Research
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 highdimensional approximation problems.
Keywords: Numerical analysis,
Partial differential equations, Nonlinear Eigenvalue Problems,
Error certification, A posteriori error estimates, Reducedorder modelling,
Quantum chemistry, Density Functional Theory (DFT),
Molecular simulation  Force fields (interatomic potentials), Tightbinding 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 highfidelity data, typically computed with electronic structure models such as Density Functional Theory (DFT).
Approximating this energy functional using datadriven methods leads to interesting problems, arising from the highdimensionality 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 rigidbody motion.
To read more:

Geneviève Dusson, Markus Bachmayr, Gábor Csányi, Ralf Drautz, Simon Etter, Cas van der Oord, and
Christoph Ortner,
Approximation of atomic interactions
with spherical harmonics,
preprint.

Cas van der Oord, Geneviève Dusson, Gábor Csányi, and Christoph Ortner.
Regularised atomic bodyordered permutationinvariant polynomials
for the construction of interatomic potentials,
Machine Learning: Science and Technology, 1 (2020).
preprint,
article.
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.
To read more:

Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralík,
Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clusters,
to appear in Mathematics of Computations,
preprint.

Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralík,
Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified
framework,
Numerische Mathematik, 140(4), 1033–1079 (2018).
preprint,
article.

Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralík,
Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors:
conforming approximations,
SIAM Journal on Numerical Analysis, 55 (2017), pp. 2228–2254.
preprint,
article.

Geneviève Dusson, Yvon Maday,
A Posteriori Analysis of a NonLinear GrossPitaevskii type Eigenvalue Problem,
IMA Journal of Numerical Analysis (2016), drw001.
preliminary version,
article.
Postprocessing 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.
To read more:

Eric Cancès, Geneviève Dusson,
Discretization error cancellation in electronic structure calculation:
toward a quantitative study,
ESAIM: Mathematical Modelling and Numerical Analysis, 51 (2017), pp. 1617–1636.
preprint,
article.

Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralík,
A perturbationmethodbased
postprocessing for the planewave discretization of KohnSham models,
Journal of Computational Physics 307 (2016) 446–459.
preprint,
article.

Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralík,
Postprocessing of the planewave approximation
of Schrödinger equations. Part I: linear operators.
preprint.

Geneviève Dusson,
Postprocessing of the planewave approximation
of Schrödinger equations. Part II: KohnSham models.
preprint.