The Morozov's principle applied to data assimilation problems

TL;DR

This paper explores the application of Morozov's principle to regularize ill-posed data assimilation problems for PDEs, proposing a duality-based solution approach and providing numerical results for the Laplace equation.

Contribution

It extends Morozov's principle to cases where the associated operator lacks a dense range, with a focus on data assimilation for PDEs and a duality-based solution method.

Findings

Regularization via Morozov's principle is effective for ill-posed PDE data assimilation.

Duality in optimization helps compute solutions satisfying Morozov's principle.

Numerical experiments demonstrate the approach's viability in two dimensions.

Abstract

This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation.