Distributionally robust shape and topology optimization

Keywords :

• incertitude
• distance de Wasserstein
• expériences numériques
• densité
• ligne de niveaux
• maillages.

This thesis introduces the paradigm of distributionally robust optimization, originally developed
in the field of convex optimization, into optimal design, a discipline focused on finding the best
design according to a physical performance criterion while satisfying constraints. By incorporating
distributional robustness, this work addresses parameter uncertainty issues, ensuring that designs
remain reliable even under variations in system parameters.
We discuss the sensitivity of optimal design results with respect to parameter perturbations and
provide a brief overview of several means to introduce robustness, leading to the introduction of the
concept of distributionally robust optimization in this context. We thus consider realistic situations in
which both the cost function to be minimized and the underlying physical model depend on uncertain
parameters, whose probability law itself is imperfectly known. In such cases, the only available
information about the latter is limited to a nominal law, reconstructed from a few observed samples.
This distributionally robust problem is then an a priori intricate bilevel optimization problem, in
which we minimize the worst-case value of a statistical quantity of the cost function (typically, its
expectation) over an ambiguity set of probability laws that are “close” to the nominal law.
Within the wide range of ambiguity sets, this thesis focuses on three particular classes of such
problems. First, the ambiguity set at hand is made of the probability laws whoseWasserstein distance
to the nominal law is less than a given threshold. Second, this ambiguity set is based on the firstand
second-order moments of the nominal probability law. Finally, a statistical quantity of the cost
function, namely its conditional value at risk, rather than its expectation is made robust with respect
to the law of the parameters. Using techniques from convex duality, we derive tractable, single-level
reformulations of these distributionally robust problems with respect to those ambiguity sets, framed
over an augmented set of variables. These methods are essentially agnostic of the optimal design
framework; we describe them in a unifying abstract framework.
We then numerically solve these distributionally robust problems to study multiple situations in
density-based and geometric-based optimal design. Several examples considering finite-dimensional
uncertainties about the loads are discussed in two and three space dimensions to appraise the features
of the proposed techniques. We provide few examples when the finite-dimensional assumption is relax
in order to deal with infinite-dimensional uncertainties about material coefficients or the geometry
itself.

President:

Emmanuel Maitre (Université Grenoble Alpes)

Directors:

• Boris Thibert (Université Grenoble Alpes )
• Charles Dapogny (Sorbonne Université )

Reporters:

• Fabien Caubet (Université de Pau )
• Beniamin Bogosel (Aurel Vlaicu University of Arad )

Examinators:

• Franck Iutzeler (Institut mathématiques de Toulouse )
• Virginie Ehrlacher (Ecole Nationale des Ponts et Chaussées )
• Emmanuel Maitre (Université Grenoble Alpes )