Parameter calibration of simulation codes is an essential step in the modeling and simulation of complex physical systems. These codes, often resulting from long multiphysics computation chains, include numerous parameters whose values can only be partially determined through measurements. Calibration therefore consists in adjusting these parameters to achieve the best possible consistency between model predictions and available experimental data.

In this context, accounting for parameter uncertainty is crucial. Experimental data are often noisy or limited, and the models themselves involve approximations. The Bayesian framework provides a rigorous approach to handle these uncertainties by representing parameters as random variables. The objective then becomes to estimate their a posteriori distribution, given the observations and the prior knowledge.

When the model cannot be linearized, when the likelihood function is complex, or when the a priori distributions are non-Gaussian, direct sampling methods become inapplicable. In such cases, Markov Chain Monte Carlo (MCMC) methods are recognized as reference tools. They enable efficient exploration of the parameter space and allow obtaining representative samples from the a posteriori distribution.

However, there is today a wide variety of MCMC algorithms (Metropolis-Hastings, Gibbs Sampling, Hamiltonian Monte Carlo, Adaptive MCMC, Differential Evolution MCMC, among others), whose performances can vary significantly depending on the nature of the problem (dimension, parameter correlations, likelihood topology, etc.). It is therefore necessary to compare these different approaches to identify their strengths and limitations.

The proposed internship fits into this perspective. Its objective is to perform a thorough comparison of the main MCMC methods applied to Bayesian parameter calibration. The main tasks will be :

- Conduct a state of the art of current MCMC methods in the scientific literature and
existing reference tools/libraries ;
- Select a representative set of MCMC algorithms covering various application cases
(complex distributions, strong correlations, etc.) ;
- Evaluate these methods on benchmark datasets, analyzing their performance in terms
of convergence, computational cost, and robustness ;
- Implement the best-performing methods in the Uranie platform, ensuring proper documentation and integration into the platform’s testing environment ;
- Propose possible improvements or new MCMC algorithm variants, as part of a methodological innovation approach.

The internship will be conducted at the Service of Software Engineering and for Simulation (SGLS - Service de Génie Logiciel pour la Simulation) within the Energy Division (DES - Direction des Énergies). The service develops and qualifies the simulation tools for enhancing the precision and reliability of the models used to design and analyze the safety of the French nuclear power systems. 

Master’s level (M2) student enrolled in an engineering school or master’s program with a specialization in statistics. The candidate should have :

- Strong knowledge in probability and statistics, ideally complemented by coursework in Bayesian statistics and MCMC methods ;
- Software development skills, particularly in C++ and Python (main languages for the internship). Good knowledge or familiarity with R, or ROOT would be an asset ;
- Scientific rigor, ability to work independently, and a strong interest in learning (state of the art, method comparison, in-depth understanding of algorithms).