## Section: New Results

### Minimax filtering

Participants : Vivien Mallet, Sergiy Zhuk.

In air quality modeling, the model error is supposed to take into account the uncertainty on the meteorological fields (winds and vertical diffusivities), the segregation and mixing in emission areas that affect the effective kinetic rates of reactions, the boundary condition fields, all physical parameterizations (dry deposition, wet scavenging), etc. All the above sources of error have bounded energy and typically are not normally distributed or independent.

In order to take this into account in the data assimilation process, we applied the Minimax State Estimation (MSE) approach. It is well known that a bottle-neck of minimax estimation algorithms as well as of the family of Kalman-type filters is the dimension issue. To solve it, we applied a powerful version of the minimax filter developed for the so-called differential-algebraic equations. This filter works for any linear ordinary differential equation with time-dependent coefficients on any linear manifold, which can also change in time. Based on this novel approach, we derived a computationally tractable reduced version of the minimax filter. The derivation was made in a new and rigorous framework. In addition to the reduction, the new filter shows all the interesting properties inherited from the minimax setting, especially the description of the model and observational errors, which only need to have bounded energy. The later is important in the context of applications because the errors are always bounded. In contrast, most high-dimensional statistical filters are designed for unbounded random errors with special distribution function.

The algorithm, already implemented in the data assimilation library Verdandi, was further developed to compute a better reduction base.

The algorithm was in addition applied for ensemble sequential aggregation. The minimax filter computes weights for each model in the ensemble and a forecast is generated as the weighted linear combination of the ensemble members. In this case, the dimension is small so that no reduction is needed. The approach shows two noteworthy advantages: the observational errors can be taken into account and a dynamics can be given for the weights.

#### A posteriori minimax motion estimation

Participants : Sergiy Zhuk, Isabelle Herlin.

Data assimilation algorithms based on the 4D-Var formulation look for the so-called conditional mode estimate. The latter maximizes the conditional probability density function, provided the initial condition, model error and observation noise are realizations of independent Gaussian random variables. However this Gaussian assumption is often not satisfied for geophysical flows. Moreover, the estimation error of the conditional mode estimate is not a first-hand result of these methods. The issues above can be addressed by means of the Minimax State Estimation (MSE) approach. It allows to filter out any random (with bounded correlation operator) or deterministic (with bounded energy) noise and assess the worst-case estimation error.

The iterative MSE algorithm was developed for the problem of optical flow estimation from a sequence of 2D images. The main idea of the algorithm is to use the "bi-linear" structure of the Navier-Stokes equations and optical flow constraint in order to iteratively estimate the optical flow. The algorithm consists of the following parts:

1) we construct the pseudo-observations that is the estimate of the image brightness function $I(x,y,t)$ solving the optical flow constraint such that $I$ fits (in the sense of least-squares) the observed sequence of images; to do so we set the velocity field in the optical flow constraint to be the current minimax estimate of the velocity field $\mathbf{v}$, obtained at the previous iteration of the algorithm, and construct the minimax estimate $\widehat{I}$ of the solution of the resulting linear advection equation using image sequence as discrete measurements;

2) we plug the estimate of the image gradient, obtained out of pseudo-observations $\widehat{I}$ in 1), into the optical flow constraint and the current minimax estimate $\mathbf{v}$ of the velocity field into the non linear part of Navier-Stokes equations so that we end up with a system of linear PDEs, which represents an extended state equation: it contains a linear parabolic equation for the velocity field and linear advection equation for the image brightness function; we construct the minimax estimate of the extended state equation using the image sequence as discrete measurements of the brightness function;

3) we use the minimax estimate of the velocity field obtained in 2) in order to start 1) again.

Currently numerical experiments are carried out in order to study the convergence rate of the algorithm.