François-Xavier Le Dimet

University of Grenoble · Laboratoire Jean Kuntmann and INRIA

We introduced a structure learning-type sparsity regularization in the framework of 4D-Var for 2D velocity reconstruction. The result shows promising performance, such as fast convergence and more consistency with the observation.

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find simultaneously unknown parameters and initial state of the model. A response function is considered as a functional of the optimal solution after assimilation. The sensitivity of the response function to the observation d...

The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to find simultaneously unknown parameters and the initial state of the model. The response function is considered as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to...

The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to simultaneously find unknown parameters and the initial state of the model. The response function is treated as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to obs...

This article presents a correction method for a better resolution of the problem of estimating and predicting pollution, governed by Burgers' equations. The originality of the method consists in the introduction of an error function into the system's equations of state to model uncertainty in the model. The initial conditions and diffusion coeffici...

This work combines a level-set approach and the optimal transport-based Wasserstein distance in a data assimilation framework. The primary motivation of this work is to reduce assimilation artifacts resulting from the position and observation error in the tracking and forecast of pollutants present on the surface of oceans or lakes. Both errors lea...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters of the model. The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the...

The problem of variational data assimilation (estimation) for a nonlinear model is considered in general operator formulation. Hessian-based methods are presented to compute the estimation error covariances. The importance of dynamic formulation and the role of the Hessian and its inverse are discussed.

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters of the model. The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Based on the...

In this paper a 2D pollution water model with an improved numerical method is considered. In order to reduce the approximation errors of the numerical scheme, a new approximation method is introduced to calculate the concentration flux between two cells (j-cell and l-cell) in the direction of the normal vector \(\vec {n}\) orthogonal to their commo...

In the last few years due to a constant increase in the need for more precise forecasting and nowcasting.

In this paper, we present a novel method for assimilating geometric information from observed images. Image assimilation technology fully utilizes structural information from the dynamics of the images to retrieve the state of a system and thus to better predict its evolution. Level-set method describing the evolution of the geometry shapes of a gi...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. Bas...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. If the model is 'perfect,' the optimal solution (analysis) error rises because of the presence of the input data errors (background and observation errors). Then, this error is quantified by the cov...

Understanding the impact of the changes in pollutant emission from a foreign region onto a target region is a key factor for taking appropriate mitigating actions. This requires a sensitivity analysis of a response function (defined on the target region) with respect to the source(s) of pollutant(s). The basic and straightforward approach to sensit...

Models and methods of the numerical modeling of ocean hydrodynamics dating back to the pioneering works of A.S. Sarkisyan are considered, with emphasis on the formulation of problems and algorithms of mathematical modeling and the four-dimensional variational assimilation of observational data. An algorithm is proposed for studying the sensitivity...

The problem of variational data assimilation for a model of ocean thermodynamics is formulated as an optimal control problem to find the boundary heat flux. The sensitivity of functionals of the optimal solution with respect to observations is studied. Computing the gradient of the functionals is reduced to the solution of a non-standard problem wh...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. The optimal solution (analysis) error arises due to the errors in the input data (background and observation errors). Under the Gaussian assumption the optimal solution error covariance can be const...

Survey paper on Data assimilation in 1988

The problem of the variational assimilation of observational data is stated for a nonlinear evolution model as a problem of optimal control in order to find the function of initial condition. The operator of the model, and consequently the optimal solution, depend on parameters that may contain uncertainties. A functional of the solution of the pro...

Sensitivity Analysis in presence of data. Sensitivity on the Optimality System

This book gathers notes from lectures and seminars given during a three-week school on theoretical and applied data assimilation held in Les Houches in 2012. Data assimilation aims at determining as accurately as possible the state of a dynamical system by combining heterogeneous sources of information in an optimal way. Generally speaking, the mat...

The equations that govern geophysical fluids (namely atmosphere, ocean and rivers) are well known but their use for prediction requires the knowledge of the initial condition. In many practical cases, this initial condition is poorly known and the use of an imprecise initial guess is not sufficient to perform accurate forecasts because of the high...

This book gathers notes from lectures and seminars given during a three-week school on theoretical and applied data assimilation held in Les Houches in 2012. Data assimilation aims at determining as accurately as possible the state of a dynamical system by combining heterogeneous sources of information in an optimal way. Generally speaking, the mat...

The equations that govern geophysical fluids (namely atmosphere, ocean and rivers) are well known but their use for prediction requires the knowledge of the initial condition. In many practical cases, this initial condition is poorly known and the use of an imprecise initial guess is not sufficient to perform accurate forecasts because of the high...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The operator of the model, and hence the optimal solution, depend on the parameters which may contain uncertainties. A response function is considered as a functional of the solu...

In order to lower the computational cost of the variational data assimilation process, we investigate the use of multigrid methods to solve the associated optimal control system. On a linear advection equation, we study the impact of the regularization term of the optimal control and the impact of discretization errors on the efficiency of the coar...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The data contain errors (observation and background errors); hence there is an error in the analysis. For mildly nonlinear dynamics the analysis error covariance can be approxima...

Recent advances in variational and optimization methods applied to increasingly complex numerical weather prediction models with larger numbers of degrees of freedom mandate to take a perspective view of past and recent developments in this field, and present a view of the state of art in the field. Variational methods attempt to achieve a best fit...

Representation of the system Taking errors into account Simplified approach to optimum static estimation theory Generalization in the multidimensional case The different data assimilation techniques Sequential assimilation method: the Kalman filter Extension to non-linear models: the extended Kalman filter Assessment of the Kalman filter Variationa...

Automatic Differentiation methods are often applied to codes that solve partial differential equations, e.g. in the domains of geophysical sciences, such as meteorology or oceanography, or Computational Fluid Dynamics. In agronomy, the differentiation of crop model has never been performed since the models are not fully deterministic but much more...

The problem of variational data assimilation (DA) for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition, boundary conditions and/or model parameters. The input data contain observation and background errors, hence there is an error in the optimal solution. For mildly nonlinear dynamics, the covari...

An optimal control methodology is proposed for plant growth. This methodology is demonstrated by solving a water supply problem for optimal sunflower fruit filling. The functional–structural sunflower growth is described by a dynamical system given soil water conditions. Numerical solutions are obtained through an iterative optimization procedure,...

At the present time, the Earth is observed by dozens of satellites giving a permanent information on the evolution of the atmosphere and of the ocean. This information is partly used by transforming radiances into state variables of the models then performing a regular method of Data Assimilation. But the dynamics of these images also contains an i...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The data contain errors (observation and background errors), hence there will be errors in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution...

International audience In order to limit the computational cost of the variational data assimilation process, we investigate the use of multigrid methods to solve the associated optimal control system. On a linear advection equation, we study the impact of the regularization term and the discretization errors on the efficiency of the coarse grid co...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The equation for the error of the optimal solution (analysis) is derived through the errors of the input data (background and observation errors). The numerical algorithm is developed to co...

Since several decades the ocean and the atmosphere are permanently observed by satellites which provides many measurement in various channels, this information is used in modern operational data assimilation centers. Satellites provide another kind of information on the move of masses of air and water. This information, of lagrangian type, is of gr...

The problem of four-dimensional variational data assimilation (4D-Var) seeks to find an optimal initial field minimizing a cost function defined as the squared distance between model solutions and observations within an assimilation window. It requires minimization algorithms along with adjoint models to compute gradient information needed for the...

In order to forecast the evolution of a dynamical system such as geophysical fluids ocean, atmosphere, continental waters), all the available information have to be accounted for. They are of very different nature: set of non linear PDE (mathematical-type information), in situ measurements and remote sensing (physical-type information), statistical...

In order to limit the computational cost of the variational data assimilation process, we investigate the use of multigrid methods to solve the associated optimal control system. On a linear advection equation, we study the impact of the regularization term of the optimal control and also the impact of the discretization errors on the efficiency of...

Optical flow motion estimation from two images is limited by the aperture problem. A method to deal with this problem is to use regularization techniques. Usually, one adds a regularization term with appriopriate weighting parameter to the optical flow cost funtion. Here, we suggest a new approach to regularization for optical flow motion estimatio...

Due to the ill-posedness of inverse problems, it is important to make use of most of the \textit{a priori} informations while solving such a problem. These informations are generally used as constraints to get the appropriate solution. In usual cases, constrains are turned into penalization of some characteristics of the solution. A common constrai...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is considered in a reduced control space to...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters such as distributed model coefficients or boundary conditions. The equation for the optimal solution error is derived through the errors of the input data (background and observation errors), and the op...

We present a Lagrangian data assimilation exper- iment in an open channel flow above a broad-crested weir. The observations consist of trajectories of particles trans- ported by the flow and extracted from a video film, in ad- dition to classical water level measurements. However, the presence of vertical recirculations on both sides of the weir actua...

Two general algorithms for solving constrained minimization problems are presented and discussed in the context of analysis and assimilation of meteorological observations. In both algorithms, the original constrained problem is transformed by appropriate modifications into one unconstrained problem, or into a sequence of unconstrained problems. Th...

Numerical methods in hydrology

Understanding and forecasting the evolution of geophysical fluids is a major scientific and societal challenge. Forecasting algorithms should take into account all the available informations on the considered dynamical system. The Variational Data Assimilation (VDA) technique combines in a consistent way all these informations in an Optimality Syst...

The ultimate purpose of environmental studies is the forecast of its natural evolution. A prerequisite before a prediction is to retrieve at best the state of the environment. Data assimilation is the ensemble of techniques which, starting from heterogeneous information, permit to retrieve the initial state of a flow. In the first part, the mathema...

In this paper, after a brief review of curvelets and their relation to classical wavelet transform, multiscale geometric analysis is systematically applied to turbulent flows in two and three dimensions. The analysis is based on the constrained minimization of a total variation functional representing the difference between the data and its represe...

We present a method to use lagrangian data from remote sensing observation in a variational data assimilation process for a river hydraulics model based on the bidimensional shallow water equations. The trajectories of particles advected by the flow can be extracted from video images and are used in addition to classical eulerian observations. This...

Variational methods are widely used for the analysis and control of computationally intensive spatially distributed systems. In particular, the adjoint state method enables a very efficient calculation of the derivatives of an objective function (response function to be analysed or cost function to be optimised) with respect to model inputs. In t...

When we take photos, we often get blurred pictures because of hand shake, motion, insufficient light, unsuited focal length, or other disturbances. Recently, a compressed-sensing (CS) theorem which provides a new sampling theory for data acquisition has been applied for medical and astronomic imaging. The CS makes it possible to take superresolutio...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find some unknown parameters of the model. The equation for the error of the optimal solution is derived through the statistical errors of the input data (background, observation, and model errors). A numerical algorithm is de...

Since several decades satellites have been launched for the observation of the atmosphere and the ocean. They provide sequences of photographic images. These sequences of images contain information about the dynamics of the observed systems. We present an extension of the data assimilation techniques targeted to the assimilation of image sequences...

The problem of data assimilation for a nonlinear evolution model is considered with the aim to identify the initial and/or boundary conditions. An approach to solve the data assimilation process is described. This approach is based on the Newton's Method. An application to the data assimilation problem in hydrology is presented. Numerical results a...

Predicting the evolution of the components of the water cycle is an important issue both from the scientific and social points of view. The basic problem is to gather all the available information in order to be able to retrieve at best the state of the water cycle. Among some others methods variational methods have a strong potential to achieve th...

Regularization is a common procedure when dealing with inverse problems. Because of the ill-posedness of many inverse problems, one needs to add some constraints as regularization to the problem in order to get a satisfactory solution. A difficulty when using multiple constraints is to properly choose a weighting parameter for each constraint. We p...

Predicting the evolution of the environment is an important and difficult task. To achieve this goal we need to consider all the available information: mathematical information under the form of models, observations, statistics, images from satellites. We will see how to use optimal control methods in order to link together theses sources of inform...

In four-dimensional variational data assimilation (4D-Var), the model equations are treated as strong constraints on an optimization problem. In reality, the model does not represent the system behaviour exactly and errors arise due to physical approximations, discretization, variability of physical parameters, and inaccuracy of initial and boundar...

Predicting the evolution of geophysical fluids (ocean, atmosphere, continental water) is a ma jor scientific and societal challenge. Achieving this goal requires to consider all the available information: numerical models, observations, error statistics... In order to combine these heterogeneous source of information one uses the data assimilation te...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find some unknown parameters of the model. The equation for the error of the optimal solution is derived through the statistical errors of the input data. The covariance operator of the optimal solution error is obtained using...

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is used to show that in a nonlinear case th...

In this article we propose a new method to estimate ocean mesoscale structures of the surface current velocity by processing sea surface satellite images. Assuming that the intensity level can be described by a transport–diffusion equation, the proposed approach is based on variational assimilation of image observations within a simple transport–di...

In the last few years, encouraging results using radiative transfer model inversion techniques were obtained for land biophysical variables retrieval. However, the inversion of radiative transfer models is a severely ill-posed problem that may lead to significant uncertainties in the biophysical variables estimates. Improvement of performances of t...