Parameter Estimation in a Model for Tracer Transport in One-Dimensional Fractals - Selected Topics of Computational and Experimental Fluid Mechanics - page 237

3.3 Optimization Algorithm

There exist diverse optimization algorithms that can be used to minimize functions
like Eq. ( 23 ). However, the particularities of our case, i.e., noisy data and solutions
that are found numerically, make gradient-free optimization algorithms more appro-
priate even though the gradient-based methods are more efficient. The reason is
that the numerical evaluation of the Jacobian and the Hessian matrices from noisy
data increases the uncertainty. The gradient-free algorithm used in the optimization
process is a modified Nelder-Mead method, which requires neither Jacobian nor
Hessian matrix evaluation and allows constraints in parameter values. This method
evaluates the OF at the vertices of a n -dimensional simplex ( n

3 in this work and the
simplex is a tetrahedron) so that the search moves away from the poorest value and
the minimum is enclosed in a simplex which continuously changes in size. The final
simplex is reached when the OF or the parameters fulfill some prescribed stopping
criterion.

=

4 Synthetic Data Generation

To generate the synthetic data, (i) we first choose a set of model parameter values,
(ii) then we numerically solve the advection-dispersion equation for a given distance
to the production well and a selected set of tracer breakthrough times, and finally
(iii) we add uniformly distributed noise to the concentration values obtained. Two
levels of noise are taken into account: 5 and 10%with respect to the maximum tracer
concentration and three levels of the amount of data points: 10, 20 and 40. The time
data are sampled in a non-uniform way in order to construct the breakthrough curve
in a prescribed time range. The data will consist in a series of pair values: time and
concentration { t i , C i }. In this specific case, the synthetic data were generated by
setting: α = 0 . 75, D ad = 0 . 0133 and ʸ = 0 . 05 and the following fixed parameters:

ʔ 1m. As mentioned above several sets of tracer

concentration versus time were constructed from the model. For each noise level (5
and 10%) three different amount of data points were generated (approximately 10,
20 and 40). The amount of points is not constant since negative concentration values
are eliminated. Therefore, we will work with six data sets. By using themwe analyse
the effect of the amount of data, the noise level and the amount of fitting parameters.
To measure the fitting quality, the percentage of relative error (PRE) and the OF
value are used.
In the optimization process a random search was used to get the starting
parameters. Ten trials per parameter are tossed so that 1,000 function evaluations
are performed. The parameter set with the lowest FO value is chosen. By this way,
the starting point will presumably be close to the global minimum of the OF.

t = 10 days, L = 400m and x w = 0 .