Sébastien Martin

Francisco Castro, Jian Gao, Sébastien Martin

When working with generative artificial intelligence (AI), users may see productivity gains, but the AI-generated content may not match their preferences exactly. To study this effect, we introduce a Bayesian framework in which heterogeneous users choose how much information to share with the AI, facing a trade-off between output fidelity and communication cost. We show that the interplay between these individual-level decisions and AI training may lead to societal challenges. Outputs may become more homogenized, especially when the AI is trained on AI-generated content, potentially triggering a homogenization death spiral. And any AI bias may propagate to become societal bias. A solution to the homogenization and bias issues is to reduce human-AI interaction frictions and enable users to flexibly share information, leading to personalized outputs without sacrificing productivity.

Silvia Bertoluzza, Astrid Decoene, Loïc Lacouture et al.

The solutions of elliptic problems with a Dirac measure in right-hand side are not H1 and therefore the convergence of the finite element solutions is suboptimal. Graded meshes are standard remedy to recover quasi-optimality, namely optimality up to a log-factor, for low order finite elements in L2-norm. Optimal (or quasi-optimal for the lowest order case) convergence has been shown in L2-seminorm, where the L2-seminorm is defined as the L2-norm on a subdomain which excludes the singularity. Here we show a quasi-optimal convergence for the Hs-seminorm, s \textgreater{} 0, and an optimal convergence in H1-seminorm for the lowest order case, on a family of quasi- uniform meshes in dimension 2. This question is motivated by the use of the Dirac measure as a reduced model in physical problems, and a high accuracy at the singularity of the finite element method is not required. Our results are obtained using local Nitsche and Schatz-type error estimates, a weak version of Aubin-Nitsche duality lemma and a discrete inf-sup condition. These theoretical results are confirmed by numerical illustrations.