• The Inria's Research Teams produce an annual Activity Report presenting their activities and their results of the year. These reports include the team members, the scientific program, the software developed by the team and the new results of the year. The report also describes the grants, contracts and the activities of dissemination and teaching. Finally, the report gives the list of publications of the year.

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FLUMINANCE

FLUMINANCE - 2021

2021
Activity report
Project-Team
FLUMINANCE
RNSR: 200919001U
Research center
In partnership with:
INRAE, Université Rennes 1
Team name:
Fluid Flow Analysis, Description and Control from Image Sequences
In collaboration with:
Institut de recherche mathématique de Rennes (IRMAR)
Domain
Digital Health, Biology and Earth
Theme
Earth, Environmental and Energy Sciences
Creation of the Project-Team: 2009 July 01

Keywords

• A3. Data and knowledge
• A3.3. Data and knowledge analysis
• A3.4. Machine learning and statistics
• A5.3. Image processing and analysis
• A5.9. Signal processing
• A6.1. Methods in mathematical modeling
• A6.1.2. Stochastic Modeling
• A6.1.4. Multiscale modeling
• A6.2. Scientific computing, Numerical Analysis & Optimization
• A6.2.1. Numerical analysis of PDE and ODE
• A6.2.7. High performance computing
• A6.3. Computation-data interaction
• A6.3.1. Inverse problems
• A6.3.2. Data assimilation
• A6.3.3. Data processing
• A6.3.4. Model reduction
• A6.3.5. Uncertainty Quantification
• B3.2. Climate and meteorology
• B3.3. Geosciences
• B5. Industry of the future
• B5.2. Design and manufacturing

1 Team members, visitors, external collaborators

Research Scientists

• Etienne Mémin [Team leader, Inria, Senior Researcher, HDR]
• Jocelyne Chaux [Inria, Emeritus, HDR]
• Christophe Collewet [Institut national de recherche pour l'agriculture, l'alimentation et l'environnement, Researcher, HDR]
• Evgueni Dinvay [Inria, Starting Research Position]
• Camilla Fiorini [Inria, Starting Research Position, until Aug 2021]
• Dominique Heitz [Institut national de recherche pour l'agriculture, l'alimentation et l'environnement, Researcher, HDR]
• Noe Lahaye [Inria, Researcher]
• Long Li [Inria, Starting Research Position, from Apr 2021]
• Gilles Tissot [Inria, Researcher]

Faculty Member

• Roger Lewandowski [Univ de Rennes I, Professor, HDR]

Post-Doctoral Fellows

• Pierre Marie Boulvard [Inria, from Oct 2021]
• Louis Thiry [Inria, from Sep 2021]

PhD Students

• Adrien Bella [Inria, from Nov 2021]
• Mohamed Yacine Ben Ali [Inria, until Mar 2021]
• Zoé Caspar-Cohen [Ifremer]
• Benjamin Dufee [Inria]
• Berenger Hug [Inria, until Aug 2021]
• Francois Legeais [Univ de Rennes I, from Sep 2021]
• Long Li [Inria, until Mar 2021]
• Igor Maingonnat [Inria, from Oct 2021]
• Manolis Perrot [U. Grenoble Alpes]
• Francesco Tucciarone [Inria]

Interns and Apprentices

• Romain Moulignier [Inria, from Mar 2021 until Aug 2021]

• Caroline Tanguy [Inria]

2 Overall objectives

The research group that we have entitled fluminance from a contraction between the words “Fluid” and “Luminance” is dedicated to the extraction of information on fluid flows from image sequences and to the development of tools for the analysis and control of these flows. The objectives of the group are at the frontiers of several important domains that range from fluid mechanics to geophysics. One of the main originality of the fluminance group is to combine cutting-edge researches on data-assimilation and flow numerical modeling with an ability to conduct proper intensive experimental validations on prototype flows mastered in laboratory. The scientific objectives decompose in four main themes:

• Fluid flows characterization from images

In this first axis, we aim at providing accurate measurements and consistent analysis of complex fluid flows through image analysis techniques.The application domain ranges from industrial processes and experimental fluid mechanics to environmental sciences. This theme includes also the use of non-conventional imaging techniques such as Schlieren techniques, Shadowgraphs, holography. The objective will be here to go towards 3D dense velocity measurements.

• Coupling dynamical model and image data

We focus here on the study, through image data, of complex and partially known fluid flows involving complex boundary conditions, multi-phase fluids, fluids and structures interaction problems. Our credo is that image analysis can provide sufficiently fine observations on small and medium scales to construct models which, applied at medium and large scale, account accurately for a wider range of the dynamics scales. The image data and a sound modeling of the dynamical uncertainty at the observation scale should allow us to reconstruct the observed flow and to provide efficient real flows (experimental or natural) based dynamical modeling. Our final goal will be to go towards a 3D reconstruction of real flows, or to operate large motion scales simulations that fit real world flow data and incorporate an appropriate uncertainty modeling.

• Control and optimization of turbulent flows

We are interested on active control and more precisely on closed-loop control. The main idea is to extract reliable image features to act on the flow. This approach is well known in the robot control community, it is called visual servoing. More generally, it is a technique to control a dynamic system from image features. We plan to apply this approach on flows involved in various domains such as environment, transport, microfluidic, industrial chemistry, pharmacy, food industry, agriculture, etc.

• Numerical models for geophysical flows simulation and analysis Numerical models are very useful for environmental applications. Several difficulties must be handled simultaneously, in a multidisciplinary context. For example, in geophysics, media are highly heterogeneous and only few data are available. Stochastic models are often necessary to describe unresolved physical processes. Computational domains are characterized by complex 3D geometries, requiring adapted space discretization. Equations modeling flow and transport are transient, requiring also adapted time discretization. Moreover, these equations can be coupled together or with other equations in a global nonlinear system. These large-scale models are very time and memory consuming. High performance computing is thus required to run these types of scientific simulations. Supercomputers and clusters are quite powerful, provided that the numerical models are written with a parallel paradigm.

3 Research program

3.1 Estimation of fluid characteristic features from images

The measurement of fluid representative features such as vector fields, potential functions or vorticity maps, enables physicists to have better understanding of experimental or geophysical fluid flows. Such measurements date back to one century and more but became an intensive subject of research since the emergence of correlation techniques 41 to track fluid movements in pairs of images of a particles laden fluid or by the way of clouds photometric pattern identification in meteorological images. In computer vision, the estimation of the projection of the apparent motion of a 3D scene onto the image plane, referred to in the literature as optical-flow, is an intensive subject of researches since the 80's and the seminal work of B. Horn and B. Schunk 53. Unlike to dense optical flow estimators, the former approach provides techniques that supply only sparse velocity fields. These methods have demonstrated to be robust and to provide accurate measurements for flows seeded with particles. These restrictions and their inherent discrete local nature limit too much their use and prevent any evolutions of these techniques towards the devising of methods supplying physically consistent results and small scale velocity measurements. It does not authorize also the use of scalar images exploited in numerous situations to visualize flows (image showing the diffusion of a scalar such as dye, p ollutant, light index refraction, fluorescein,...). At the opposite, variational techniques enable in a well-established mathematical framework to estimate spatially continuous velocity fields, which should allow more properly to go towards the measurement of smaller motion scales. As these methods are defined through PDE's systems they allow quite naturally constraints to be included such as kinematic properties or dynamic laws governing the observed fluid flows. Besides, within this framework it is also much easier to define characteristic features estimation procedures on the basis of physically grounded data model that describes the relation linking the observed luminance function and some state variables of the observed flow. The Fluminance group has allowed a substantial progress in this direction with the design of dedicated dense estimation techniques to estimate dense fluid motion fields. See 7 for a detailed review. More recently problems related to scale measurement and uncertainty estimation have been investigated 46. Dynamically consistent and highly robust techniques have been also proposed for the recovery of surface oceanic streams from satellite images 43. Very recently parameter-free approaches relying on uncertainty concept has been devised 44. This technique outperforms the state of the art.

3.2 Data assimilation and Tracking of characteristic fluid features

Real flows have an extent of complexity, even in carefully controlled experimental conditions, which prevents any set of sensors from providing enough information to describe them completely. Even with the highest levels of accuracy, space-time coverage and grid refinement, there will always remain at least a lack of resolution and some missing input about the actual boundary conditions. This is obviously true for the complex flows encountered in industrial and natural conditions, but remains also an obstacle even for standard academic flows thoroughly investigated in research conditions.

This unavoidable deficiency of the experimental techniques is nevertheless more and more compensated by numerical simulations. The parallel advances in sensors, acquisition, treatment and computer efficiency allow the mixing of experimental and simulated data produced at compatible scales in space and time. The inclusion of dynamical models as constraints of the data analysis process brings a guaranty of coherency based on fundamental equations known to correctly represent the dynamics of the flow (e.g. Navier Stokes equations) 11. Conversely, the injection of experimental data into simulations ensures some fitting of the model with reality.

To enable data and models coupling to achieve its potential, some difficulties have to be tackled. It is in particular important to outline the fact that the coupling of dynamical models and image data are far from being straightforward. The first difficulty is related to the space of the physical model. As a matter of fact, physical models describe generally the phenomenon evolution in a 3D Cartesian space whereas images provides generally only 2D tomographic views or projections of the 3D space on the 2D image plane. Furthermore, these views are sometimes incomplete because of partial occlusions and the relations between the model state variables and the image intensity function are otherwise often intricate and only partially known. Besides, the dynamical model and the image data may be related to spatio-temporal scale spaces of very different natures which increases the complexity of an eventual multiscale coupling. As a consequence of these difficulties, it is necessary generally to define simpler dynamical models in order to assimilate image data. This redefinition can be done for instance on an uncertainty analysis basis, through physical considerations or by the way of data based empirical specifications. Such modeling comes to define inexact evolution laws and leads to the handling of stochastic dynamical models. The necessity to make use and define sound approximate models, the dimension of the state variables of interest and the complex relations linking the state variables and the intensity function, together with the potential applications described earlier constitute very stimulating issues for the design of efficient data-model coupling techniques based on image sequences.

On top of the problems mentioned above, the models exploited in assimilation techniques often suffer from some uncertainties on the parameters which define them. Hence, a new emerging field of research focuses on the characterization of the set of achievable solutions as a function of these uncertainties. This sort of characterization indeed turns out to be crucial for the relevant analysis of any simulation outputs or the correct interpretation of operational forecasting schemes. In this context, stochastic modeling play a crucial role to model and process uncertainty evolution along time. As a consequence, stochastic parameterization of flow dynamics has already been present in many contributions of the Fluminance group in the last years and will remain a cornerstone of the new methodologies investigated by the team in the domain of uncertainty characterization.

This wide theme of research problems is a central topic in our research group. As a matter of fact, such a coupling may rely on adequate instantaneous motion descriptors extracted with the help of the techniques studied in the first research axis of the fluminance group. In the same time, this coupling is also essential with respect to visual flow control studies explored in the third theme. The coupling between a dynamics and data, designated in the literature as a Data Assimilation issue, can be either conducted with optimal control techniques 47, 54 or through stochastic filtering approaches 48, 51. These two frameworks have their own advantages and deficiencies. We rely indifferently on both approaches.

3.3 Optimization and control of fluid flows with visual servoing

Fluid flow control is a recent and active research domain. A significant part of the work carried out so far in that field has been dedicated to the control of the transition from laminarity to turbulence. Delaying, accelerating or modifying this transition is of great economical interest for industrial applications. For instance, it has been shown that for an aircraft, a drag reduction can be obtained while enhancing the lift, leading consequently to limit fuel consumption. In contrast, in other application domains such as industrial chemistry, turbulence phenomena are encouraged to improve heat exchange, increase the mixing of chemical components and enhance chemical reactions. Similarly, in military and civilians applications where combustion is involved, the control of mixing by means of turbulence handling rouses a great interest, for example to limit infra-red signatures of fighter aircraft.

Flow control can be achieved in two different ways: passive or active control. Passive control provides a permanent action on a system. Most often it consists in optimizing shapes or in choosing suitable surfacing (see for example  45 where longitudinal riblets are used to reduce the drag caused by turbulence). The main problem with such an approach is that the control is, of course, inoperative when the system changes. Conversely, in active control the action is time varying and adapted to the current system's state. This approach requires an external energy to act on the system through actuators enabling a forcing on the flow through for instance blowing and suction actions 57, 50. A closed-loop problem can be formulated as an optimal control issue where a control law minimizing an objective cost function (minimization of the drag, minimization of the actuators power, etc.) must be applied to the actuators 42. Most of the works of the literature indeed comes back to open-loop control approaches 56, 52, 55 or to forcing approaches 49 with control laws acting without any feedback information on the flow actual state. In order for these methods to be operative, the model used to derive the control law must describe as accurately as possible the flow and all the eventual perturbations of the surrounding environment, which is very unlikely in real situations. In addition, as such approaches rely on a perfect model, a high computational costs is usually required. This inescapable pitfall has motivated a strong interest on model reduction. Their key advantage being that they can be specified empirically from the data and represent quite accurately, with only few modes, complex flows' dynamics. This motivates an important research axis in the Fluminance group.

3.4 Numerical models applied to hydrogeology and geophysics

The team is strongly involved in numerical models for hydrogeology and geophysics. There are many scientific challenges in the area of groundwater simulations. This interdisciplinary research is very fruitful with cross-fertilizing subjects.

In geophysics, a main concern is to solve inverse problems in order to fit the measured data with the model. Generally, this amounts to solve a linear or nonlinear least-squares problem.

Models of geophysics are in general coupled and multi-physics. For example, reactive transport couples advection-diffusion with chemistry. Here, the mathematical model is a set of nonlinear Partial Differential Algebraic Equations. At each timestep of an implicit scheme, a large nonlinear system of equations arise. The challenge is to solve efficiently and accurately these large nonlinear systems.

3.5 Numerical algorithms and high performance computing

Linear algebra is at the kernel of most scientific applications, in particular in physical or chemical engineering. The objectives are to analyze the complexity of these different methods, to accelerate convergence of iterative methods, to measure and improve the efficiency on parallel architectures, to define criteria of choice.

4 Application domains

By designing new approaches for the analysis of fluid image sequences, data -model coupling and stochastic representation of fluid flows the Fluminance group contrinutes to several application domains of great interest for the community and in which the anaysis of complex turbulent flow is key. The group focuses on two broad application domains:

• Environmental sciences
• Experimental fluid mechanics and industrial flows

More recently a focus on ocean dynamics and indoor environmental flow has been operated.

5 Social and environmental responsibility

5.1 Footprint of research activities

The team dedicates entirely its research effort to environmental problems related to the climate change issue and its consequences for the future of our planet. Our activities concerns mainly Mathematics for planet Earth.

5.2 Impact of research results

The team's research results are principally published in journals dedicated to Mathematics for planet Earth and environmental sciences issues. As such we intend to contribute with our stength and skills to these questions.

6 New software and platforms

6.1 New software

6.1.1 2DLayeredMotion

• Name:
Estimation of 2D independent mesoscale layered atmospheric motion fields
• Functional Description:
This software enables to estimate a stack of 2D horizontal wind fields corresponding to a mesoscale dynamics of atmospheric pressure layers. This estimator is formulated as the minimization of a global energy function. It relies on a vertical decomposition of the atmosphere into pressure layers. This estimator uses pressure data and classification clouds maps and top of clouds pressure maps (or infra-red images). All these images are routinely supplied by the EUMETSAT consortium which handles the Meteosat and MSG satellite data distribution. The energy function relies on a data model built from the integration of the mass conservation on each layer. The estimator also includes a simplified and filtered shallow water dynamical model as temporal smoother and second-order div-curl spatial regularizer. The estimator may also incorporate correlation-based vector fields as additional observations. These correlation vectors are also routinely provided by the Eumetsat consortium.
• URL:
• Contact:
Etienne Memin
• Participant:
Etienne Memin

6.1.2 3DLayeredMotion

• Name:
Estimation of 3D interconnected layered atmospheric motion fields
• Functional Description:
This software extends the previous 2D version. It allows (for the first time to our knowledge) the recovery of 3D wind fields from satellite image sequences. As with the previous techniques, the atmosphere is decomposed into a stack of pressure layers. The estimation relies also on pressure data and classification clouds maps and top of clouds pressure maps. In order to recover the 3D missing velocity information, physical knowledge on 3D mass exchanges between layers has been introduced in the data model. The corresponding data model appears to be a generalization of the previous data model constructed from a vertical integration of the continuity equation.
• URL:
• Contact:
Etienne Memin

6.1.3 DenseMotion

• Name:
Estimation of 2D dense motion fields
• Functional Description:
This code allows the computation from two consecutive images of a dense motion field. The estimator is expressed as a global energy function minimization. The code enables the choice of different data models and different regularization functionals depending on the targeted application. Generic motion estimators for video sequences or fluid flows dedicated estimators can be set up. This software allows in addition the users to specify additional correlation based matching measurements. It enables also the inclusion of a temporal smoothing prior relying on a velocity vorticity formulation of the Navier-Stoke equation for Fluid motion analysis applications.
• URL:
• Contact:
Etienne Memin
• Participant:
Etienne Memin

6.1.4 Low-Order-Motion

• Name:
Estimation of low order representation of fluid motion
• Functional Description:
This code enables the estimation of a low order representation of a fluid motion field from two consecutive images.The fluid motion representation is obtained using a discretization of the vorticity and divergence maps through regularized Dirac measure. The irrotational and solenoidal components of the motion fields are expressed as linear combinations of basis functions obtained through the Biot-Savart law. The coefficient values and the basis function parameters are formalized as the minimizer of a functional relying on an intensity variation model obtained from an integrated version of the mass conservation principle of fluid mechanics.
• URL:
• Contact:
Etienne Memin
• Participants:
Anne Cuzol, Etienne Memin

6.1.5 TYPHOON

• Keyword:
Fluid mechanics
• Functional Description:
Typhoon is a fluid motion estimator from image sequences. It is almost real-time dedicated to the measurement of LIDAR sequences, multi-scale, fast and precise to make a fine scale analysis of fluid flows with applications in the fields of energy, transport and environment.
• URL:
• Contact:
Etienne Memin
• Participants:
Christopher Mauzey, Etienne Memin, Pierre Dérian
• Partner:
CSU Chico

6.1.6 H2OLab

• Keywords:
Simulation, Energy, Contamination, Groundwater, Hydrogeology, Heterogeneity, Uncertainly, Multiscale
• Scientific Description:
The software platform contains a database which is interfaced through the web portal H2OWeb. It contains also software modules which can be used through the interface H2OGuilde. The platform H2OLab is an essential tool for the dissemination of scientific results. Currently, software and database are shared by the partners of the h2mno4 project.
• Functional Description:

The software platform H2OLab is devoted to stochastic simulations of groundwater flow and contaminant transport in highly heterogeneous porous and fractured geological media.

-Modeling and numerical simulation of aquifers -Porous and fractured heterogeneous media -Flow with mixed finite elements -Solute transport with a Lagrangian method -Stochastic modeling for data uncertainty.

• URL:
• Contact:
Jocelyne Erhel
• Participants:
Geraldine Pichot, Grégoire Lecourt, Jean-Raynald De Dreuzy, Jocelyne Erhel
• Partners:
Université de Rennes 1, CNRS, Université de Lyon, Université de Poitiers

6.1.7 PALMTREE

• Keyword:
Monte-Clarlo
• Functional Description:

We present an easy-to-use package for the parallelization of Lagrangian methods for partial differential equations. In addition to the reduction of computation time, the code aims at satisfying three properties:

simplicity: the user just has to add the algorithm governing the behaviour of the particles. portability: the possibility to use the package with any compiler and OS. action-replay: the ability of the package to replay a selected batch of particles.

The last property allows the user to replay and capture the whole sample path for selected particles of a batch. This feature is very useful for debugging and catching some relevant information.

• Authors:
Lionel Lenôtre, Geraldine Pichot, Lionel Lenôtre, Lionel Lenôtre
• Contact:
Geraldine Pichot

6.1.8 GRT3D

• Name:
Global Reactive Transport in 3D
• Keywords:
Geochemistry, Dispersion, Scientific calculation, Simulation, Advection
• Scientific Description:

Participants : Édouard Canot, Jocelyne Erhel [correspondant] .

Version: version 2.0, April 2014

APP: registered

Programming language: C

Abstract: Reactive transport modeling has become an essential tool for understanding complex environmental problems. It is an important issue for MoMaS and C2S@EXA partners (see sections 8.2.5 , 8.2.3 ), in particular Andra. We have developed a method coupling transport and chemistry, based on a method of lines such that spatial discretization leads to a semi-discrete system of algebraic differential equations (DAE system). The main advantage is to use a complex DAE solver, which controls simultaneously the timestep and the convergence of Newton algorithm. The approach SIA uses a fixed-point method to solve the nonlinear system at each timestep, whereas the approach SNIA uses an explicit scheme.

The software suite GRT3D has four executable modules:

SIA1D: Sequential Iterative Approach for 1D domains,

GDAE1D: Global DAE approach for 1D domains,

SNIA3D: Sequential Non Iterative Approach for 1D, 2D or 3D domains.

GDAE3D: Global DAE approach for 1D, 2D or 3D domains. This module has three variants: the original one with logarithms, an optimized one still with logarithms, an optimized one which does not use logarithms.

Current work: extension of the chemistry module and parallelization.

• Functional Description:

Reactive transport modeling has become an essential tool for understanding complex environmental problems. It is an important issue for MoMaS and C2S@EXA partners, in particular Andra. We have developed a method coupling transport and chemistry, based on a method of lines such that spatial discretization leads to a semi-discrete system of algebraic differential equations (DAE system). The main advantage is to use a complex DAE solver, which controls simultaneously the timestep and the convergence of Newton algorithm. The approach SIA uses a fixed-point method to solve the nonlinear system at each timestep, whereas the approach SNIA uses an explicit scheme.

The software suite GRT3D has four executable modules:

SIA1D: Sequential Iterative Approach for 1D domains,

GDAE1D: Global DAE approach for 1D domains,

SNIA3D: Sequential Non Iterative Approach for 1D, 2D or 3D domains.

GDAE3D: Global DAE approach for 1D, 2D or 3D domains. This module has three variants: the original one with logarithms, an optimized one still with logarithms, an optimized one which does not use logarithms.

• Contact:
Jocelyne Erhel
• Participants:
Caroline De Dieuleveult, Édouard Canot, Jocelyne Erhel, Nadir Soualem, Souhila Sabit
• Partner:
ANDRA

7 New results

7.1 Fluid motion estimation

7.1.1 Development of an image-based measurement method for large-scale characterization of indoor airflows

Participants: Dominique Heitz, Etienne Mémin.

The goal is to design a new image-based flow measurement method for large-scale industrial applications. From this point of view, providing in situ measurement technique requires: (i) the development of precise models relating the large-scale flow observations to the velocity; (ii) appropriate large-scale regularization strategies; and (iii) adapted seeding and lighting systems, like Hellium Filled Soap Bubles (HFSB) and led ramp lighting. This work conducted within the PhD of Romain Schuster in collaboration with the compagny ITGA has started in february 2016. The first step has been to evaluate the performances of a stochastic uncertainty motion estimator when using large scale scalar images, like those obtained when seeding a flow with smoke. The PIV characterization of flows on large fields of view requires an adaptation of the motion estimation method from image sequences. The backward shift of the camera coupled to a dense scalar seeding involves a large scale observation of the flow, thereby producing uncertainty about the observed phenomena. By introducing a stochastic term related to this uncertainty into the observation term, we obtained a significant improvement of the estimated velocity field accuracy. The technique was validated on a mixing layer in a wind tunnel for HFSB and smoke tracers and applied on a laboratory fume-hood.

7.1.2 3D flows reconstruction from image data

Participants: Dominique Heitz, Etienne Mémin.

Our work focuses on the design of new tools for the estimation of 3D turbulent flow motion in the experimental setup of Tomo-PIV. This task includes both the study of physically-sound models on the observations and the fluid motion, and the design of low-complexity and accurate estimation algorithms. We have proposed a novel method for volumetric velocity reconstruction exploring the locality of 3D object space. Under this formulation the velocity of local patch was sought to match the projection of the particles within the local patch in image space to the image recorded by camera. The core algorithm to solve the matching problem is an instance-based estimation scheme that can overcome the difficulties of optimization originated from the nonlinear relationship between the image intensity residual and the volumetric velocity. The proposed method labeled as Lagrangian Particle Image Velocimetry (LaPIV) is quantitatively evaluated with synthetic particle image data. The promising results that have been obtained indicates the potential application of LaPIV to a large variety of volumetric velocity reconstruction problems .

7.2 Tracking, Data assimilation and model-data coupling

7.2.1 Optimal control techniques for the coupling of large scale dynamical systems and image data

Participants: Mohamed Yacine Ben Ali, Dominique Heitz, Etienne Mémin, Gilles Tissot.

In collaboration with the CSTB Nantes centre and within the PhD of Yacine Ben Ali we explored the definition of efficient data assimilation schemes for wind engineering. The goal is here to couple Reynolds average model to pressure data at the surface of buildings. Several techniques have been proposed to that end. We show in particular that optimisation conducted in a Sobolev space is highly beneficial as it brings natural smoothing to the sought solutions and avoids the use of regularization penalty. The techniques proposed consists in correcting the equations related to turbulent kinetic energy and dissipation. This work is thoroughly detailed in the PhD manuscript of Yacine Ben Ali 33. Two journal papers are currently under submission.

In another line of work, we addressed the study of variational data assimilation from a learning point of view. Data assimilation aims to reconstruct the time evolution of some state given a series of observations, possibly noisy and irregularly-sampled. Using automatic differentiation tools embedded in deep learning frameworks, we introduce end-to-end neural network architectures for data assimilation. It comprises two key components: a variational model and a gradient-based solver both implemented as neural networks. A key feature of the proposed end-to-end learning architecture is that we may train the NN models using both supervised and unsupervised strategies. This work has been published in 21

7.2.2 Ensemble data assimilation of large-scale dynamics with uncertainty

Participants: Benjamin Dufée, Etienne Mémin.

We investigated the application of a physically relevant stochastic dynamical model in ensemble Kalman filter methods. Ensemble Kalman filters are very popular in data assimilation because of their ability to handle the filtering of high-dimensional systems with reasonably small ensembles (especially when they are accompanied with so called localization techniques). The stochastic framework used in this study relies on Location Uncertainty (LU) principles which model the effects of the model errors on the large-scale flow components. The experiments carried out on the Surface Quasi Geostrophic (SQG) model with the localized square root filter demonstrate two significant improvements compared to the deterministic framework. Firstly, as the uncertainty is a priori built into the model through the stochastic parametrization, there is no need for ad-hoc variance inflation or perturbation of the initial condition. Secondly, it yields better MSE results than the deterministic ones. This work is currently under submission in a journal.

7.2.3 Reduced-order models for flows representation from image data

Participants: Dominique Heitz, Etienne Mémin.

The uncertainty based representation of Navier-Stokes equations proposed in 9 has been applied in the context of POD-Galerkin methods to devise stochastic reduced order models. This uncertainty modeling methodology provides a theoretically grounded technique to define an appropriate subgrid tensor as well as drift correction terms. This reduced order stochastic system has been evaluated on wake flow at moderate Reynolds number. For this flow the system has shown to provide very good uncertainty quantification properties as well as meaningful physical behavior with respect to the simulation of the neutral modes of the dynamics. This study is pursued within a strong collaboration with the industrial partner: SCALIAN 274032.

7.2.4 Learning of the dynamics of large scale geophysical systems using semi-group theory for data assimilation

Participants: Berenger Hug, Etienne Mémin, Gilles Tissot.

A methodological framework for ensemble-base estimation and simulation of high dimensional dynamical systems such as the oceanic or atmospheric flows is proposed. To that end, the dynamical system is embedded in a manifold of reproducible kernel Hilbert spaces with kernel functions driven by the dynamics. This manifold is nicknamed Wonderland for its appealing properties. In Wonderland the Koopman and Perron-Frobenius operator (also referred to in the literature as the composition and transfer operators, respectively) are unitary and uniformly continuous. They can be safely expressed in exponential series of diagonalizable bounded infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. Wonderland enables us the devise of strikingly simple ensemble data assimilation methods for trajectory reconstructions in terms of constant-in-time linear combinations of trajectory samples. Such an embarrassingly simple strategy is made possible through a fully justified superposition principle ensuing from several fundamental theorems. Numerical proofs of concept for data assimilation and trajectory recovery have been performed with a quasi-geostrophic flow model.

7.2.5 Estimation and control of amplifier flows

Participants: Gilles Tissot.

A class of flows, denoted "oscillator flows", are characterised by unstable modes of the linearised operator. A consequence is the dominance of relatively regular oscillations associated with a nonlinear saturation. Despite its non-linear behaviour, the associated structures and dynamical evolution are relatively easy to predict. Canonical configurations of this type of flows are the cylinder wake flow or the flow over an open cavity.

By opposition to that, "amplifier flows" are linearly stable with regard to the linearised operator. However, due to their convective nature, a wide range of perturbations are amplified in time and convected away such that it vanishes at long time. As a consequence there is a high sensitivity to perturbations together with a broad band response that forbid any low rank representation. Jets and mixing layers show this behaviour and a wide range of industrial applications and geophysical flows are affected by these broad band perturbations. It constitutes then a class of problems that are worth to treat separately since it is one of the scientific locks that make the estimation of observation-driven flow reconstruction by data assimilation in realistic configurations.

The stochastic framework of the modelling under location uncertainty (LU) considers a separation between a resolved velocity field and unresolved incoherent turbulent fluctuations. Amplifier flows are particularly sensitive to these perturbations and we believe that LU would improve significantly the predictions of turbulent amplifier flows.

In a deterministic context, there exists a type of models, termed as "parabolised", that enable to efficiently represent amplifier flows. These models, such as parabolised stability equations and one-way Navier-Stokes propagate, in the frequency domain, hydrodynamic instability waves over a given turbulent mean flow. The extension of parabolised models to a stochastic version is a promising approach which is perfectly adapted to represent the evolution and the variability of an instability propagating within a turbulent flow.

7.3 Analysis and modeling of turbulent flows and geophysical flows

7.3.1 Quantification of high-frequency surface dynamics using models and observations

Participants: Noé Lahaye.

The dynamics of upper-ocean high-frequency motions — internal waves and, more specifically, internal tides, has been addressed using different strategies. As part of F. Le Guillou PhD work (IGE, Grenoble) and in collaboration with J. Le Sommer, E. Cosme, S. Metref (MEOM team, IGE, Grenoble), C. Ubelmann (Ocean Next, Grenoble), E. Blayo, A. Vidard (Inria, Grenoble) and A. Ponte (LOPS, Brest), a joint algorithm for estimating the vortices as well as first mode internal tide from future wide-swath altimeter data has been proposed and tested in an idealized framework 23. This work is being pursued in the Fluminance team. In the context of Z. Caspar-Cohen PhD work, the signature of internal tides collected by surface drifters has been investigated using idealized numerical simulations. It has been shown that an apparent loss of coherence (time regularity of the oscillations at the tidal frequencies) arises in the Lagrangian perspective, resulting from the drifters being advected by the slow flow 36, 29. A statistical model based on the statistical properties of the slow flow has been proposed and validated against the data in this framework. It allows to predict this « apparent » loss of coherence.

7.3.2 Reduced dynamical models for geophysical flows

Participants: Noé Lahaye.

In this task, the dynamics of coherent structures in reduced models of the ocean upper layer was addressed. First, in collaboration with V. Zeitlin and T. Dubos (LMD, Paris), we investigated the dynamics of coherent dipolar analytical solutions in the Thermal Rotating Shallow Water model. This model is a vertically averaged model of the hydrostatic Navier-Stokes equations under the Boussinesq approximation and with Coriolis term, and retains the horizontal variations of density (or temperature). In the strong rotation regime, a low Rossby number asymptotic model can be derived : the Thermal Quasi-Geostrophic model. It exhibited a surprising small scale instability, leading to mixing of the thermal anomaly carried by the dipole. This research activity is pursued to better understand the dynamics of these models, in connexion with the STUOD project, which involves such models. In a second part of this task, in collaboration with A. Paci and S. Llewellyn Smith, we investigated the destabilization of surface eddies in a two layer with outcropping isopycnal layer — a different approach for representing horizontal variations of temperature and density in the ocean. This work highlighted the stabilization of surface eddies by a weak co-rotating lower-layer flow, which was shown to result from the suppression of one of the mode triggering instability through wave resonance 22.

7.3.3 Low order modelling of the interactions between internal tides and currents

Participants: Igor Maingonnat, Noé Lahaye, Gilles Tissot, Etienne Mémin.

This study focuses on the interactions between internal tides and oceanic currents. The goal is to develop new modelling strategies to extract, understand and predict physical features involved in the interactions between internal waves and flow structures. The phenomenon is isolated in a simplified numerical simulation of the rotating shallow water model, where a single wave propagates through a jet flow. Spectral proper orthogonal decomposition (SPOD) allows to identify from data purely coherent structures evolving at a given frequency. It allows to perform physical analyses and it constitutes reference observations. Linearising the model over the mean flow, resolvent analysis allows to predict the most responsive non-linearities and the associated responses. The framework will allow to predict coherence decay and to explicitly compute triadic interactions responsible of wave scattering by the flow. On this basis, extension to stochastic modelling under location uncertainty will refine the predictions. A potential direction is the development of data assimilation procedures defined in the frequency domain.

7.3.4 Physical understanding of internal tides dynamics using high fidelity numerical simulations

Participants: Adrien Bella, Noé Lahaye, Gilles Tissot, Etienne Mémin.

This research focuses on physical analysis of internal tides from very large databases issued from high fidelity numerical simulations. After a projection onto vertical modes, the energy exchanges at the tidal frequencies are quantified. The role of the topography and the currents in the exchange between modes are explored. In a second step, loss of coherence of internal tides due to currents will be explored. Three regions increasing in complexity are targeted: the Açores where the topography plays a major role, a region where the gulf stream and internal tildes interact strongly, and equatorial regions where very strong non-linearities occur.

7.3.5 Effects of smooth divergence-free flows on tracer gradients and spectra: Eulerian prognosis description

Participants: Etienne Mémin.

To predict the tracer deformations by ocean eddies and the evolution of their 2nd-order statistics, an efficient proxy has been proposed in collaboration with Bertrand Chapron (Ifremer) and Valentin Resseguier (Scalian) 25. Applied to a single velocity snapshot, this proxy extends the Okubo-Weiss criterion. For the Lagrangian-advection-based downscaling methods, it successfully predicts the evolution of tracer spectral energy density after a finite time, and the optimal time to stop the downscaling operation. A practical estimation can then be proposed to define an effective parameterization of the horizontal eddy diffusivity.

7.3.6 Geophysical flows modeling under location uncertainty

Participants: Noe Lahaye, Long Li, Etienne Mémin, Gilles Tissot, Francesco Tucciarone.

In this research axis we have devised a principle to derive representation of flow dynamics under location uncertainty 9. Such an uncertainty is formalized through the introduction of a random term that enables taking into account large-scale approximations or truncation effects performed within the dynamics analytical constitution steps. Rigorously derived from a stochastic version of the Reynolds transport theorem, this framework, referred to as modeling under location uncertainty (LU), encompasses several meaningful mechanisms for turbulence modeling. It indeed introduces without any supplementary assumption the following pertinent mechanisms: (i) a dissipative operator related to the mixing effect of the large-scale components by the small-scale velocity;(ii) a multiplicative noise representing small-scale energy backscattering; and (iii) a modified advection term related to the so-called turbophoresis phenomena, attached to the migration of inertial particles in regions of lower turbulent diffusivity.

Within the PhDs of Long Li and Valentin Resseguier 34, 20, 26 we have shown how LU modeling can be applied to provide stochastic representations of a variety of classical geophysical flows dynamics. Numerical simulations and uncertainty quantification have been performed on Quasi Geostophic approximation (QG) of oceanic models. It has been shown that LU leads to remarkable estimation of the unresolved errors opposite to classical eddy viscosity based models. The noise brings also an additional degree of freedom in the modeling step and pertinent diagnostic relations and variations of the model can be obtained with different scaling assumptions of the turbulent kinetic energy (i.e. of the noise amplitude). For a wind forced QG model in a square box, which is an idealized model of north-Atlantic circulation, we have shown that for different versions of the noise the QG LU model leads to improve long-terms statistics when compared to classical large-eddies simulation strategies. For a QG model we have demonstrated that the LU model allows conserving the global energy. We have also shown numerically that Rosby waves were conserved and that inhomogeneity of the random component triggers secondary circulations. This feature enabled us to draw a formal bridge between a classical system describing the interactions between the mean current and the surface waves and the LU model in which the turbophoresis advection term plays the role of the classical Stokes drift.

In collaboration with Ruediger Brecht, PhD student at Memorial University of Newfoundland, we worked on the incorporation of a stochastic representation of the small-scale velocity component of a fluid flow in a variational integrator for the rotating shallow-water equations on the sphere, already developed within the first part of its PhD work. This work has been published in JAMES 20.

A study of a stochastic version of the primitive equations model is currently investigated within the PhD of Francesco Tucciarone. An investigation of such model in a realistic ocean numerical code with various noise assumptions is undergoing.

7.3.7 Hamiltonian formulation of the stochastic wave problem

Participants: Evgueni Dinvay, Etienne Mémin.

We study here a Hamiltonian stochastic formulation of the water wave problem in the setting of the modelling under location uncertainty. Starting from reduction of the stochastic fluid motion equations to the free surface, we show how one can naturally deduce Hamiltonian structure under a small noise assumption. Moreover, as in the classical water wave theory, the non-local Dirichlet-Neumann operator appears explicitly in the energy functional. This, in particular, allows us to conduct in a natural way the systematic approximation of the Dirichlet-Neumann operator and to devise different simplified wave models including noise. Well posedness analysis of some of such models is currently under investigation.

7.3.8 Parameterization for coarse-resolution ocean modeling

Participants: Louis Thiry, Long Li, Etienne Mémin.

We work on simple parameterization for coarse-resolution oceanic models to replace computationally expensive high-resolution ocean models. We focus on the eddy-permitting scale (grid step   Rossby radius) and computationally cheap parameterization. We are currently investigating the modification of the diffusion (friction) operator to reproduce the mean velocity observed via measurements or a high-resolution reference solution. To test this new parameterization on a double-gyre quasi-geostrophic model, we are implementing a fast and portable python implementation of the multilayer quasi-geostrophic model. Preliminary results are encouraging as they show that we can fairly reproduce the reference mean velocity and significantly improve energy statistics. This method shall serve as a deterministic basis for future coarse-resolution stochastic parameterizations.

7.3.9 Variational principles for structure-preserving discretizations in stochastic fluid dynamics

Participants: Long Li, Etienne Mémin.

In collaboration with Rudiger Brecht (MPI, Hamburg) and Werner Bauer (Imperial College) we introduced in 20 a physically relevant stochastic representation of the rotating shallow water equations. This derivation relies mainly on a stochastic transport principle and on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. As for the classical (deterministic) system, this scheme, referred to as modeling under location uncertainty (LU), conserves the global energy of any realization and provides the possibility to generate an ensemble of physically relevant random simulations with a good trade-off between the model error representation and the ensemble's spread. To maintain numerically the energy conservation feature, we combine an energy (in space) preserving discretization of the underlying deterministic model with approximations of the stochastic terms that are based on standard finite volume/difference operators. The LU derivation, built from the very same conservation principles as the usual geophysical models, together with the numerical scheme proposed can be directly used in existing dynamical cores of global numerical weather prediction models. The capabilities of the proposed framework is demonstrated for an inviscid test case on the f-plane and for a barotropically unstable jet on the sphere.

7.3.10 Higher order temporal numerical schemes in time for dynamics under location uncertainty

Participants: Pierre-Marie Boulvard, Camilla Fiorini, Long Li, Etienne Mémin.

In this work we studied Milstein-type schemes for models under location uncertainty. We focus in particular on study the surface quasi-geostrophic (SQG) system under location uncertainty (LU). For this model we devised efficient scheme based on a second order Runge-Kutta scheme with the Levy area term set to zero. This scheme has been numerically compared to Milstein scheme in which the Levy area was fully taken into account. It is shown that the proposed scheme leads to better results with a comparable computational cost.

7.3.11 Stochastic compressible fluid dynamics

Participants: Etienne Mémin, Gilles Tissot.

We are currently working on the extension of the stochastic formulation under location uncertainty to compressible flows. The interest is to extend the formulation on the one hand to compressible fluids (for instability mechanisms involved in areoacoustics for instance, or for thermal effects in mixing layers) and on the other hand to geophysical flows where the Boussinesq equation is not valid anymore (density variations due to temperature or salinity gradients). A theoretical study has been performed that opens the door to numerical validations. In particular a baroclinic torque term has been identified that could have major effects in some situations.

7.3.12 Stochastic hydrodynamic stability under location uncertainty

Participants: Etienne Mémin, Gilles Tissot.

In order to predict instability waves propagating within turbulent flows, eigenmodes of the linearised operator is not well suited since it neglects the effect of turbulent fluctuations on the wave dynamics. To cope this difficulty, resolvent analysis has become popular since it represents the response of the linearised operator to any forcing representing the generalised stress tensors. The absence of information on the non-linearity is a strong limitation of the method. In order to refine these models, we propose to consider a stochastic model under location uncertainty expressed in the Fourier domain, to linearise it around the corrected mean-flow and to study resulting eigenmodes. The stochastic part represents the effect of the turbulent field onto the instability wave. It allows to specify a structure of the noise and then to improve existing models. Improvements compared to the resolvent analysis have been found for turbulent channel flow data at $R{e}_{\tau }=180$, $R{e}_{\tau }=550$ and $R{e}_{\tau }=1000$. A paper has been published in Journal of Fluid Mechanics 28, and a second one is in preparation for Journal of Fluid Mechanics. This work is in collaboration with André Cavalieri (Instituto Tecnologico de Aeronautica, SP, Brésil).

7.3.13 Singular and regular solutions to the Navier-Stokes equations (NSE) and relative turbulent models

Participants: Francois Legeais, Roger Lewandowski, Etienne Mémin.

The common thread of this work is the problem set by J. Leray in 1934 : does a regular solution of the Navier- Stokes equations (NSE) with a smooth initial data develop a singularity in finite time, what is the precise structure of a global weak solution to the Navier-Stokes equations, and are we able to prove any uniqueness result of such a solution. This is a very hard problem for which there is for the moment no answer. Nevertheless, this question leads us to reconsider the theory of Leray for the study of the Navier-Stokes equations in the whole space with an additional eddy viscosity term that models the Reynolds stress in the context of large- scale flow modelling. It appears that Leray's theory cannot be generalized turnkey for this problem, and must be reconsidered from the beginning. This problem is approached by a regularization process using mollifiers, and particular attention must be paid to the eddy viscosity term. For this regularized problem and when the eddy viscosity has enough regularity, we have been able to prove the existence of a global unique solution that is of class C1 in time and space and that satisfies the energy balance. Moreover, when the eddy viscosity is of compact support in space, uniformly in time, we recently shown that this solution converges to a turbulent solution of the corresponding Navier-Stokes equations carried when the regularizing parameter goes to 0. These results are described in a paper published in JMAA.

In the framework of the collaboration with the University of Pisa (Italy), namely with Luigi Berselli collaboration, we considered the three dimensional incompressible Navier-Stokes equations with non stationary source terms chosen in a suitable space. We proved the existence of Leray-Hopf weak solutions and that it is possible to characterize (up to sub-sequences) their long-time averages, which satisfy the Reynolds averaged equations, involving a Reynolds stress. Moreover, we showed that the turbulent dissipation is bounded by the sum of the Reynolds stress work and of the external turbulent fluxes, without any additional assumption, than that of dealing with Leray-Hopf weak solutions. This is a very nice generalisation to non stationnary source terms of a famous results by Foais. In the same work, we also considered ensemble averages of solutions, associated with a set of different forces and we proved that the fluctuations continue to have a dissipative effect on the mean flow. These results have been published in Nonlinearity. We have studied in 17 the rate of convergence of the weak solutions ${u}^{\alpha }$ of $\alpha$-regularization models to the weak solution u of the Navier-Stokes equations in the two-dimensional periodic case, as the regularization parameter $\alpha$ goes to zero. More specifically, we have considered the Leray-$\alpha$, the simplified Bardina, and the modified Leray-$\alpha$ models. We have improved known results in terms of convergence rates and also to show estimates valid over long-time intervals. The results also hold in the case of bounded domain with homogeneous Dirichlet boundary conditions. We have derived in 18 from the laws of the turbulence and Leray’s energy inequality for weak solutions of the Navier-Stokes equations, a back-scatter rotational Large Eddy Simulation model, which is the extension of the Baldwin and Lomax model to non-equilibrium problems. The model is particularly designed to mathematically describe a fluid filling a domain with solid walls and consequently the differential operators appearing in the smoothing terms are degenerate at the boundary. After the derivation of the model, we have prove some of the mathematical properties coming from the weighted energy estimates and which allow to prove existence and uniqueness of a class of regular weak solutions.

In 35 we show that the rotational Smagorinsky model for turbulent flows, can be put, for a wide range of parameters in the setting of Bochner pseudo-monotone evolution equations. This allows to prove existence of weak solutions a) identifying a proper functional setting in weighted spaces and b) checking some easily verifiable assumptions, at fixed time. We also will discuss the critical role of the exponents present in the model (power of the distance function and power of the curl) for what concerns the application of the theory of pseudo-monotone operators.

Another study in collaboration with B. Pinier, P. Chandramouli and E. M/'emin has been undertaken. This work takes place within the context of the PhD work of B. Pinier. We have tested the performances of an incompressible turbulence Reynolds-Averaged Navier-Stokes one-closure equation model in a boundary layer, which requires the determination of the mixing length $\ell$. A series of direct numerical simulation have been performed, with flat and non trivial topographies, to obtain by interpolation a generic formula $l=l\left(Re\delta ,z\right)$, $Re\delta$ being the frictional Reynolds number, and z the distance to the wall. Numerical simulations have been carried out at high Reynolds numbers with this turbulence model, in order to discuss its ability to properly reproduce the standard profiles observed in neutral boundary layers, and to assess its advantages, its disadvantages and its limits. We also proceeded to a mathematical analysis of the model. The study has been published in 24

7.3.14 Dissipation mechanisms in perforated liners with grazing flow

Participants: Gilles Tissot.

Perforated liners are a technology implanted in the nacelles of aircraft engines, in order to absorb noise coming from the fan and the combustion chamber. It is constituted by honeycomb cavities covered by a perforated plate. The cavities produce resonance, thus inducing a flow through the perforations where viscous dissipation occurs, necessary for sound absorption. These perforations cause drag in the air intake, which can be reduced by considering micro perforates. However, existing models are not able to predict correctly the impedance of such plates. Understanding the dissipation mechanisms and improving the impedance predictions for microperforated plates was the objective of the thesis of Robin Billard, successfully defended this year and leaded in collaboration with Gwénaël Gabard (LAUM – laboratoire d'acoustique de l'université du Mans) and Safran Nacelles. The challenge was to account for non-linear effects and grazing flow in the developed models. Resolvent analysis has been explored to identify relevant non-linearities that impact the impedance, with and without flow. Part of this study has been pubished in 19.

7.4 Coupled models in hydrogeology

7.4.1 Reactive transport in multiphase flow

Participants: Jocelyne Erhel.

Groundwater resources are essential for life and society, and should be preserved from contamination. Pollutants are transported through the porous medium and a plume can propagate. Reactive transport models aims at simulating this dynamic contamination by coupling advection dispersion equations with chemistry equations. If chemistry is at thermodynamic equilibrium, then the system is a set of partial differential and algebraic equations (PDAE). Space discretization leads to a semi-discrete DAE system which should be discretized in time. An explicit time scheme allows an easy decoupling of transport and chemistry, but very small timesteps should be taken, leading to a very large CPU time. Therefore, an implicit time scheme is preferred, coupling transport and chemistry in a nonlinear system. The special structure of linearized systems can be used in preconditioned Newton-Krylov methods in order to improve efficiency. Some experiments illustrate the methodology and show also the need for an adaptive timestep and a control of convergence in Newton's iterations.

7.4.2 Characterizations of Solutions in Geochemistry at equilibrium

Participants: Jocelyne Erhel.

Geochemistry at thermodynamic equilibrium involves aqueous reactions and mineral precipitation or dissolution. Quantities of solute species are assumed to be strictly positive, whereas those of minerals can vanish. The mathematical model is expressed as the minimization of Gibbs energy subject to positivity of mineral quantities and conservation of mass. Optimality conditions lead to a complementarity problem. We show that, in the case of a dilute solution, this problem can also be considered as optimality conditions of another minimization problem, subject to inequality constraints. This new problem is easier to handle, both from a theoretical and a practical point of view. Then we define a partition of the total quantities in the mass conservation equation. This partition builds a precipitation diagram such that a mineral is either precipitated or dissolved in each subset. We propose a symbolic algorithm to compute this diagram. Simple numerical examples illustrate our methodology.

7.4.3 Mathematical models of kinetic reactions in geochemistry

Participants: Jocelyne Erhel.

In geochemistry, kinetic reactions can lead to the appearance or disappearance of minerals or gas. We defined two mathematical models based first on a differential inclusion system and second on a projected dynamical system. We proposed a regularization process for the first model and a projection algorithm for the second one.

This work, supported by IFPEN, has been published in 37 .

7.5 Sparse Linear solvers

7.5.1 Parallel GMRES

Participants: Jocelyne Erhel.

Sparse linear systems $Ax=b$ arise very often in computational science and engineering. Krylov methods are very efficient iterative methods, and restarted GMRES is a reference algorithm for non-symmetric systems. A first issue is to ensure a fast convergence, by preconditioning the system with a matrix $M$. Preconditioning must reduce the number of iterations, and be easy to solve. A second issue is to achieve high performance computing. The most time-consuming part in GMRES is to build an orthonormal basis $V$. With the Arnoldi process, many scalar products involve global communications. In order to avoid them, s-step methods have been designed to find a tradeoff between parallel performance and stability. Also, solving a system with the matrix $M$ and for multiplying a vector by the matrix $A$ should be efficient. A domain decomposition approach involves mainly local communications and is frequently used. A coarse grid correction, based on deflation for example, improves convergence. These techniques can be combined to provide fast convergence and fast parallel algorithms. Numerical results illustrate various issues and achievements.

8 Bilateral contracts and grants with industry

8.1 Bilateral contracts with industry

8.1.1 Contract ITGA

Participants: Dominique Heitz, Etienne Mémin.

duration 36 months. This partnership between Inria, Irstea and ITGA funds the PhD of Romain Schuster. The goal of this PhD is to design new image-based flow measurement methods for the study of industrial fluid flows. Those techniques will be used in particular to calibrate industrial fume hood.

8.1.2 Contract CSTB

Participants: Mohamed Yacine Ben Ali, Dominique Heitz, Etienne Mémin.

duration 36 months. This partnership between Inria, Irstea and CSTB funds the PhD of Yacine Ben Ali. This PhD aims to design new data assimilation scheme for Reynolds Average Simulation (RANS) of flows involved in wind engineering and buildings construction. The goal pursued here consists to couple RANS models and surface pressure data in order to define data driven models with accurate turbulent parameterization.

9 Partnerships and cooperations

9.1 International initiatives

9.1.1 Participation in other international programs

Noé Lahaye and Etienne Mémin are part of the SWOT Science Team. They participate to the DIEGO project (CNES-NASA) headed by Aurélien Ponté (Ifremer). Noé Lahaye is leader of WP4: Internal tide mapping and predictability.

9.2 International research visitors

Etienne Mémin is Visiting Professor at the Mathematics department of Imperial College London.

9.3 European initiatives

9.3.1 ERC Synergy Grant STUOD: Stochastic Transport in Upper Ocean Dynamics

PI: Etienne Mémin

Duration: 01/03/2020 - 01/03/2026

Participants: Evgueni Dinvay, Benjamin Dufée, Camilla Fiorini, Beranger Hug, Long Li, Etienne Mémin, Louis Thiry, Francesco Tucciarone.

Partners: IFREMER, IMPERIAL COLLEGE London ((UK))

Summary: 71 percent of Earth is covered by ocean. The ocean has absorbed 93 percent of the heat trapped by human’s greenhouse gas emissions. The ocean’s future responses to continued warming are uncertain. Our project will deliver new capabilities for assessing variability and uncertainty in upper ocean dynamics. It will provide decision makers a means of quantifying the effects of local patterns of sea level rise, heat uptake, carbon storage and change of oxygen content and pH in the ocean. Its multimodal monitoring will enhance the scientific understanding of marine debris transport, tracking of oil spills and accumulation of plastic in the sea. Our approach accounts for transport on scales that are currently unresolvable in computer simulations, yet are observable by satellites, drifters and floats. Four scientific capabilities will be engaged: (i) observations at high resolution of upper ocean properties such as temperature, salinity, topography, wind, waves and velocity; (ii) large scale numerical simulations; (iii) data-based stochastic equations for upper ocean dynamics that quantify simulation error; and (iv) stochastic data assimilation to reduce uncertainty. These four scientific capabilities will tackle a network of joint tasks achieved through cooperation of three world-calibre institutions: IFREMER (ocean observations, reanalysis); INRIA (computational science); and Imperial College (mathematics, data assimilation). Our complementary skill sets comprise a single systemic effort: (1) Coordinate and interpret high-resolution satellite and in situ upper ocean observations (2) Extract correlations from data needed for the mathematical model (3) Perform an ensemble of computer simulations using our new stochastic partial differential equations (SPDE) which are derived by matching the observed statistical properties (4) Apply advanced data assimilation and computer simulations to reduce model uncertainty The key to achieving these goals will be synergy in our combined expertise.

9.4 National initiatives

9.4.1 Comins'lab: SEACS : Stochastic modEl-dAta-Coupled representationS for the analysis, simulation and reconstruction of upper ocean dynamics

Participants: Etienne Mémin.

duration 48 months. The SEACS project whose acronym stands for: “Stochastic modEl-dAta-Coupled representationS for the analysis, simulation and reconstruction of upper ocean dynamics” is a Joint Research Initiative between the three Britanny clusters of excellence of the "Laboratoires d'Excellence" program: Cominlabs, Lebesgue and LabexMer centered on numerical sciences, mathematics and oceanography respectively. Within this project we aim at studying the potential of large-scale oceanic dynamics modeling under uncertainty for ensemble forecasting and satellite image data assimilation.

9.4.2 ANR BECOSE : Beyond Compressive Sensing: Sparse approximation algorithms for ill-conditioned inverse problems.

Participants: Dominique Heitz.

duration 48 months. The BECOSE project aims to extend the scope of sparsity techniques much beyond the academic setting of random and well-conditioned dictionaries. In particular, one goal of the project is to step back from the popular L1-convexification of the sparse representation problem and consider more involved nonconvex formulations, both from a methodological and theoretical point of view. The algorithms will be assessed in the context of tomographic Particle Image Velocimetry (PIV), a rapidly growing imaging technique in fluid mechanics that will have strong impact in several industrial sectors including environment, automotive and aeronautical industries. The consortium gathers the Fluminance and Panama Inria research teams, the Research Center for Automatic Control of Nancy (CRAN), The Research Institute of Communication and Cybernetics of Nantes (IRCCyN), and ONERA, the French Aerospace Lab.

9.4.3 IFPEN project

Participants: Jocelyne Erhel, Bastien Hamlat.

Contract with IFPEN (Institut Français du Pétrole et Energies Nouvelles) Duration: three years from October 2016. Title: Fully implicit Formulations for the Simulation of Multiphase Flow and Reactive Transport Coordination: Jocelyne Erhel. Contract with IFPEN (Institut FranÃ§ais du Pétrole et Energies Nouvelles). Duration: three years October 2016-September 2019. Title: Fully implicit Formulations for the Simulation of Multiphase Flow and Reactive Transport. Coordination: Jocelyne Erhel. Abstract: Modeling multiphase flow in porous media coupled with fluid-rock chemical reactions is essential in order to understand the origin of sub-surface natural resources and optimize their use. This project focused on chemistry models, with kinetic reactions. We developed a mathematical tool, which can be embedded into a reactive transport code.

9.4.4 LEFE MANU: MSOM

Participants: Long Li, Etienne Mémin.

Title: Multiple Scale Ocean Model

Duration: From 2018 to 2021

Coordination: Bruno Deremble (CNRS LMD/ENS Paris)

Abstract: The objective of this project is to propose a numerical framework of a multiscale ocean model and to demonstrate its utility in the understanding of the interaction between the mean current and eddies.

Participants: Gilles Tissot, Etienne Mémin.

Title: Apprentissage de la Dynamique de modèles Océaniques par des approches issues de la Théorie des Semi-groupes pour l’Assimilation de Données (ADOTSAD). Duration: From 2019 to 2021

9.5 Regional initiatives

9.5.1 ARED COMIOE

Participants: Adrien Bela, Noé Lahaye, Gilles Tissot, Etienne Mémin.

The Britany ARED project "COmpréhension et Modélisation de mécanismes non-linéaires dans l'océan : les Interactions entre Ondes internes et Ecoulement" funds 50 percent of the PhD thesis of Adrien Bela.

10 Dissemination

10.1 Promoting scientific activities

10.1.1 Journal

Member of the editorial boards

• Jocelyne Erhel
• member of the editorial board of ETNA.
• member of the editorial board of ESAIM:Proceedings and Surveys.

Reviewer - reviewing activities

• Dominique Heitz
• Reviewer for Exp. in Fluids, Physics of Fluids, EDF « Prix Paul Caseau »
• Etienne Mémin
• Reviewer for Journ. of Fluid Mech., Ocean Modelling, J. of Heat and Mass Transfer, J. Comp. Phys., Siam Review, Comp. and Fluids
• Gilles Tissot
• Reviewer for Journ. of Fluid Mech., J. Comp. Phys.

Dominique Heitz

• Member of INRAE OPAALE research unit scientific council
• Member of INRAE OPAALE research unit council
• Head of INRAE ACTA Team

Roger Lewandowski

• President du Comité de liaison du groupe GAMNI-SMAI
• President of the Blaise Pascal award jury
• President of GAMNI-SMAI PhD thesis award
• Corresponding person of the SMAI in Rennes
• Responsible of the group "Mathematical modeling" of IRMAR
• Member of the scientific council of IRMAR,
• Member of Mathematical teaching council of U. Rennes I
• Member of the scientific council of the Henri Lebesgue Centre

Etienne Mémin

• Member of the scientific council of LEFE-MANU action of CNRS INSU
• Member of the comity GAMNI-SMAI

10.2 Teaching - Supervision - Juries

10.2.1 Teaching

• Licence: Jocelyne Erhel, Optimisation, 12h, niveau L3, ENSAI Rennes
• Licence : Dominique Heitz, Mécanique des fluides, 30h, niveau L2 INSA Rennes
• Licence: Noé Lahaye, thermo-energetics, 50h, INSA Rennes
• Master: Jocelyne Erhel, arithmétique flottante, 4h, niveau M1, INSA Rennes
• Master : Dominique Heitz, Mécanique des fluides, 25h, niveau M1, Dep GMA INSA Rennes
• Master: Roger Lewandowski, Euler and the Navier-Stokes equations, M2, master « fondamental mathematics».
• Master : Etienne Mémin, Motion analysis , 9h, Master 2 SISEA Université de Rennes 1.
• Master : Gilles Tissot, mathematics for acoustics, 20h, niveau M1, Université du Mans.

10.2.2 Supervision

• PhD in progress: Adrien Bella, Understanding interactions between internal tides and currents in the ocean using high-fidelity numerical simulations, started October 2021, supervised by Noé Lahaye, Gilles Tissot, Etienne Mémin
• PhD in progress: Francois Legeais, Mathematical and numerical modeling of the sea atmosphere interface, started october 2021, supervised by Roger Lewandowski
• PhD in progress: Igor Maingonnat, Understanding and modelling nonlinear mechanisms in the ocean: internal waves / background flow interactions. Started November 2021, supervised by Noé Lahaye, Gilles Tissot, Etienne Mémin
• PhD in progress: Manolis Perrot, student at U. Grenoble Alpes, Consistent modelling of subgrid scale for ocean climate models. Started October 2021, supervised by Eric Blayo, Florian Lemarié, Etienne Mémin
• PhD in progress : Berenger Hug, analysis of stochastic models under location uncertainty started November 2020, supervisors: Etienne Mémin, Arnaud Debussche.
• PhD in progress: Benjamin Dufée, Particle filters in high dimentional spaces, started November 2020, supervisors: Dan Crisan, Etienne Mémin
• PhD in progress: Francesco Tuccarone, Stochastic models for high resolution oceanic models, started November 2020 Long Li, Etienne Mémin.
• PhD in progress : Z. Caspar-Cohen, PhD student at Ifremer, LOPS, Signatures des ondes de marée internes et des mouvements en équilibre quasi-géostrophique sur les courants de surface : perspectives eulérienne vs lagrangienne. Started 15/10/2019. Supervised by Noé Lahaye with X. Carton (UBO) and A. Ponte (Ifremer).
• PhD successfully defended: Robin Billard, Modelling of non-conventional perforated acoustic liners, Université du Mans, supervised by Gilles Tissot, defended 14/04/2021.
• PhD successfully defended: Bastien Hamlat, University of Rennes 1, co-advisors Jocelyne Erhel and A. Michel. 21 march 2021
• PhD successfully defended: Yacine Ben Ali, Variational assimilation of RANS models for wind engineering, supervised by Dominique Heitz, Etienne Mémin, Gilles Tissot, defended 24/03/2021.
• PhD successfully defended: Long Li, Data assimilation and stochastic transport for the upper ocean dynamics, supervisor: Etienne Mémin, defended 25/03/2021.

10.2.3 Juries

Dominique Heitz

• Walid MRAD, PhD, IMT Atlantique (Examinateur)
• Mathias LEMKE, HDR, TU Berlin (Rapporteur)

Etienne Mémin

• Said Ouala, IMT Atlantic (President), 17/03/2021
• Nishant Kupar, Institut P', U. Poitiers (Examinateur), 08/09/2021
• Vincent Le Guen Cnam Paris (Examinateur), 30/11/2021

10.3 Popularization

Jocelyne Erhel

• was scientific coordinator of the website Interstices (June 2012 - September 2019). She is now member of the editorial board, from October 2019.

11 Scientific production

11.1 Major publications

• 1 articleK.K. Burrage and J.Jocelyne Erhel. On the performance of various adaptive preconditioned GMRES.Numerical Linear Algebra with Applications51998, 101-121
• 2 articleJ.Jérôme Carrayrou, J.Joachim Hoffmann, P.Peter Knabner, S.Serge Kräutle, C. D.Caroline De Dieuleveult, J.Jocelyne Erhel, J. V.Jan Van der Lee, V.V. Lagneau, K.K.U. Mayer and K.K.T.B. MacQuarrie. Comparison of numerical methods for simulating strongly non-linear and heterogeneous reactive transport problems. The MoMaS benchmark case.Computational Geosciences1432010, 483-502
• 3 bookT.T. Chacón-Rebollo and R.R. Lewandowski. Mathematical and Numerical Foundations of Turbulence Models and Applications.Modeling and Simulation in Science, Engineering and TechnologyBirkhäuser Basel2014
• 4 articleT.Thomas Corpetti, D.Dominique Heitz, G.Georges Arroyo, E.Etienne Mémin and A.Alina Santa-Cruz. Fluid experimental flow estimation based on an optical-flow scheme.Experiments in fluids402006, 80--97
• 5 articleJ.-R. D.Jean-Raynald De Dreuzy, A.Anthony Beaudoin and J.Jocelyne Erhel. Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations.Water Resource Research43W10439, doi:10.1029/2006WR0053942007
• 6 articleA.Alejandro Gronskis, D.Dominique Heitz and E.Etienne Mémin. Inflow and initial conditions for direct numerical simulation based on adjoint data assimilation.Journal of Computational Physics2422013, 480-497
• 7 articleD.Dominique Heitz, E.Etienne Mémin and C.C. Schnoerr. Variational Fluid Flow Measurements from Image Sequences: Synopsis and Perspectives.Experiments in fluids4832010, 369--393
• 8 articleH.H. Hoteit, J.Jocelyne Erhel, R.R. Mosé, B.Bernard Philippe and P.P. Ackerer. Numerical Reliability for Mixed Methods Applied to Flow Problems in Porous Media.Computational Geosciences622002, 161-194
• 9 articleE.E. Mémin. Fluid flow dynamics under location uncertainty.Geophysical & Astrophysical Fluid Dynamics10822014, 119-146
• 10 bookN.N. Nassif, J.J. Erhel and B.B. Philippe. Introduction to computational linear Algebra.CRC Press2015
• 11 articleN.Nicolas Papadakis and E.Etienne Mémin. A variational technique for time consistent tracking of curves and motion.Journal of Mathematical Imaging and Vision3112008, 81--103
• 12 articleV.V. Resseguier, E.E. Mémin and B.B. Chapron. Geophysical flows under location uncertainty, Part I Random transport and general models.Geophys. & Astro. Fluid Dyn.11132017, 149-176
• 13 articleV.Valentin Resseguier, E.Etienne Mémin and B.Bertrand Chapron. Geophysical flows under location uncertainty, Part II Quasi-geostrophy and efficient ensemble spreading.Geophysical and Astrophysical Fluid Dynamics1113April 2017, 177-208
• 14 articleV.Valentin Resseguier, E.Etienne Mémin and B.Bertrand Chapron. Geophysical flows under location uncertainty, Part III SQG and frontal dynamics under strong turbulence conditions.Geophysical and Astrophysical Fluid Dynamics1113April 2017, 209-227
• 15 articleV.V. Resseguier, E.E. Mémin, D.D. Heitz and B.B. Chapron. Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis.J. Fluid Mech.8282017, 29
• 16 articleY.Y. Saad, M.M. Yeung, J.Jocelyne Erhel and F.F. Guyomarc'h. A deflated version of the Conjugate Gradient Algorithm.SIAM Journal on Scientific Computing2152000, 1909-1926

11.2 Publications of the year

International journals

• 17 articleL. C.Luigi C. Berselli, A. A.Argus Adrian Dunca, R.Roger Lewandowski and D.Dinh Duong Nguyen. Modeling Error of -Models of Turbulence on a Two-Dimensional Torus.Discrete and Continuous Dynamical Systems - Series B2692021, 4613-4643
• 18 articleL. C.Luigi C. Berselli, R.Roger Lewandowski and D. D.Dinh Duong Nguyen. Rotational forms of Large Eddy Simulation turbulence models: modeling and mathematical theory.Chinese Annals of Mathematics - Series B4212021, 1-24
• 19 articleR.Robin Billard, G.Gilles Tissot, G.Gwénaël Gabard and M.Marc Versaevel. Numerical simulations of perforated plate liners: Analysis of the visco-thermal dissipation mechanisms.Journal of the Acoustical Society of America1491January 2021, 16-27
• 20 articleR.Rüdiger Brecht, L.Long Li, W.Werner Bauer and E.Etienne Mémin. Rotating shallow water flow under location uncertainty with a structure-preserving discretization.Journal of Advances in Modeling Earth SystemsOctober 2021
• 21 articleR.Ronan Fablet, B.Bertrand Chapron, L.Lucas Drumetz, E.Etienne Mémin, O.Olivier Pannekoucke and F.François Rousseau. Learning Variational Data Assimilation Models and Solvers.Journal of Advances in Modeling Earth Systems132021, article n° e2021MS002572
• 22 articleN.Noé Lahaye, A.Alexandre Paci and S.Stefan Llewellyn Smith. Instability of Lenticular Vortices: Results from Laboratory Experiments, Linear Stability Analysis and Numerical Simulations.Fluids611November 2021, 1-25
• 23 articleF.Florian Le Guillou, N.Noé Lahaye, C.Clément Ubelmann, S.Sammy Metref, E.Emmanuel Cosme, A.Aurélien Ponte, J.Julien Le Sommer, E.Eric Blayo and A.Arthur Vidard. Joint Estimation of Balanced Motions and Internal Tides From Future Wide‐Swath Altimetry.Journal of Advances in Modeling Earth Systems1312December 2021, 1-17
• 24 articleB.Benoît Pinier, R.Roger Lewandowski, E.Etienne Mémin and P.Pranav Chandramouli. Testing a one-closure equation turbulence model in neutral boundary layers.Computer Methods in Applied Mechanics and Engineering3762021, article n° 113662
• 25 articleV.Valentin Resseguier, B.Bertrand Chapron and E.Etienne Mémin. Effects of smooth divergence-free flows on tracer gradients and spectra: Eulerian prognosis description.Journal of Physical Oceanography2021
• 26 articleV.Valentin Resseguier, L.Long Li, G.Gabriel Jouan, P.Pierre Dérian, E.Etienne Mémin and B.Bertrand Chapron. New trends in ensemble forecast strategy: uncertainty quantification for coarse-grid computational fluid dynamics.Archives of Computational Methods in Engineering281January 2021, 215-261
• 27 articleV.Valentin Resseguier, A. M.Agustin M Picard, E.Etienne Mémin and B.Bertrand Chapron. Quantifying truncation-related uncertainties in unsteady fluid dynamics reduced order models.SIAM/ASA Journal on Uncertainty Quantification932021, 1152–1183
• 28 articleG.Gilles Tissot, A. .André V. G. Cavalieri and E.Etienne Mémin. Stochastic linear modes in a turbulent channel flow.Journal of Fluid Mechanics9122021, 1-33
• 29 articleX.Xiaolong Yu, A.Aurélien Ponte, N.Noé Lahaye, Z.Zoé Caspar‐Cohen and D.Dimitris Menemenlis. Geostrophy Assessment and Momentum Balance of the Global Oceans in a Tide‐ and Eddy‐Resolving Model.Journal of Geophysical Research. Oceans12610October 2021, 1-19

International peer-reviewed conferences

• 30 inproceedingsL.Long Li, D.Deremble Bruno, N.Noé Lahaye and E.Etienne Mémin. Stochastic modeling of the oceanic mesoscale eddies.6th Sandbox STUOD Sandbox WorkshopLondon, United KingdomFebruary 2021, 1-24

Conferences without proceedings

• 31 inproceedingsL.Long Li, B.Bertrand Chapron, B.Bruno Deremble, N.Noé Lahaye and E.Etienne Mémin. Statistically data-driven modelling Location Uncertainty in mesoscale dynamics.STUOD 2021 - 2nd Stochastic Transport in Upper Ocean Dynamics Annual WorkshopLondon, United KingdomSeptember 2021
• 32 inproceedingsA. M.A M Picard, M.M Ladvig, V.Valentin Resseguier, D.D Heitz, E.E Mémin and B.B Chapron. Real-time flow estimation from reduced order models and sparse measurements.AERO2020+1﻿ - International Conference on Applied AerodynamicsVirtual, FranceApril 2021, 1-20

Doctoral dissertations and habilitation theses

• 33 thesisM. Y.Mohamed Yacine Ben Ali. Investigating Data-Model coupling using adjoint techniques for wind engineering.Université de Rennes 1March 2021
• 34 thesisL.Long Li. Stochastic modeling and numerical simulation of ocean dynamics.Université de Rennes 1March 2021

Reports & preprints

• 35 miscL. C.Luigi C Berselli, A.Alex Kaltenbach, R.Roger Lewandowski and M.Michael Růžička. On the existence of weak solutions for a family of unsteady rotational smagorinsky models.September 2021
• 36 miscZ.Zoé Caspar-Cohen, A.Aurélien Ponte, N.Noé Lahaye, X.Xavier Carton, X.Xiaolong Yu and S.Sylvie Le Gentil. Characterization of internal tide incoherence : Eulerian versus Lagrangian perspectives.January 2022
• 37 miscB.Bastien Hamlat, J.Jocelyne Erhel, A.Anthony Michel and T.Thibault Faney. Analysis and numerical computation of geochemical systems with kinetic precipitation and dissolution reactions involving several minerals.March 2021

Other scientific publications

• 38 miscC.Camilla Fiorini, L.Long Li and É.Étienne Mémin. Higher order schemes in time for the surface quasigeostrophic system under location uncertainty.Vienna / Virtual, AustriaApril 2021, 1
• 39 miscL.Long Li, B.Bruno Deremble, N.Noé Lahaye and E.Etienne Mémin. Stochastic modeling of mesoscale eddies.Vienne, AustriaApril 2021
• 40 miscV.Valentin Resseguier, M.Matheus Ladvig, A. M.Agustin Martin Picard, E.Etienne Mémin, D.Dominique Heitz, D.Dimitry Voisin and C.Caroline BRAUD. Real-time unsteady air flow prediction to reduces mechanic load variations and wind turbine maintenance costs.Nantes, Saint-Nazaire, FranceSeptember 2021, 1

11.3 Cited publications

• 41 articleR.R. Adrian. Particle imaging techniques for experimental fluid mechanics.Annal Rev. Fluid Mech.231991, 261-304
• 42 articleT.T. Bewley. Flow control: new challenges for a new Renaissance.Progress in Aerospace Sciences372001, 21--58
• 43 articleS.S. Beyou, A.A. Cuzol, S.S. Gorthi and E.E. Mémin. Weighted Ensemble Transform Kalman Filter for Image Assimilation.TellusA6518803January 2013
• 44 articleS.S. Cai, E.E. Mémin, P.P. Dérian and C.C. Xu. Motion Estimation under Location Uncertainty for Turbulent Fluid Flow.Exp. in Fluids5982017
• 45 articleH.H. Choi, P.P. Moin and J.J. Kim. Direct numerical simulation of turbulent flow over riblets.Journal of Fluid Mechanics2551993, 503--539
• 46 articleT.T. Corpetti and E.E. Mémin. Stochastic Uncertainty Models for the Luminance Consistency Assumption.IP2122012, 481--493
• 47 articleF.-X. L.F.-X. Le Dimet and O.O. Talagrand. Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects.Tellus38A1986, 97--110
• 48 articleG.G. Evensen. Sequential data assimilation with a non linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics.J. Geophys. Res.99 (C5)101994, 143--162
• 49 phdthesisJ.J. Favier. Contrôle d'écoulements~: approche expérimentale et modélisation de dimension réduite.Institut National Polytechnique de Toulouse2007
• 50 inproceedingsJ.J. Favier, A.A. Kourta and G.G. Leplat. Control of flow separation on a wing profile using PIV measurements and POD analysis.IUTAM Symposium on Flow Control and MEMSLondon, UKSeptember 19-22, 2006
• 51 articleN.N. Gordon, D.D. Salmond and A.A. Smith. Novel approach to non-linear/non-Gaussian Bayesian state estimation.IEEE Processing-F1402April 1993
• 52 articleA.A. Guégan, P.P. Schmid and P.P. Huerre. Optimal energy growth and optimal control in swept Hiemenz flow.J. Fluid Mech.5662006, 11--45
• 53 articleB.B.K.P. Horn and B.B.G. Schunck. Determining Optical Flow.Artificial Intelligence171-3August 1981, 185--203
• 54 bookJ.J.L. Lions. Optimal Control of Systems Governed by Partial Differential Equations.Springer-Verlag1971
• 55 articleL.L. Mathelin and O.O. Le Maître. Robust control of uncertain cylinder wake flows based on robust reduced order models.Computer and Fluids382009, 1168--1182
• 56 articleB.B. Protas and J.J.E. Wesfreid. Drag force in the open-loop control of the cylinder wake in the laminar regime.Physics of Fluids142February 2002, 810--826
• 57 inproceedingsI.I. Wygnanski. Boundary layer flow control by periodic addition of momentum.4th AIAA Shear Flow Control ConferenceUSAJune 29-July 2, 1997