Many mathematical models involve input parameters, which are not precisely known. Globalsensitivity analysis aims to identify the parameters whose uncertainty has the largestimpact on the variability of a quantity of interest (output of the model). One of thestatistical tools used to quantify the influence of each input variable on the output isthe Sobol sensitivity index. We consider the statistical estimation of this index from afinite sample of model outputs: we present two estimators and state a central limittheorem for each. We show that one of these estimators has an optimal asymptotic variance.We also generalize our results to the case where the true output is not observable, and isreplaced by a noisy version.

We present a reduced basis offline/online procedure for viscous Burgers initial boundaryvalue problem, enabling efficient approximate computation of the solutions of thisequation for parametrized viscosity and initial and boundary value data. This procedurecomes with a fast-evaluated rigorous error bound certifying the approximation procedure.Our numerical experiments show significant computational savings, as well as efficiency ofthe error bound.

In this paper, we consider the back and forth nudging algorithm that has been introducedfor data assimilation purposes. It consists of iteratively and alternately solving forwardand backward in time the model equation, with a feedback term to the observations. Weconsider the case of 1-dimensional transport equations, either viscous or inviscid, linearor not (Burgers’ equation). Our aim is to prove some theoretical results on theconvergence, and convergence properties, of this algorithm. We show that for non viscousequations (both linear transport and Burgers), the convergence of the algorithm holdsunder observability conditions. Convergence can also be proven for viscous lineartransport equations under some strong hypothesis, but not for viscous Burgers’ equation.Moreover, the convergence rate is always exponential in time. We also notice that theforward and backward system of equations is well posed when no nudging term is considered.

We consider an optimal control problem for the three-dimensional non-linear Primitive Equations of the ocean in a vertically bounded and horizontally periodic domain. We aim to reconstruct the initial state of the ocean from Lagrangian observations. This inverse problem is formulated as an optimal control problem which consists in minimizing a cost function representing the least square error between Lagrangian observations and their model counterpart, plus a regularization term. This paper proves the existence of an optimal control for the regularized problem. To this end, we also prove new energy estimates for the Primitive Equations, thanks to well-chosen functional spaces, which distinguish the vertical dimension from the horizontal ones. We illustrate the result with a numerical experiment.