Robust and distributionally robust shape and topology optimization

Shape and topology optimization generally aims to minimize a cost function of the domain, possibly under constraints. In applications, this cost function involves the solution to a boundary-value problem encoding its physical behavior; the latter typically involves data, such as the applied forces or the properties of the constituent material in the realm of mechanical structures, the viscosity in uid mechanics, etc. Most often, these physical parameters are known imperfectly, be it because they are measured with error-prone processes, or because they are altered with time. This raises the need to incorporate a degree of awareness about such uncertainties into the optimal design problem. In this presentation, we describe three different paradigms to achieve this purpose: 

• Worst-case approaches are suitable when no information is available about the uncertain parameters, except for an upper bound on their amplitude around a given mean value. We then minimize the worst (i.e. largest) value of the cost functional thanks to a linearization of the dependence of the cost with respect to these parameters.

• Probabilistic approaches describe the uncertain parameters as random variables, and rely on the knowledge of their law. Using again a linearization strategy, we construct approximate counterparts for the mean value, or the standard deviation of the cost function under the possible uncertainties.

• The recent paradigm of distributionally robust optimization springs from the realization that in realistic situations, the law of the uncertain parameters is itself unknown: at best, it can be (imperfectly) reconstructed from a few observed samples. We then minimize the worst possible value of the expected cost, when the law of the uncertain parameters lies \within a certain distance" from this nominal law. This leverages recent findings in optimal transport theory and convex optimization. 

This presentation is based on several works, in collaboration with G. Allaire, F. Iutzeler, J. Prando and B. Thibert.