Partial Differential Equations - Modelling and ... - ResearchGate

252 J.-M. Brun and B. Mohammadi
Once the model is established, the second inverse problem of interest is
the identification of possible sources of an observed pollution. This region is
defined where J p ′ is large. In this case, the parameter p is the location of the
different sources (cultures).
To solve the minimization problems, we use a semi-deterministic global
optimization algorithm based on the solution of the following boundary value
problem [MS03, Ivo06, IMSH06]:
{
p ζζ + p ζ = −J p(p(ζ)),
′ (7)
p(0) = p 0 , J(p(1)) = J m =0,
where ζ ∈ [0, 1] is a fictitious parameter. J m is the infimum of our inverse
problems (here taken as 0). This can be solved using solution techniques for
BVPs with free surface to find p(1) realizing the infimum (i.e. J(p(1)) = J m ).
An analogy can be given with the problem of finding the interface between
water and ice which is only implicitly known through the iso-value of zero
temperature. In case a local minima is enough, the second boundary condition
can be replaced by J ′ p(p(1)) = 0.
This algorithm requires the sensitivity of the functional with respect to
independent variables p. An interesting feature of the present low-cost modelling
is that gradients are also available at very low calculation cost. Indeed,
sensitivity evaluation for large dimension minimization problems is not an
easy task. The most efficient approach is to use an adjoint variable with the
difficulty that it requires the development of a specific software. Automatic
differentiation brings some simplification, but does not avoid the main difficulty
of intermediate states storage, even though check-pointing technique
brings some relief [Gri01, CFG + 01]. By simplifying the solution of the transport
problem, the present approach also addresses this issue.
4 Numerical Results
The application of low complexity transport model to several flow condition
is shown. Typical fields of 0.01 ∼ 0.1km 2 have been considered in a region of
400 km 2 . Rows are spaced by about 1.5 m. The source of the treatment moves
at a speed of around 1 m/s and the injection velocity is taken at 7 to 10 m/s for
a typical treatment of 100 kg/km 2 . Mono and multi sources situations (Figs.
3 and 4) are considered and examples of the constructed flow field are shown
together with the wind measurement points assimilated by the model (Figs. 1
and 4). The transport-based and the Euclidean distances have been reported
for a given point in Fig. 2. The impact of ground variations on the advected
species is shown in Fig. 5. An example of source identification problem is
shown in Fig. 6.