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.

翻译：经典参数化问题的模型降阶（MOR）可能因所需投影基规模过大而计算效率低下，尤其针对具有缓慢衰减Kolmogorov n-宽度的系统。此外，动力系统的哈密顿结构可能存在且应在降阶过程中予以保持。本文针对这两个问题，提出了一种基于字典的在线自适应MOR方法。该方法需要为状态变量、非线性项及离散经验插值（DEIM）点构建字典。在线仿真过程中，在保持在线高效性的前提下执行局部基扩展/简化操作，即基修正与降阶模型在线仿真的运行时复杂度均不依赖于全阶状态维度。通过线性波动方程与非线性正弦-戈登方程的数值算例验证了该方法的有效性。