Post-doctoral Research Visit F/m Cwi-inria Postdoctoral Fellowship On Structure-preserving Model Reduction of Nonlinear Conservation Laws

Le descriptif de l’offre ci-dessous est en Anglais_

**Type de contrat**: CDD

**Niveau de diplôme exigé**: Thèse ou équivalent

**Fonction**: Post-Doctorant

**Niveau d'expérience souhaité**: De 3 à 5 ans

**A propos du centre ou de la direction fonctionnelle**:
The Inria center at the University of Bordeaux is one of the nine Inria centers in France and has about twenty research teams.. The Inria centre is a major and recognized player in the field of digital sciences. It is at the heart of a rich R&D and innovation ecosystem: highly innovative SMEs, large industrial groups, competitiveness clusters, research and higher education players, laboratories of excellence, technological research institute...

**Contexte et atouts du poste**:
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).

**Mission confiée**:
**Principales activités**:
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.

**Compétences**:
**Avantages**:

- Subsidized meals
- Partial reimbursement of public transport costs
- Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
- Possibility of teleworking and flexible organization of working hours
- Professional equipment available (videoconferencing, loan of computer equipment, etc.)
- Social, cultural and sports