A new sampling technique for the application of proper orthogonal decomposition to a set of snapshots has been recently developed by the authors to facilitate a variety of data processing tasks (J. Comput. Phys. 335, 2017). According to it, robust modal expansions result from performing the decomposition on a limited number of relevant snapshots and a limited number of discretization mesh points, which are selected via Gauss elimination with double pivoting on the original snapshot matrix containing the given data. In the present work, the sampling method is adapted and combined with low-dimensional modeling. This combination yields a novel adaptive algorithm for the simulation of time-dependent non-linear dynamics in pattern-forming systems. Convenient snapshot sets, computed on demand over the evolution, are stored to record local temporal events whose underlying mechanisms are essential for the approximations. Also, a collection of sparse grid points, which are used to construct the mode basis and the reduced system of equations, is adaptively sampled according to unlinked spatial structures. The outcome is a reduced order model of the problem that (i) yields reliable approximations of the dynamical transitions, (ii) is well-suited to describe localized spatio-temporal complexity, and (iii) provides fast computations. Robustness, accuracy, and computational efficiency of the proposed algorithm are illustrated for some relevant pattern-forming systems, in both one and two spatial dimensions, exhibiting solutions with a rich spatio-temporal structure.

作者最近开发了一种新的采样技术，用于将适当的正交分解应用于一组快照，以促进各种数据处理任务（J. Comput. Phys.335, 2017)。根据它，鲁棒模态扩展是通过对有限数量的相关快照和有限数量的离散化网格点执行分解而产生的，这些网格点是通过高斯消除和包含给定数据的原始快照矩阵的双枢轴选择的。在目前的工作中，采样方法被改编并与低维建模相结合。这种组合产生了一种新颖的自适应算法，用于模拟模式形成系统中的时间相关非线性动力学。在进化过程中按需计算的方便的快照集被存储以记录本地时间事件，其底层机制对于近似是必不可少的。此外，一组稀疏网格点，用于构建模式基和简化的方程组，根据未链接的空间结构自适应采样。结果是问题的降阶模型，即 (i) 产生动态转换的可靠近似，(ii) 非常适合描述局部时空复杂性，以及 (iii) 提供快速计算。对于一些相关的模式形成系统，在一个和两个空间维度上说明了所提出算法的稳健性、准确性和计算效率，展示了具有丰富时空结构的解决方案。