The optimization of complex systems, based on numerical simulations, is currently growing strongly in the industrial field, for example in aeronautics or telecommunications. The approach consists in coupling an optimization algorithm, which will seek for the optimal value of a set of parameters, to a simulator estimating the value of the cost function for each set of parameters proposed. A major difficulty lies in the computational time required for each simulation, which can amount to several hours when fine numerical models are used. Optimization must therefore take into account a highly constrained computational budget, which, in practice, is often limited to a few dozen simulations.
In this difficult context, Bayesian optimization methods have recently demonstrated their ability to provide interesting results. The approach consists in building, on the basis of some observations of the cost function, a statistical model of Gaussian Process type, which is then enriched iteratively by determining the parameters maximizing an acquisition function and by simulating the corresponding configurations.
In the perspective of relying on extremely expensive simulations, we are interested in this thesis in extending this method to multi-fidelity optimization in the context of large-scale computational infrastructures. The idea is to mix several estimation levels of the cost function during the optimization, to progress more quickly. Indeed, assuming that there are different methods for the estimation of the cost function, hierarchical in terms of accuracy and computational cost, the algorithm can certainly sometimes rely on less accurate, but also less expensive, estimates, if these are sufficiently correlated with the fine estimates. The objective is then to converge towards the optimum, for the finest estimate, by using as much as possible coarser estimates. An important point of the algorithm is the selection of the level of fidelity to use for each new simulation, via a multi-fidelity acquisition function. For this, we seek to determine which level is the most relevant, in terms of information provided and computational cost. In the context of large-scale computing facilities, additional criteria should also be considered, such as simulator scalability, available resources or energy consumption.
An expected result of this thesis is the definition of different formulations for this key step, and their comparison for a set of computational scenarios. Some advanced cases, e.g. multi-criteria optimization problems and asynchronous algorithms will also be investigated.
The algorithms developed will be validated on benchmark problems, and then applied to aerodynamic design exercises. In the context of aerodynamic simulations, the different fidelity levels can be defined by adjusting the space/time discretization, the flow model or by using surrogate models from pre-computed databases. In such case the hierarchy between fidelities is unknown and the multi-information source framework would be considered. Finally, additional test-cases (geophysics, micro-swimmers) will be considered in the framework of Sage-HPC PEPR project, in collaboration with research partners.
