Strong- and weak-constraint 4DVAR with a finite gradient run

Fix $T\ge 1$, positive dimensions $d_0,\dots ,d_T$ and $p_1,\dots ,p_T$, open sets $U_t\subseteq {\mathbb{R}}^{d_t}$, and continuously differentiable maps ${\mathrm{\Psi }}_{t-1}:U_{t-1}\to U_t$ and $G_t:U_t\to {\mathbb{R}}^{p_t}$. Let $m_b\in {\mathbb{R}}^{d_0}$, $B\in {\mathbb{R}}^{d_0\times d_0}$ be symmetric positive definite, $y_t\in {\mathbb{R}}^{p_t}$, and ${\mathrm{\Gamma }}_t\in {\mathbb{R}}^{p_t\times p_t}$ be symmetric positive definite. In weak-constraint mode also require symmetric positive-definite $Q_t\in {\mathbb{R}}^{d_t\times d_t}$. The strong-constraint decision domain is $U_0$; the weak-constraint decision domain is $U_0\times \cdots \times U_T$. Both objectives are continuously differentiable on their stated open decision domains. For a positive-definite covariance $M$, write $\mathrm{\parallel }v{\mathrm{\parallel }}_{M^{-1}}^2=v^T{M}^{-1}v$.

In strong-constraint 4DVAR, choose the initial state $u_0$ and generate $u_t={\mathrm{\Psi }}_{t-1}(u_{t-1})$ for $1\le t\le T$. Minimize the trajectory objective $J_s(u_0)=\frac{1}{2}\mathrm{\parallel }{u}_0-m_b{\mathrm{\parallel }}_{B^{-1}}^2+\frac{1}{2}{\sum }_{t=1}^T\mathrm{\parallel }{y}_t-G_t(u_t){\mathrm{\parallel }}_{{\mathrm{\Gamma }}_t^{-1}}^2\mathrm{.}$

In weak-constraint 4DVAR, the decision variable is the full path $(u_0,\dots ,u_T)$. Given model-error covariances $Q_t\succ 0$, minimize $J_w(u_0,\dots ,u_T)=\frac{1}{2}\mathrm{\parallel }{u}_0-m_b{\mathrm{\parallel }}_{B^{-1}}^2+\frac{1}{2}{\sum }_{t=1}^T\mathrm{\parallel }{y}_t-G_t(u_t){\mathrm{\parallel }}_{{\mathrm{\Gamma }}_t^{-1}}^2+\frac{1}{2}{\sum }_{t=1}^T\mathrm{\parallel }{u}_t-{\mathrm{\Psi }}_{t-1}(u_{t-1}){\mathrm{\parallel }}_{Q_t^{-1}}^2\mathrm{.}$

Select one objective, an initial decision vector in its stated decision domain, a finite iteration limit $M\ge 0$, and either positive fixed steps or the backtracking parameters required by Gradient descent. Run that algorithm on the selected objective for at most $M$ updates and return its last iterate (or report its domain failure). In strong mode return the trajectory generated from the last initial-state iterate; in weak mode return the last path iterate. This is a finite numerical procedure, not a guarantee that a global or local minimizer is reached. The page's BCS-20 hypotheses for gradient descent supply no general guarantee for these objectives.