Thesis

I defended my thesis, High-Dimensional Bayesian Multi-Objective Optimization on October 28th 2019. The subject was the high dimensional optimization of the internal and external aerodynamics of a vehicle. It aimed at developping methods and optimization algorithms for the multi-criteria optimization of expensive-to-evaluate blackbox functions, depending on a high number of parameters. In the industrial context of Groupe PSA, one seeks to find designs that optimize several objectives (such as Drag, in external aerodynamics, or Engine Efficiency in internal aerodynamics). If you do not directly belong to the scientific community and already feel lost with this introduction, I suggest you to visit the page "For other audiences", where the subject of my thesis and my research are popularized (English and French) ;)

Flow around an Ahmed-Body (left),and inside a combustion chamber (right).

The shape and the settings of these objects are typical parameters to be optimized.

Since a car or an engine are depending on a high number of parameters, and as the evaluation of the objectives for one single design is performed using costly high-fidelity Computational Flud Dynamics (CFD) codes, the simulator can only be employed a limited number of times. Indeed, it takes between 12 and 24 hours to run such a single simulation. Moreover, as not only one, but several conflicting objectives are considered, there does not exist one single optimal design, being the best in each objective. Instead, several designs that offer different degrees of trade-off do exist. Those designs are optimal in the sense that it is not possible to improve their performance in one objective (i.e. minimize it) without deteriorating it in the others. We therefore are looking for designs that offer compromise between the competing quantities to optimize, called Pareto optimal solutions. Thereafter, an end-user will choose the design that matches the best his requirements, among all optimal designs delivered by the optimization algorithm.

Example of objective values found by a multi-objective optimization algorithm, within a restricted number of runs of the CFD simulator.

Here, two objectives are to be minimized. The blue triangles correspond to the objective values of designs that are Pareto optimal.

Optimal designs are those for which we have not found any design being better with respect to both objectives.

The goal of my thesis is to devise and validate a multi-objective method, that leads to designs which offer the best compromise between the objectives, using the CFD code only a limited and prescribed number of times. Mathematically speaking, my work focusses on the use of surrogate modelling (Gaussian Processes) in a multi-objective optimization setting. The aim is to find the best approximation of the compromise solutions (aka Pareto Front) within a very small number of iterations (thus calls to the simulator). For the purpose of tackling the restricted optimizaton budget, I propose a methodology to directly target well-balanced (that is to say, that do not favour one objective at the expense of the other objectives) solutions of the Pareto Front. This both enables to find compromise solutions quicker than if all Pareto solutions were sought, and alleviates problems arrising when considering a great number of objectives simultaneously. Furthermore, it ensures that solutions delivered by the algorithm are well-balanced and therefore attractive for the end-user ; solutions that belong to the Pareto Front but which favour one objective over the others may not interest him/her as much.

Instead of converging moderately well towards the whole Pareto Front (right column, classical method), the approach I developped (left column) estimates and targets the center of the PF (top).

Once convergence towards it has been detected (here at iteration #22), the algorithm targets a broader part of the PF (red square), determined by forecasting the width that can be accurately discovered within the remaining (18) iterations (bottom).

The resulting approximation front exhibits a faster, closer and a more dense convergence to the central part of the true Pareto Front, which corresponds to solutions that are meaningful for practitioners (equilibrium between the objectives)

The following scheme illustrates the methodology I developped for directing the optimization towards the most well-balanced solutions in a multi-objective problem under restricted budget

Sketch of the algorithm: at each step, estimate the center of the PF (a) and optimize over it (b).

If convergence has been detected (c), determine and target a wider part of the Pareto Front for the remaining iterations; otherwise, return to (a) and continue trying to improve over the currently estimated center