Design of experiments for the calibration of costly computer codes

Keywords :

• design of physical experiments
• design of numerical experiments
• Gaussian processes
• Bayesian optimization.

In this thesis, we study the problem of Bayesian calibration of expensive computer codes with scalar, vector,
or functional outputs using a limited amount of physical data. A computer code is considered expensive when its
evaluation requires significant computation time. In this context, Bayesian inference on its parameters necessitates the
use of an emulator or surrogate model.
We introduce a two-step strategy that combines the optimal selection of the design of physical experiments (for the
physical measurements to be carried out in the field) and the design of numerical experiments (for constructing the
Gaussian process emulator). The first phase consists of building an initial Gaussian process emulator, which will be
used to compute the optimality criteria for the design of physical experiments. Two types of criteria are introduced. On
one hand, Bayesian criteria based on the posterior distribution of the calibration parameters, which have the advantage
of accounting for all uncertainties, whether they are related to physical observations, numerical observations, or
the computer code parameters. On the other hand, a criterion that uses the variation of the code with respect to its
parameters combined with the distribution of the design in the experimental space. The latter is distinguished by its
reduced optimization cost. To optimize these criteria, four algorithms are proposed : simulated annealing, a greedy
algorithm, a genetic algorithm, and a stochastic optimization algorithm by simultaneous perturbation. After acquiring
the physical data, the emulator is improved through a sequential planning of numerical experiments, guided by an
acquisition criterion aimed at progressively reducing the calibration uncertainty. Two acquisition criteria, exclusively
dedicated to calibration, have been defined : the first is based on the sum of the posterior variances of the calibration
parameters, and the second is based on the prediction error on the physical observations. The final emulator is then
used to approximate the posterior density of the parameters using a Markov chain Monte Carlo sampling method,
coupled with a non-parametric kernel estimation. A numerical application on analytical functions with scalar outputs
has allowed us to evaluate the relevance of the criteria and the strategies for planning numerical experiments, by
comparing them to other state-of-the-art approaches. Moreover, an adaptation of the calibration methodology to
computer codes with vector or functional outputs is proposed.
Finally, the methodology was applied to the calibration of a numerical simulator dedicated to geological CO2
storage, using synthetic data.

President:

DR Nathalie BARTOLI (ONERA)

Directors:

• PR François BACHOC (Université Toulouse 3 )
• PR Clémentine PRIEUR (UGA )

Reporters:

• PR Pierre BARBILLON (AgroParisTech )
• DR Olivier Le Maître (Ecole Polytechnique )

Examinators:

• DR Luc PRONZATO (CNRS Côte d'Azur )
• Invité Sarah BOUQUET (IFP Energies Nouvelles )
• Invité Miguel MUNOZ ZUNIGA (IFP Energies Nouvelles )