The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4D ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, in which system nonlinearity is significant, are solved by applying this method iteratively. However, the ensemble-based methods basically seek the solution within a lower-dimensional subspace spanned by the ensemble members. It is not necessarily trivial how high-dimensional problems can be solved with the ensemble-based algorithm which employs the lower-dimensional approximation based on the ensemble. In the present study, an ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in high-dimensional nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in a high-dimensional context. It is shown that the ensemble-based algorithm can solve high-dimensional problems by distributing the ensemble in different subspace at each iteration. The findings as the results of the present study were also experimentally supported.

中文翻译：

---

非线性问题中基于迭代集成的变分方法的行为

讨论了基于迭代集成的数据同化算法的行为。基于集合的变分数据同化方法，称为4D集合变分方法（4DEnVar），是解决数据同化问题的有用工具。尽管4DEnVar是基于线性逼近得出的，但通过迭代应用此方法可以解决系统非线性非常重要的高度不确定的问题。但是，基于整体的方法基本上是在整体成员跨越的低维子空间内寻求解决方案的。使用基于整体的低维近似的基于整体的算法如何解决高维问题并不一定很简单。在目前的研究中，重新设计了基于整体的迭代算法，以使我们能够分析其在高维非线性问题中的行为。在高维上下文中讨论了单调收敛到目标函数的局部最大值的条件。结果表明，基于集合的算法可以通过在每次迭代中将集合分布在不同的子空间中来解决高维问题。作为本研究结果的发现也得到了实验支持。