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I am using variational data assimilation (4D-VAR) to estimate emissions of anthropogenic greenhouse gases using a rather complex atmospheric transport model. Hence, the optimal solution to my problem is obtained by minimizing a (non-linear) cost functional $J$.

The model is weakly nonlinear so I can expect the inverse of Hessian of the 4D-VAR cost functional - let's call it $\nabla^2 J^{-1}$ - at the optimal (analysis) point $x_a$ to be a decent approximation to the posterior error covariance matrix $P$. As per standard 4D-VAR theory, the leading eigenpairs of $\nabla^2 J$ (approximately) span the maximum uncertainty space after the assimilation. This all makes sense when we are at a local optimum - i.e., $\nabla^2 J$ is (semi-)positive definite, but what to do if we have not yet reached a local minimum and $\nabla^2 J$ and some of its leading eigenvalues are negative? Is it "safe" to exclude the corresponding eigenpairs from my approximation to the uncertainty space?

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    $\begingroup$ If you haven't yet reached a region where $J$ is locally convex (which must be the case if the Hessian has negative eigenvalues) then you aren't close enough to a local optimum for this to make any sense at all. $\endgroup$ – Brian Borchers Aug 19 '15 at 22:00

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