应的基于字典的在线自适应 MOR 方法来解决这两方面的问题。该方法需要状态变量、非线性和离散经验插值 (DEIM) 点的字典。在在线仿真期间，以在线有效的方式执行局部基础扩展/简化，即，基础修改和简化模型的在线仿真的运行时复杂性不依赖于完整的状态维度。线性波动方程和非线性 Sine-Gordon 示例的实验证明了该方法的效率。