Burgers方程变分同化中的全广义变分正则化
Total Generalized Variation Regularization in Variational Data Assimilation for Burgers' Equation
摘要：
提出了一种二阶全广义变分（TGV）正则化方法，用于变分资料同化问题中初始条件的重建。在证明了TGV正则化和MAP估计器的贝叶斯方法之间的等效性之后，我们重点研究了无粘Burgers的数据同化问题。由于控制双曲守恒律的困难结构，我们考虑了一个离散然后优化的方法，并推导出一阶最优性条件的问题。对于数值解，我们提出了一个整体化的约化牛顿型方法，并证明了算法收敛到稳定点。本文完成了一些数值实验，其中包括TGV正则化的性能相比，TV正则化进行了测试。
英文摘要：
We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and the Bayesian method for the MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize-then-optimize approach and derive first-order optimality conditions for the problem. For the numerical solution, we propose a globalized reduced Newton-type method and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where among others, the performance of TGV-regularization compared to the TV-regularization is tested.
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