Regularización de variación total generalizada para el problema mal condicionado de asimilación de datos

Abstract

In this paper we study the second order total generalized variation (TGV) regularization applied to the well-known data assimilation problem using the Burgers equation as the state equation. It has been experimentally shown that the TGV regularization preserve the sharp fronts and recovers the solutions better compared to the TV regularization, mainly because it eliminates the staircase effect. We perform the derivation of the first order optimality conditions. For the numerical solution of the Burgers equation, a semi-implicit time discretization was performed and for the spatial variables the finite differences scheme with Upwinding was used for the first order spatial derivative. The solution of the problem was made using a globalized Newton method and for this method we show theoretical results that guarantee the convergence to stationary points of the problem. The numerical experiments section is devoted to show the way each regularization recover the solutions. In particular, we show the staircase effect produced by the TV regularization and the way how the TGV regularization eliminates it.