Total generalized variation regularization in data assimilation for Burgers' equation

Juan Carlos De Los Reyes,Estefanía Loayza-Romero

doi:10.3934/ipi.2019035

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.

Highlights

Data assimilation can be described as the process in which one aims to find an estimate of the state of a system using observations and background information, in order to obtain improved numerical forecasts of the system
We focus our attention on the variational data assimilation framework, whose main idea consists in solving an optimization problem that fits the observations, on the one hand, and uses background information of previous forecasts, on the other
Variational data assimilation problems arise from a robust Bayesian estimation of the initial condition [14, 17]

Summary

Introduction

Data assimilation can be described as the process in which one aims to find an estimate of the state of a system using observations and background information, in order to obtain improved numerical forecasts of the system. In order to prove the local existence of solutions of the state equation we will use the implict function theorem (see, e.g., [7]) For this purpose, we consider hereafter the following assumption. Let us notice that Assumption 1 is of importance for the analysis, and for the numerical solution of both the state equation and the data assimilation problem This hypothesis may be constructively guaranteed by using an upwinding scheme. In a similar manner as in the proof of Theorem 3.4, we obtain existence of a Lagrange multiplier pγ (adjoint state) such that the following optimality system characterizing the optimal solutions of (28) holds:.

5: Solve the system

18: End While

10 Iterations