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Total generalized variation regularization in data assimilation for burgers’ equation

Abstract:

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 a Bayesian 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 rigorously derive a first–order optimality condition for the problem. For the numerical solution, we propose a globalized reduced Newton–type method together with a polynomial line–search strategy, 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 TV-regularization is tested.