## Section: Overall Objectives

### Overall Objectives

**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 *Modulopt library*, which gathers optimization
pieces of software produced by the team, and the *Libopt environment*, 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 $F{\left(x\right)}^{\phantom{\rule{-0.55542pt}{0ex}}\top \phantom{\rule{-1.111pt}{0ex}}}G\left(x\right)=0$, $F\left(x\right)\ge 0$, and
$G\left(x\right)\ge 0$, where $F$ and $G:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ 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
to 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: $min\left(F\right(x),G(x\left)\right)=0$.