基于POD/DEIM的降阶策略用于高效四维变分数据同化
POD/DEIM Reduced-Order Strategies for Efficient Four Dimensional Variational Data Assimilation
AI总结：
本文研究了降阶建模方法以加速大规模非线性动力系统变分数据同化问题，提出基于POD和DEIM的降阶策略，通过伽辽金和皮尔洛-伽辽金投影构建降阶数据同化系统，验证了理论框架的有效性，提升了计算效率。
摘要：
本文研究了降阶建模（ROM）方法以加速变分数据同化问题的求解，特别是针对大规模非线性动力系统。研究表明，成功实现降阶解的关键在于降阶Karush-Kuhn-Tucker条件能准确表示其全阶对应条件。特别地，需要精确的降阶近似来表示正向和共轭动力学模型以及降阶梯度。本文开发了新的基于降阶的策略，用于使用伽辽金和皮尔洛-伽辽金投影的POD ROM数据同化。首次将POD、张量POD和离散经验插值法（DEIM）应用于二维浅水方程的降阶数据同化系统。数值实验验证了伽辽金投影的理论框架。在皮尔洛-伽辽金投影的情况下，必须考虑降阶模型的稳定化策略。新的降阶浅水数据同化系统在计算时间仅为全分辨率系统的十分之一的情况下，提供了与全分辨率数据同化系统相似的分析结果。
英文摘要：
This work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for Proper Orthogonal Decomposition (POD) ROM data assimilation using both Galerkin and Petrov-Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.
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