论文标题
莫罗佐夫的原则适用于数据同化问题
The Morozov's principle applied to data assimilation problems
论文作者
论文摘要
本文的重点是莫罗佐夫的原理应用于抽象的数据同化框架，特别注意三个简单的示例：拉普拉斯方程的数据同化问题，拉普拉斯方程的库奇问题和热量方程的数据同化问题。这些不良问题的问题是在混合型配方的帮助下进行正规化的，事实证明，这些问题等同于适用于精心挑选的操作员的Tikhonov正则化。主要问题是，该操作员可能没有密集的范围，这使得有必要将与Morozov选择正规化参数有关的众所周知的结果扩展到这种异常情况。满足Morozov原理的解决方案是在优化方面的二元性的帮助下计算的，这可能是通过强迫解决方案来满足先验约束的。在拉普拉斯方程的数据同化问题的情况下，提出了两个维度的一些数值结果。
This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation.
