Distributionally robust shape and topology optimization

The present Ph.D. project deals with the modelling, the mathematical analysis and the numerical implementation of the notion of distributional robustness, in the context of shape and topology optimization problems. Briefly, the main goal is to consider shape and topology optimization problems (posed in the context of structural mechanics, for instance), whose data are plagued with uncertainty, and to incorporate into their formulation a degree of robustness with respect to these uncertainty. As these are only known through observations, a probability distribution describing their law should first be reconstructed, as a preliminary step before expressing the worst mean value of the optimized function over all probability laws ``close’’ to this nominal object.

The candidate is expected to:

• Learn the basics about the theoretical and numerical aspects of shape optimization, optimal transport, and robust optimization which will be needed for getting acquainted with the topic.
• Getting familiar with shape optimization algorithms and with robust optimization algorithms.
• From the knowledge developed about robust optimization, devise a ``distributionally robust’’ formulation of a model shape optimization problem, which is realistic from the numerical implementation perspective.
• Appraise its numerical method on realistic examples in structural optimization, for different types of uncertain data (loads, material coefficients, geometry, etc.).
• Propose ditributionally robust formulation that lend themselves to further mathematical analysis in particular cases (quadratic objective function with respect to the uncertain data, etc.).

The Laboratoire Jean Kuntzmann from Université Grenoble-Alpes, Unité Mixte de Recherche (UMR 5224) of CNRS, is an applied mathematics and computer science department. It gathers about 100 researchers, attached to Université Grenoble Alpes, Institut National Polytechnique de Grenoble, CNRS, or INRIA. Their research is concerned with the theoretical and applied aspects of domains so diverse as the analysis of partial differential equations, optimization, image and signal processing, cybersecurity, statistics.

Within the Laboratoire Jean Kuntzmann, the EDP team (Equations aux Dérivées Partielles) is attached to the AMAC division (Algorithmes, Modèles, Analyse et Calcul). It is chaired by Christophe Picard, and it is composed of 14 permanent researchers, 11 graduate students, and 2 post-doctoral fellows. The keywords of its scientific activity are the theoretical and numerical analyses of partial differential equations, high performance scientific computing, aiming at applications in biology, fluid and structure mechanics.

• Trade skills/ expertise

The candidate is expected to have a solid training in applied mathematics, and notably in functional analysis, measure theory, partial differential equations, but also to have technical skills in programming: the acquaintance with a language such as Python or Julia is strongly recommended, as well as some familiarity with a Finite Element library such as FreeFem or Fenics.

In addition to this mandatory background, some knowledge of shape optimization, optimal transport, convex optimization will be particularly appreciated.

• Personal skills

• Enthusiasm
•  Autonomy
• Creativity, originality
• Appetite for risk
• Critical judgement
• Pugnacity

Previous formation, diplomas:

Masters diploma in mathematics or applied mathematics, or equivalent.