# Gaussian processes-based excursion set estimation for scalar or vector black box functions. Application to the calibration of a numerical wind turbine simulator.

#### Speciality : Mathématiques Appliquées

21/11/2024 - 10:00
Clément Duhamel *(Université Grenoble Alpes)*
Bâtiment Imag Salle 306

Keywords :

- Excursion set estimation
- Sequential design of experiments.

Many industrial issues are related to excursion set estimation problems, in the form of estimating a set of feasible model input values, corresponding for example to feasible optimal design schemes. The excursion set we seek to estimate is defined as the set of model values in the input space (design space) satisfying a given constraint(s) on the model outputs. In the following, we consider the framework of a black box numerical simulator, which can be evaluated at any point in the design space, but for a high computational time. An efficient way of estimating an excursion set is to model the black box function as the realization of a Gaussian process. The sequential construction of a design of experiments, i.e. a set of points in the design space and the associated evaluations by the black box function, enables the progressive learning of the surrogate model. The points added to the sequential design of experiments are chosen according to the optimization of an acquisition criterion, that depends on the surrogate model at the current stage. Many types of acquisition criteria can be considered, depending on the objective set for the black box function (global knowledge, optimization, estimation of an excursion set, etc.). There is also a class of acquisition criteria called ”Stepwise Uncertainty Reduction” (SUR), whose aim is to anticipate the impact of adding the next evaluation to the experimental design on a well-chosen uncertainty measure. The aim of this thesis is to study acquisition criteria suitable for estimating excursion sets, for black box functions with scalar or vector outputs, and to apply these contributions to the calibration of a numerical simulator for wind turbine design. The first part focuses, in the case of scalar outputs, on the improvement of an acquisition criterion adapted to the estimation of excursion sets into a SUR version. The aim of this approach is to combine the robustness of the chosen criterion due to its exploration property with good performance in terms of the exploitation of SUR-type strategies. The second part focuses on the framework of black box models with vector output, and for which all output components are evaluated simultaneously (isotopic data). The aim of this second part is to develop several criteria adapted to the simultaneous estimation of each of the excursion sets for each output component of the model. Among the proposed criteria, we distinguish between criteria inspired by the scalar framework using a surrogate model on each model output component, and another criterion using a Multi- Output Gaussian Process (MOGP) type surrogate model whose aim is to take into account the correlation between the different output components of the black box function. The different proposed strategies are applied to the calibration of a numerical simulator for wind turbine design. The aim of this application is to find a set of input parameters of the simulator (stiffness coefficients for certain materials), such that the deformation modes calculated by the simulator are not too far apart, for an adapted norm and in relation to fixed thresholds, from the observed modes derived from experimental data.

#### Directors:

- Clémentine Prieur
*(Université Grenoble Alpes )* - Céline Helbert
*(Ecole Centrale de Lyon )* - Miguel Munoz Zuniga
*(IFP Energies Nouvelles )* - Delphine Sinoquet
*(IFP Energies Nouvelles )*

#### Raporteurs:

- Rodolphe Le Riche
*(École des Mines de Saint-Etienne, Université Clermont Auvergne )* - David Ginsbourger
*(University of Bern )*

#### Examinators:

- Adeline Leclercq Samson
- Josselin Garnier
*(École Polytechnique )* - Julien Bect
*(CentraleSupélec, Université Paris-Saclay )*