next up previous contents
Next: Transport equations Up: Theoretical convergence results Previous: Theoretical convergence results   Contents

Linear case

We consider here a linear situation, although simple, that describes quite well how the BFN algorithm works. We assume that the observation operator $ C$ is equal to the identity, and that the model $ F$ is linear. We also assume that $ F$ and $ K$ commute. Note that this assumption is valid in our experiments as $ K$ is set proportional to the identity matrix. In this pretty simple situation, we can explicit the BFN trajectories. For the sake of concision and clarity, we assume that $ K'=K$ , but the following results remain valid if $ K'\ne K$ . We finally assume that the length of the assimilation period is $ T>0$ .

Then, for all $ n>1$ ,

$\displaystyle X_n(0)\!\!$ $\displaystyle =$ $\displaystyle \left( I-e^{-2KT} \right)^{-1} \left( I-e^{-2nKT} \right) \int_0^T \left( e^{-(K+F)s}+e^{-2KT}e^{(K-F)s} \right) K X_{obs}(s) ds$  
    $\displaystyle + e^{-2nKT} x_0$ (3.19)

and for all $ t\in[0,T]$ ,

$\displaystyle X_n(t)=e^{-(K-F)t} \int_0^t e^{(K-F)s}KX_{obs}(s)ds + e^{-(K-F)t}X_n(0).$ (3.20)

The following result proves the existence of a limit trajectory [20]:

Theorem 3.2   If $ n \to +\infty$ , then $ X_n(0)$ converges and

$\displaystyle \displaystyle \lim_{n\to +\infty}X_n(0)=X_{\infty}(0)= \left( I-e... ...{-1} \int_0^T \!\!\left( e^{-(K+F)s}+e^{-2KT}e^{(K-F)s} \right) K X_{obs}(s)ds.$ (3.21)

Moreover, if $ T>0$ , for any $ t\in[0,T]$ ,

$\displaystyle \displaystyle \lim_{n\to +\infty}X_n(t)=X_{\infty}(t)= e^{-(K-F)t} \int_0^t e^{(K-F)s} K X_{obs}(s) ds + e^{-(K-F)t} X_{\infty}(0).$ (3.22)

Under the same hypothesis, we have a similar result for backward trajectories, i.e. there exists a function $ \tilde{X}_{\infty}(t)$ such that $ \displaystyle \lim_{n \to +\infty} \tilde{X}_n(t)=\tilde{X}_{\infty}(t)$ , for all $ t\in[0,T]$ . This proves the convergence of the BFN algorithm in such a situation.

Note that the limit function $ X_{\infty}$ (resp. $ \tilde{X}_{\infty}$ ) is totally independent of the initial condition $ x_0$ of the algorithm.

Moreover, if the observations are perfect, i.e. $ X_{obs}$ satisfies the direct model equation (3.1), then for all $ t\in[0,T]$ ,

$\displaystyle X_{obs}(t) = e^{Ft} X_{obs}(0).$ (3.23)

It is then straightforward to see in equations (3.21) and (3.22) that

$\displaystyle \lim_{n\to\infty} X_n(t) = X_{obs}(t), \quad \forall t\in [0,T].$ (3.24)

The BFN algorithm also has a similar behaviour on linear parabolic operators in infinite dimension (e.g. the heat operator). A Fourier decomposition of the trajectories allows us to study only first order ordinary differential equations, and gives then the convergence of the algorithm.

next up previous contents
Next: Transport equations Up: Theoretical convergence results Previous: Theoretical convergence results   Contents
Back to home page