**Multidomain simulation:**When simulating phenomena on a
large scale, it is natural to try to divide the domain of
calculation into subdomains with different physical
properties. According to these properties one may think of
using in the subdomains different discretizations in space
and time, different numerical schemes and even different
mathematical models. Research toward this goal includes the
study of interface problems, subdomain time discretization,
implementation using high level programming languages and
parallel computating. Applications are mostly drawn from
environmental problems from hydrology and hydrogeology, such
as studies for a deep underground nuclear waste disposal and
for the coupling of water tables with surface flow.

**Flow and transport in porous media with
fractures:**Looking at a scale where the fractures can be
represented individually and considering the coupling of
these fractures with the surrounding matrix rock, various
numerical models where the fracture is represented as an
interface between subdomains are proposed and analyzed.
Transmission conditions are then nonlocal. One phase and
twophase flow are studied.

**Interphase problems for twophase flow in porous
media:**Twophase flow is modeled by a system of nonlinear
equations which is either of parabolic type or of hyperbolic
type depending on whether capillary pressure is taken into
account or not. Interface problems occur when the physical
parameters change from one rock type to the other, including
the nonlinear coefficients (relative permeabilities and
capillary pressure). The study of these interface problems
leads to the modeling of twophase flow in a porous medium
with fractures.

**Reactive transport:**Efficient and accurate numerical
simulation is important in several situations: the need to
predict the fate of contaminated sites is the primary
applications. Numerical simulation tools help to design
remediation strategies, for example by natural degradation
processes catalyzed by microbia which are present in the
earth. Another important application is the assessment of
long-term nuclear waste storage in the underground.
Multi-species reactive ow problems in porous media are
described by a set of partial differential equations for the
mobile species and ordinary differential equations for the
immobile species (which may be viewed as attached to the
interior surfaces of the soil matrix) altogether coupled
through nonlinear reaction terms. The large variety of time
scales (e.g., fast aqueous complexation in the ground water
and relatively slow biodegradation reactions and transport
processes) makes it desirable to describe fast reactions by
equilibrium conditions, i.e., by nonlinear algebraic
equations.

**Code Coupling :**As physical models become more and more
sophisticated, we start encountering situations involving
different physics. In most situations, the computer codes for
the individual components are different (they may even be
built by different groups). However, it may be desirable to
use a strongly coupled methods, in order to fuly resolve the
physics. The Newton–Krylov framework enables to build global
methods for the coupled problems, without the need to have a
monolithic solver. Again here, reactive transport is a
natural application.

**Functional Programming and scientific
computation:**Implementing subdomain coupling requires
complex programming. This can be done efficiently using
OCamlP3l, a recent development of the language OCaml which
allows for parallel computing. This provides an alternative
to Corba and MPI. Another example of implementation with
OCaml is the programming of a parameterization method
developed to estimate at the same time the zonation and the
values of the hydraulic transmissivities in groudwater
flow.

**Parameter Estimation and sensitivity analysis:**When
parameters appearing in a Partial Derivative Equation (PDE)
are not precisely known, they can be estimated from measures
of the solution. The parameter estimation problem is usually
formulated as a minimization problem for an Output
Least-Squares (OLS) function. The adjoint state technique is
an efficient tool to compute the analytical gradient of this
OLS function which can be plugged into various local
optimization codes. The Singular Value Decomposition is a
powerful tool for deterministic sensitivity analysis. It
quantifies the number of parameters which can be estimated
from the field measures. This can help in choosing a
parameterization of the searched coefficients, or even in
designing the experiments. Current applications under study
are in optometry, in hydrogeology and in reservoir
simulation.

**Optimization:**An important facet of the project deals
with the development optimization theories and algorithms.
This activity is in part motivated by the fact that parameter
estimation leads to minimization problems. Special focus is
on large scale problems, such as those encountered in
engineering applications. The developed techniques and
domains of interest include lagrangian relaxation (including
augmented Lagrangian approach and progressive hedging),
sequential quadratic programming, interior point methods,
nonsmooth methods, algebraic optimization, optimization
without derivative, decomposition methods for large scale
problems, bilevel optimization,
*etc*. There are many applications: seismic tomography
data inversion, shape optimization (aeronautic and tyre
industry), mathematical modelling in medicine and biology
(cancer chronotherapy), optimization of the electricity
production, to mention a few of those that have been
considered by the team. Outcomes of this activity are also
the
*Moduloptlibrary*, which
gathers optimization pieces of software produced by the team,
and the
*Liboptenvironment*, which
is a platform for testing and profiling solvers on
heterogeneous collections of problems.

**Complementarity problems:**Extending optimization,
*complementarity problems*occur when two systems of
equations are in competition, the one that is active being
determined by variables reaching threshold values.
Mathematically, these conditions can be expressed by
,
, and
, where
Fand
are two functions. Usually, a model will include other
equations and inequations. The full system can be viewed as a
special case of
*variational inequalities*. The numerical techniques to
solve such a problem have known a spectacular development
during these recent years and have a vast domain of
applications. Complementarity can indeed be used to model
contact problems, chemical or economical equilibria,
precipitation-dissolution phenomena,
*etc*. We have started in 2008, with the PhD thesis of
Ibtihel Ben Gharbia, to apply nonlinear complementarity
techniques to the solution of a diphasic (water and hydrogen)
flow with phase exchange in a porous medium. The
appearance/disappearance of the hydrogen gas phase can indeed
be modeled by nonlinear complementarity conditions. Special
attention is paid on the so-called Newton-min algorithm,
which may be viewed as a semismooth Newton method applied to
the following nonsmooth equivalent formulation of the
problem:
min(
F(
x),
G(
x)) = 0.

`M1qn3`(version 3.3: October, 2009): 47 downloads in 2010.

`SQPlab`(version 0.4.4: February 2009): 250 downloads in
2010.

We have achieved significant progresses dor the simulation of reactive transport phenomena. B. Gueslin and M. Kern have completed the simulation of a water–gas system, including several aqueous species and minerals, designed by IFP-Energies Nouvelles, In the framework of the SHPCO2 ANR project. A snapshot of the concentrations of some species is shown on figure .

The figure shows the initial gas bubble dissolving in water, with a corresponding increase in pH, and calcite dissolution.

**(EdF)** A. Chiche is preparing a PhD thesis (Cifre
EdF-Inria, direction J. Ch. Gilbert) on
decomposition-coordination methods for the middle-term
optimization of the electricity production. The case where
uncertainties are present is also considered, using scenario
trees, which leads to even larger deterministic optimization
problems. Improvements have been brought

on the solution of
*infeasible*convex quadratic optimization problems
using the augmented Lagrangian approach
and

on the solution of the optimization of the electricity production under uncertainties, using the progressive hedging algorithm.

**ANDRA**

Phuong Hoang Thi Thao's PhD began in October her PhD thesis on subdomain time stepping formulated as a time and space domain decomposition problem. Her thesis is supported by a contract between INRIA and ANDRA.

Another contract between INRIA and ANDRA concerns M. Kern's consulting support on high performance computing.

Groupement Momas(Mathematical Modeling and Numerical Simulation for a Deep Underground Disposal of Nuclear Waste).

Agence Nationale de la Recherche ANR Fost(Formal prOofs about Scientific compuTations), with EPI Proval from INRIA Saclay - Île-de-France, Laboratoire de Recherche en Informatique from University of Paris 11, and Laboratoire d'Informatique de l'Université Paris-Nord from University of Paris 13.

Agence Nationale de la Recherche ANR SHPCO2(Simulation Haute Performance du Stockage Géologique de CO2) with IFP, LAGA laboratory from University Paris 13, Ecole des Mines de St Etienne and BRGM.

Estime is also associated with Lamsin-ENIT in the DGRSRT(Tunisie)/INRIA STIC project “Identification de paramètres en milieu poreux : analyse mathématiques et étude numérique”. From 2008.

Estime is also associated with LIRNE-Equipe d'ingéniérie mathématiques, université Ibn Tofaïl, Kenitra, Maroc (PHC Volubilis) in the project “Techniques multi-échelles adaptatives pour la résolution des problèmes d'écoulement et de transport en milieux poreux hétérogènes”. From 2010.

There is also a cooperation with the Tata Institute of Fundamental Research (TIFR) in Bangalore through the CEFIPRA project “Conservation Laws and Hamilton Jacobi equations”. From 1/09/2006.

A. Taakili, Univ. of Errachidia (Marrocco), 09/09/2010 to 30/07/2010.

A. Fumagalli, PhD student, MOX, Politecnico di MIlano (Itlay), 06/09/2010 to 24/09/2010.

M. Kern is a member of the Scientific Board of Groupement MoMaS.

M. Kern is a member of the Scientific Board of UNIT, l'Université Numérique Ingénierie et Technologie.

Université Paris Dauphine, License
2nd year,
*Calcul matriciel*, 54 h.

Université Paris Dauphine, License
1st year,
*Linear algebra*, 39 h.

ENSTA, 2nd year,
*Optimisation différentiable – théorie et
algorithmes*, 26 h.

École des Mines de Paris, 1st year:
*Differential Calculus*, 20 h.

ENSTA, 2nd year,
*Optimisation différentiable – théorie et
algorithmes*, 42 h.

École Nationale d'Ingénieurs de Tunis
(ENIT), Tunisia, Mastère Mathématiques Appliquées,
*Volumes finis et éléments finis mixtes*, 20 h
with J. E. Roberts.

Mines-ParisTech,
*Introduction au calcul scientifique*, 2nd year
students, 10 h,
*Eléments finis*, 2nd year students, 30 h,
*Approximation et évolution : aspects numériques*,
2nd year students, 20 h.

École Supérieure d'Ingénieurs Léonard
de Vinci,
*Approximation methods*, 4th year students, 20
h.

École Nationale d'Ingénieurs de Tunis
(ENIT), Tunisia, Mastère Mathématiques Appliquées,
*Volumes finis et éléments finis mixtes*, 20 h
with J. Jaffré.

(with J. Ch. Gilbert and J.
Jaffré)
*Henry's law and gas phase appearance and
disappearance modeled as a complementarity
problem*, European Geosciences Union General
Assembly 2010 (EGU), Vienna, Austria, May 02-07,
2010.

(with J. Ch. Gilbert and J.
Jaffré)
*Formulation avec contraintes de complémentarité
pour un modèle diphasique en milieux poreux avec
échange entre les phases*, Journée scientifique du
GNR MOMAS, Paris, 3 décembe 2010.

*How the augmented Lagrangian algorithm deals with an
infeasible convex quadratic optimization problem*, The
International Conference on Continuous Optimization
(ICCOPT) 2010, Santiago, Chili, 26-29 juillet 2010.

*Comportement et usage de l'algorithme de Lagrangien
augmenté dans le cas d'un problème quadratique convexe
non-réalisable*, Groupe de travail de l'équipe
Commands (Inria-Ensta-Cmap-Cnrs), Journée des doctorants,
Paris, France, 18 juin 2010.

(with M. Kern)
*Simulation numérique du stockage du CO2*, 10èmes
Journées d'Étude sur les Milieux Poreux, Nancy (France),
October 20-21, 2010.

*How the augmented Lagrangian algorithm can deal with
an infeasible convex quadratic optimization problem*,
Erice, Sicile, Italy, July 2-10, 2010.

*Flow in porous media with fractures: a discrete
fracture model using cell-centered discretization
methods*, Premières Journées Scientifiques du
Laboratoire Euro-Maghrébin de Mathématiques et de
leurs interaction (LEM2I), Tipaza, Algérie, 13-22
juin 2010.

*Henry's law and gas phase disappearance solved as
a complementarity problem*(with I. Ben Gharbia and
J.-C. Gilbert), 2ième Congrès de la Société Marocaine
de Mathématiques Appliquées (SM2A), Rabat, Maroc,
28-30 juin 2010.

*Mixed finite elements on hexahedral
meshes*(with J. E. Roberts). Workshop on Advanced
methods for the diffusion equation on general meshes,
5,6 juillet, 2010, Paris, France.

*Composite mixed finite elements for deformed
cubes*. 2010 Full SIAM meeting, Pittsburg, USA,
July 12-16, 2010.

*On the upstream mobility finite difference
scheme*. 12th European Conference on Mathematics
for Oil Recovery (ECMOR XII), Oxford, England, 6-9
septembre 2010.

One week visit to Prof. Zoubida Mghazli, université Ibn Tofaïl, Kenitra, Maroc (PHC Volubilis).

(with L. Amir, A. Taakili)
*Reactive transport in porous media*, visit to
MOX, Politecnico di Milano (Italy), April 14
2010.

(with A. Taakili)
*Linear and nonlinear preconditioning for a model
of transport with sorption*, XVIII Conference on
Computational Methods in Water Resources, Barcelona
(Spain), June 21-24, 2010.

(with A. Michel) organized the SHPCO2 workshop, St Lambert des Bois (France), June 14-15 2010 (30 participants, 15 invited talks).

(with L. Amir, B. Gueslin, A.
Taakili),
*Reactive transport in porous media: formulations,
non-linear solvers and preconditioners*, High
Performance Computing for CO2 Geological Storage, St
Lambert des Bois (France), June 14-16, 2010,

(with L. Amir, B. Gueslin, A.
Taakili)
*Coupled transport and chemistry*, thematic day
“Groupement MoMaS, Numerical Methods”, Paris (Fance),
October 6, 2010.

(with L. Amir, B. Gueslin)
*Coupled formulations and coupling algorithms for
reactive transport in porous media*, DyCap
Workshop "Microbiology and Reactive Transport in the
Capillary Fringe", University of Heidelberg,
(Germany), October 7-8, 2010, invited lecture.

(with B. Gueslin)
*Coupled formulations and coupling algorithms for
reactive transport in porous media*, seminar at
LMA, Université Technologique de Compiègne (France),
November 16, 2010.

*Comment calcule un ordinateur ?*, pedagogical
presentation to first year university students,
Université René Descartes, Paris (France), September
17, 2010.

(with N. Frih, V. Martin and A.
Saada)
*Some numerical results for modeling fractures as
interfaces with nonconforming grids*(poster
presentation), 2010 InterPore Conference and Annual,
College Station,Texas, USA, March 15-17, 2010.

*Flow in heterogeneous porous media: a discrete
fracture model with cell centered elements*.
International congress in Mathematical Fluid Dynamics
and its applications (MFD 2010), Rennes, June 21-24,
2010.

*Single phase flow in porous media with fractures
modeling Forchheimer fractures as interfaces*(with
Peter Knabner) and organized with Z. Mghazli the
minisymposium
*Méthodes et Outils pour les Milieux Poreux*,
2ième Congrès de la Société Marocaine de
Mathématiques Appliquées (SM2A), Rabat, Maroc, 28-30
juin 2010.

One week visit to Prof. Zoubida Mghazli, université Ibn Tofaïl, Kenitra, Maroc (PHC Volubilis).