莫罗佐夫原理应用于数据同化问题

IF 1.9 3区 数学 Q2 Mathematics
L. Bourgeois, J. Dardé
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引用次数: 0

摘要

本文重点讨论了Morozov原理在抽象数据同化框架中的应用,并特别注意了三个简单的例子:拉普拉斯方程的数据同化问题、拉普拉斯方程的Cauchy问题和热方程的数据同化问题。这些不适定问题在混合型公式的帮助下被正则化,该公式被证明等同于应用于选定算子的吉洪诺夫正则化。主要的问题是,这样的算子可能没有一个密集的范围,这使得有必要将众所周知的与Morozov选择正则化参数相关的结果扩展到这种不寻常的情况。在优化的对偶性的帮助下,可能通过强迫解满足给定的先验约束来计算满足Morozov原理的解。对于拉普拉斯方程的数据同化问题,给出了二维上的一些数值结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The Morozov's principle applied to data assimilation problems
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.
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来源期刊
CiteScore
2.70
自引率
5.30%
发文量
27
审稿时长
6-12 weeks
期刊介绍: M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem. Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.
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