# The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations

In many areas of science and engineering, it is important to estimate the initial (or the current) state of a complex system from output functions measured over some finite time interval. In oceanography and meteorology, this problem is called data assimilation and it is mainly concerned with determining the initial state of the system from existing measures. On the other hand, one of the important concepts of control theory, introduced by Kalman and Luenberger, is the notion of observer. An observer is a new dynamical system in which one injects the measured functions, so that the current state of this new system approaches the current state of the original one for large times. One of the funding ideas of back and forth nudging, introduced in Auroux and Blum (2005, 2008), is to consider both forward and backward in time observers to recover the initial state of the system. However, this procedure requires long time measures, which are generally not available in applications to oceanography and meteorology. The second leading idea of back and forth nudging methodology is to iterate forward and backward observers using only measures on a short time interval. This paper brings one of the first mathematically rigorous justifications of this method in the context of partial differential equations. The main considered applications are a linear viscous transport equation and the viscous and non-viscous Burgers' equation. A very interesting discussion on the competing roles of viscous and convective terms is provided in the concluding section.

**The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations**

Didier Auroux and Maëlle Nodet

ESAIM: COCV 18 (2012) 318-342

http://dx.doi.org/10.1051/cocv/2011004