Research topics and projects 📖

My research fits in the domain of numerical analysis. The work done during my PhD and post-doctorates concerned the development of numerical methods for the simulation of multicomponent flows [A1-A7,T1]. Since I arrived in Centrale Nantes, my research work has mainly focused on the development of approximation and model order reduction methods for partial differential equations [A8-A14,P1-P3,T2].

Keyworks: Numerical Analysis, PDEs, model order reduction, probabilistic and deterministic methods

Summary of contributions

Partial Differential Equations (PDEs) are commonly considered to model complex physical systems. In some situations, such equations may dependent on some parameters, possibly unknown, through involved physical quantities (e.g., diffusivity, viscosity, source terms) or the geometry of the physical domain. In that setting, two kinds of problems may be considered, namely, forward and inverse problems.  

Forward problems appear for example in optimization, uncertainty quantification to compute some quantity of interest of the solution or either in parametric studies. In such contexts, it is often required to evaluate the solution of the PDE for many instances of the parameter. As the exact solution is not available, these equations are addressed using classical discretization techniques such as, finite difference, finite element, finite volume etc. Generally, such approximations lead to costly numerical models, that cannot be addressed rapidly unless the overall complexity has been appropriately reduced. Inverse problems occur when the parameter value is unknown. Instead, the only information available is a vector of linear measurements of the solution of the original problem. For such problems, the goal is to recover the unknown solution from these measurements (state estimation) or/and the underlying parameter value (parameter estimation).

Low-rank approximation methods for model order reduction of parameter-dependent problems.

Contributions [A5,A8,A9] are devoted to the resolution of parameter-dependent problems motivated by uncertainty quantification using low-rank approximation methods following two directions : low-rank approximation in tensor format and projection based low-rank approximation. More particularly tensor approximation methods based on ideal minimal residual formulations for the solution of high-dimensional problems [A5,A8] have been first proposed. Then, goal oriented projection based model order reduction methods for the estimation of vector-valued variables of interest [A9] have been developped. These contributions were done in collaboration together with A. Nouy and O. Zahm.

Low-rank approximation methods for the approximation of time and parameter-dependent problems.

Low-rank approximation methods discussed can be extended for the approximation of time and parameter-dependent problems. Here, dynamical low-rank approximation methods have been considered following two subsequent directions. First, dynamical reduced basis method for parameter-dependent dynamical systems that can be interpreted as a dynamical low-rank approximation approach with a subspace point of view is presented [A10]. As the proposed method is shown to perform better than usual reduced basis for transport (dominated) problems, possible extension to parameter-dependent hyperbolic conservation laws is discussed. In this direction a reconstruction method in finite volume setting has been devel- oped and validated for linear parameter-dependent transport problems [P1]. Secondly, we consider a dynamical low-rank approximation method that works directly in the set of fixed rank matrices. Based on a suitable geometric parametrization of this set [A12], a new splitting integration scheme for the approximation of the solution of matrix dynamical systems [A13], arising from discretization of parameter-dependent dynamical systems, has been proposed. 

The contributions [A10,A12,A13] are the result of collaborations with A. Falco and A. Nouy on dynamical low-rank approximation methods for the solution of parameter dependent dynamical systems and T. Heuzé for extension to hyperbolic parameter dependent conservation laws [P1].

The work concerning dynamical RB methods has been partially supported by the GdR MOMAS through the project REDYN (2015). The work concerning application to hyperbolic conservation laws has been founded through the project PEPS : DROME by the Cellule Energie du CNRS (2019), co-driven with T. Heuzé.

Probabilistic approximation methods for PDEs.

We consider the problem of constructing an approximation of the solution of a partial differential equation (possibly parameter-dependent) using approaches that rely on sampled estimates of the function to approximate. To that goal, probabilistic approaches are considered. The main corner stone of the proposed contributions is the Feynman-Kac representation formula of the solution of a partial differential equation. Using this key ingredient, a probabilistic sparse polynomial interpolation method has been proposed to deal with high dimensional problem [A11]. Then reduced basis method using pointwise estimates is discussed [P2] for solving parameter- dependent partial differential equations. Especially, it relies on a greedy procedure with a probabilistic error estimate gathered with a probably approximately correct bandit algorithm proposed in [A14]. These contributions [A11,A14,P2] where done within the PhD of A. Macherey supervised together with A. Nouy and C. Prieur.

Current projects

Neural Galerkin shemes for time-dependent problems (inverse and forward problems) During CEMRACS 2023, together with O. Mula, and J. Aghili), we supervised a project on “Data assimilation methods with Neural Galerkin schemes” together with PhD students (J. Atokple, G. Garnier and N. Tognon) [P3]. This project allowed preliminary discussions around developing data assimilation methods for parameter-dependent evolution equation using Neural Galerkin scheme.  

For the next steps, we are interested by a more exploratory aspects concerning geometric reinterpretation of Neural Galerkin scheme (or more general NN architectures). If so, a subsequent and interesting direction to explore is to derive integration schemes for NN weights with respect to this geometry that is suitable for data assimilation.

Randomized ROM for time-dependent problems

Recently probabilistic variants of MOR methods using randomized numerical linear algebra widely used in data science have recently emerged for improving computational performance of classical MOR methods. Also, these randomized methods can be used to construct approximation spaces that are (almost) optimal locally over time slabs. This is done by solving in parallel evolution problems with independent random initial times and initial conditions.

Withing a PhD project together with A. Nouy and K. Smetana, we propose to develop new efficient MOR methods for parameter-dependent dynamical systems or evolution equations, by combining DLRA and randomized (parallel in time) methods.