The seventy-third UQSay seminar on UQ, DACE and related topics will take place **online** on Thursday afternoon, May 16, 2024.

#### 2–3 PM — Gaël Poette (CEA DAM, CESTA - ENSEIRB-MATMECA) — [slides]

#### Building and solving reduced models for the uncertain linear Boltzmann equation (sometimes intrusiveness, a.k.a. physics-informedness, is worth it)

In this talk, I will present one way to build reduced models which aim at being able to efficiently propagate uncertainties through the linear Boltzmann equation. Only one run of a Monte-Carlo code will be enough to solve the system and accurately estimate statistical observables (mean, variance, tolerance intervals, sensitivity indices) on your physical observables of interest. From the Machine Learning community point of view, the presented reduced model can be cast in the category of 'Physics Informed' ones in the sense that the structure of the partial differential equation is intensively used in order to build the reduced model. In this context, we will show that such a physics-informed strategy (from the literature from another generation, it is also called an 'intrusive' strategy) combined with the relevant numerical scheme can be very competitive with respect to the best of the non-intrusive ones.

From the transport community point of view, the reduced model I will present can be understood, in a sense, as an attempt to make the best of two worlds: Pn reduced models and Monte-Carlo resolution schemes.The Pn model, well-known in neutronics and photonics, is built with respect to the uncertain variables (in the literature relative to uncertainty quantification, it is commonly called Polynomial Chaos) rather than with respect to the angular one. And the resulting reduced model is solved with a Monte-Carlo scheme (due to the high dimensional context we are in). The reasons for such a combination will be detailed and numerical results will be presented (for the instationary linear Boltzmann but also for eigenvalue computations). Performance comparisons will be presented on neutronics (keff computations) and photonics applications. If time allows it, discussions on the asymptotical errors (noise) will be tackled.

Some parts of this talk come from a joint work with Emeric Brun (CEA, DES).

References:

- G. Poëtte (2019) A gPC-intrusive Monte-Carlo scheme for the resolution of the uncertain linear Boltzmann equation. J. Comp. Phys. DOI:10.1016/j.jcp.2019.01.052,
- G. Poëtte (2020) Spectral convergence of the generalized Polynomial Chaos reduced model obtained from the uncertain linear Boltzmann equation. Mathematics and Computers in Simulation. 10.1016/j.matcom.2020.04.009.
- G. Poëtte (2021) Efficient uncertainty propagation for photonics: Combining implicit semi-analog monte carlo (ISMC) and monte carlo generalised polynomial chaos (MC-gPC). J. Comp. Phys. DOI:10.1016/j.jcp.2021.110807,
- G. Poëtte (2022) Numerical analysis of the Monte-Carlo noise for the resolution of the deterministic and uncertain linear Boltzmann equation (comparison of non-intrusive gPC and MC-gPC). J. of Comp. and Theor. Transport. DOI:10.1080/23324309.2022.2063900.
- G. Poëtte, E. Brun (2022) Efficient uncertain keff computations with the Monte Carlo resolution of generalised Polynomial Chaos based reduced models. J. Comp. Phys. DOI:10.1016/j.jcp.2022.111007.

**Organizing committee**: Pierre Barbillon (MIA-Paris), Julien Bect (L2S), Nicolas Bousquet (EDF R&D), Vincent Chabridon (EDF R&D), Amélie Fau (LMPS), Filippo Gatti (LMPS), Clément Gauchy (CEA), Bertrand Iooss (EDF R&D), Alexandre Janon (LMO), Sidonie Lefebvre (ONERA), Didier Lucor (LISN), Sébastien Petit (LNE), Emmanuel Vazquez (L2S), Xujia Zhu (L2S).

**Coordinators**: Julien Bect (L2S) & Sidonie Lefebvre (ONERA)

**Practical details**: the seminar will be held online using Microsoft Teams.

If you want to attend this seminar (or any of the forthcoming online UQSay seminars), and **if you do not already have access to the UQSay group on Teams**, simply send an email and you will be invited. Please specify which email address the invitation must be sent to (this has to be the address associated with your Teams account).

You will find the link to the seminar on the "General" UQSay channel on Teams, approximately 15 minutes before the beginning.

The technical side of things: you can use Teams either directly from your web browser or using the "fat client", which is available for most platforms (Windows, Linux, Mac, Android & iOS). We strongly recommend the latter option whenever possible. Please give it a try before the seminar to anticipate potential problems.