BACKGROUND Surrogate solutions and surrogate models for complex problems in many fields of science and engineering represent an important recent research line towards the construction of the best trade-off between modeling reliability and computational efficiency. Among surrogate models, hierarchical model (HiMod) reduction provides an effective approach for phenomena characterized by a dominant direction in their dynamics. HiMod approach obtains 1D models naturally enhanced by the inclusion of the effect of the transverse dynamics. METHODS HiMod reduction couples a finite element approximation along the mainstream with a locally tunable modal representation of the transverse dynamics. In particular, we focus on the pointwise HiMod reduction strategy, where the modal tuning is performed on each finite element node. We formalize the pointwise HiMod approach in an unsteady setting, by resorting to a model discontinuous in time, continuous and hierarchically reduced in space (c[M([Formula: see text])G(s)]-dG(q) approximation). The selection of the modal distribution and of the space-time discretization is automatically performed via an adaptive procedure based on an a posteriori analysis of the global error. The final outcome of this procedure is a table, named HiMod lookup diagram, that sets the time partition and, for each time interval, the corresponding 1D finite element mesh together with the associated modal distribution. RESULTS The results of the numerical verification confirm the robustness of the proposed adaptive procedure in terms of accuracy, sensitivity with respect to the goal quantity and the boundary conditions, and the computational saving. Finally, the validation results in the groundwater experimental setting are promising. CONCLUSION The extension of the HiMod reduction to an unsteady framework represents a crucial step with a view to practical engineering applications. Moreover, the results of the validation phase confirm that HiMod approximation is a viable approach.

中文翻译：

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抛物方程的时空自适应分层模型约简。

背景技术在科学和工程学的许多领域中，用于复杂问题的替代解决方案和替代模型代表了一条重要的研究路线，旨在构建建模可靠性和计算效率之间的最佳权衡。在替代模型中，分层模型（HiMod）归约为以动力学为主导方向的现象提供了一种有效的方法。HiMod方法获得一维模型，该模型通过包含横向动力学的影响而自然增强。方法HiMod约简将沿主流的有限元逼近与横向动力学的局部可调模态表示耦合在一起。特别是，我们专注于逐点HiMod减少策略，其中对每个有限元节点执行模态调整。通过采用时间上不连续，连续且空间递减的模型，我们将非定点设置中的逐点HiMod方法形式化（c [M（[Formula：see text]）G（s）]-dG（q）近似） 。模态分布的选择和时空离散化是基于全局误差的后验分析，通过自适应过程自动执行的。该过程的最终结果是一个名为HiMod查找图的表，该表设置时间分区，并为每个时间间隔设置相应的1D有限元网格以及相关的模态分布。结果数值验证的结果从准确性，对目标数量和边界条件的敏感性以及计算节省的角度证实了所提出的自适应程序的鲁棒性。最后，在地下水实验环境中的验证结果是有希望的。结论将HiMod简化扩展到不稳定的框架是实现实际工程应用的关键步骤。此外，验证阶段的结果证实了HiMod逼近是一种可行的方法。