Total Generalized Variation Regularization in Variational Data Assimilation for Burgers' Equation

TL;DR

This paper introduces a second-order TGV regularization method for initial condition reconstruction in variational data assimilation, specifically applied to Burgers' equation, demonstrating improved performance over TV regularization.

Contribution

It develops a TGV regularization framework for data assimilation, establishes its Bayesian equivalence, and proposes a convergent Newton-type algorithm for solving the resulting optimization problem.

Findings

TGV regularization outperforms TV in numerical tests

The proposed algorithm converges to stationary points

Application to Burgers' equation shows improved reconstruction quality

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 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.