Every year Inria International Relations Department offers a few postdoctoral positions in order to support Inria’s international collaborations. This postdoctoral offer will be in collaboration with the Inria-CWI (Centrum Wiskunde & Informatica) center in the Netherlands. The CWI and Inria have a long history of cooperation and scientific dialogue, and the two institutes have strengthened their collaboration and formalized their partnership by signing a new strategic agreement in 2023.

The postdoctoral contract will have a duration of between 12 to 24 months. The expected start date is November 1st, 2025 and not later than January, 1st 2026. The postdoctoral fellow will be recruited in the Inria Center of Bordeaux, France, but it is recommended to split time between France and the Netherlands (please note that the postdoctoral fellow has to start their contract being in France and visits have to follow Inria rules for missions).

Candidates for postdoctoral positions are recruited after the end of their Ph.D. or after a first post-doctoral period: for the candidates who obtained their PhD in the Northern hemisphere, the date of the Ph.D. defense shall be later than September 1, 2022; in the Southern hemisphere, later than April 1, 2022. In order to encourage mobility, the postdoctoral position must take place in a scientific environment that is truly different from the one of the Ph.D. (and, if applicable, from the position held since the Ph.D.); particular attention is thus paid to French or international candidates who obtained their doctorate abroad.

The postdoctoral project fits in the ongoing Inria-CWI Associate Team SPADES https://team.inria.fr/memphis/spades-associate-team/ . The project SPADES (Structure-Preserving Approximations of Dynamical systems in Engineering and Science) focuses on the development and the analysis of structure-preserving model reduction techniques for conservation laws. Long research stays at CWI Amsterdam will be encouraged.

The postdoctoral project will focus on the development of space-time structure-preserving model reduction of parametric compressible flows, with emphasis on the compressible Euler and the Navier-Stokes equations.  

Reduced order models introduce a further layer of approximation compared to the corresponding high-fidelity models, being designed to enable fast and efficient simulations while maintaining an acceptable level of accuracy. This is particularly important in parametric problems, when simulations need to be repeated many times under different input parameters. To allow for further reduction, we consider space-time techniques [1,2] as they enable model reduction of temporal degrees of freedom as well, as opposed to more traditional time-marching high-fidelity methods. Thus, space-time techniques have the potential to achieve much more significant speedups for parametric problems.

However, the development of reduced models with a low computational cost, which at the same time retain the physical properties and structures of high-fidelity models, remains a major challenge. In this project, we shall focus on the development of energy- and entropy-stable numerical methods, to ensure nonlinear stability and accuracy of the numerical scheme, as well as consistency with the physics of the problem [3]. Such a property is also compelling in the case of discontinuities as it allows the method to capture the correct weak solution.

We also plan to address the development of effective deterministic and/or stochastic closure terms to model the effect of the truncated modes on the dominant dynamics (truncation error).  Since we  use  standard (time-marching) high-fidelity models for snapshot generation, while using fully-implicit space-time methods for the construction of the reduced-order model, the closure model should also account for the discrepancy between the high-fidelity model and the reduced-order model (model error).

A tentative program of the project is as follows.

• Review of entropy- and energy-stable high-order methods and projection-based model reduction techniques for conservation laws.
• Design and implementation of space-time model reduction techniques for subsonic inviscid and viscous flows. We shall focus on the development of provably semi- and fully-discrete entropy- and energy-stable reduced models. We shall develop a space-time method for both high-fidelity and reduced-order computations, to facilitate the assessment of the model reduction procedure.
• Extension to shock-dominated (transonic, supersonic, all-Mach) flows. Such an extension requires to introduce limiting strategies to prevent oscillatory behaviors; it also probably requires the use of nonlinear data compression techniques.