## 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 dierential
equations for the mobile species and ordinary dierential
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 and Grid Computing:** As physical models become more
and more sophisticated, we start encountering situations involving
different physics. This leads naturally to a computer code built from
individual components, where each component simulates one of the
physical models. A natural extension is to have the individual
components running on different computers (each one possibly being
parallel). Applications include densityâ€“driven flow, modelling
seawater intrusion in aquifers and reactive transport in porous
media.

**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 developped 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 concepts 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 sequential quadratic programming, interior
point methods, the augmented Lagrangian approach, bilevel optimization,
nonlinear complementarity problems, *etc* . There are many
applications: seismic tomography data inversion, shape optimization
(aeronautic and tyre industry), mathematical modelling in medicine and
biology (chronotherapy of cancer), to name a few. An outcome of this
activity is also the *Modulopt library* , which gathers optimization
softwares produced by the team.