arXiv subjects

Guillaume Bonnet

Publications and source records attributed to Guillaume Bonnet.

4 recordsLinked to original sources

arXiv2026

Virtual element methods for a class of fully nonlinear elliptic PDEs

We study virtual element discretizations of a well-known variational formulation in $H^2$ of Hamilton-Jacobi-Bellman and Isaacs equations with Cordes coefficients. We show that the use of $H^2$ conforming virtual element spaces leads to a relatively simple analysis, bypassing the need for discrete Miranda-Talenti estimates that typically arises when using nonconforming schemes. We investigate how the polynomial degree of projection operators, especially for lower order terms, affects both the error analysis and robustness of the proposed schemes. We also show the possibility of imposing weakly the Dirichlet boundary condition, which simplifies the implementation of the method in some virtual element codes. Our results are complemented by numerical experiments in which we compare the convergence of different variants of the scheme for some test problems.

math.NA↗

arXiv2026

Quantitative Stability and Numerical Resolution of the Moment Measure Problem

The moment measure problem consists in finding a convex function $\psi$ whose moment measure, i.e., the pushforward by $\nabla \psi$ of the measure with density $e^{-\psi(\,\cdot\,)}$, is prescribed. It is highly non-linear and less understood than the related optimal transport problem. We establish a quantitative stability estimate for this problem. This estimate validates, as well as leads us to introduce, an approach to the numerical resolution of the moment measure problem inspired by semi-discrete optimal transport, consisting in approximating the prescribed measure by a finitely supported one. We describe a Newton method for solving the discrete problem thus obtained, and perform numerical experiments, studying the experimental rates of convergence of the approximation beyond the predictions of the stability estimate.

math.FA↗

arXiv2024

Conforming virtual element method for nondivergence form linear elliptic equations with Cordes coefficients

We propose and analyze an $H^2$-conforming Virtual Element Method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension $d \ge 2$. The analysis relies on the continuous Miranda-Talenti estimate for convex domains $Ω$ and is rather elementary. We prove stability and error estimates in $H^2(Ω)$, including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in $H^2(Ω)$.

math.NA↗

arXiv2022

Morphology of dark matter haloes beyond triaxiality

The morphology of haloes inform about both cosmological and galaxy formation models. We use the Minkowski Functionals (MFs) to characterize the actual morphology of haloes, only partially captured by smooth density profile, going beyond the spherical or ellipsoidal symmetry. We employ semi-analytical haloes with NFW and $αβγ$-profile and spherical or ellipsoidal shape to obtain a clear interpretation of MFs as function of inner and outer slope, concentration and sphericity parameters. We use the same models to mimic the density profile of $N$-body haloes, showing that their MFs clearly differ as sensitive to internal substructures. This highlights the benefit of MFs at the halo scales as promising statistics to improve the spatial modeling of dark matter, crucial for future lensing, Sunyaev-Zel'dovich, and X-ray mass maps as well as dark matter detection based on high-accuracy data.

astro-ph.CO↗