My research activity focuses on the development and the mathematical analysis of numerical methods for the simulation of molecular systems and materials. I am particularly interested in electronic structure models in quantum chemistry which exhibit eigenvalue problems, as well as the study of potential energy surfaces, which often lead to high-dimensional approximation problems.
Keywords: Numerical analysis, Partial differential equations, Nonlinear Eigenvalue Problems, Error certification, A posteriori error estimates, Reduced-order modelling, Quantum chemistry, Density Functional Theory (DFT), Molecular simulation - Force fields (interatomic potentials), Tight-binding models.

Approximation of potential energy surfaces

Simulating the dynamics of a molecular system requires to compute the energy of the system and the forces on the nuclei at each time step of the simulation. For this kind of applications, precise models such as electronic structure models are too expensive to be used for a large number of atomic configurations. To reduce the computational cost of the calculations, the energy functional is modeled as a functions of the positions of the particles, and fitted to some high-fidelity data, typically computed with electronic structure models such as Density Functional Theory (DFT). Approximating this energy functional using data-driven methods leads to interesting problems, arising from the high-dimensionality of the problem, as well as symmetry issues coming from the invariance of the energy with respect to the permutation of identical particles as well as rigid-body motion.

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Numerical analysis for electronic structure calculation

To describe matter at the molecular scale, there exists a large range of models offering different levels of description. Some of these models are partial differential equations, which require several approximations to compute a numerical (approximate) solution, such as the choice of a discretization or the use of an iterative algorithm stopped when a certain criterion is satisfied. In this context, one of my interests is to quantify these different errors for two reasons. First, guaranteed estimates on all components of the error make it possible to certify the numerical result, by supplementing the value of the observable calculated in the numerical simulation with guaranteed error bars. Second, knowing these errors allows to adapt the parameters of the simulation (approximate model, discretization parameters, criteria for stopping the algorithms, etc.) optimally, and to minimize the computational cost required to achieve the desired accuracy.

So far, I have mostly studied linear and nonlinear eigenvalue problems arising from electronic structure models. In collaboration with Eric Cancès, Yvon Maday, Benjamin Stamm and Martin Vohralik, I provided guaranteed error bounds based on an a posteriori analysis for several eigenvalue problems.

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Post-processing and study of quantities of interest

At some point, I became interested in an electronic structure calculation model that is widely used in practice: the Kohn–Sham model. For this model, I proposed a postprocessing method which improves the solutions of the discrete problem at low computational cost. I am also interested in the approximation of quantities of interest. In this direction, I observed an error cancellation phenomenon for energy differences, which I explained for a toy model.

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