Bayesian inversion for hybrid deterministic-stochastic kinetic solvers

• Applicant: Dr. Emil Løvbak
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 563450842

Kinetic equations are a core modelling tool across many domains of science and engineering, including fusion reactor design, radiation therapy planning and nuclear waste analysis. These equations model particle dynamics in a position-velocity phase space, whose high-dimensionality makes grid-based discretization expensive in practice. Often, one therefore uses particle-based Monte Carlo methods for their simulation. These methods have the drawback of producing simulation results with a stochastic sampling error, due to tracing a finite number of particles. The stochastic nature of this error presents challenges when performing, e.g., parameter estimation where one wishes to find the correct solver inputs to produce a simulation result that matches a measurement under given assumptions on measurement noise. Applying a Bayesian framework to such estimation problems, one assumes a prior distribution on the parameters to be identified. One then aims to compute a corresponding posterior distribution that takes into account how likely it is that the solver output for a given parameter value matches the provided measurement. In this project we consider sampling methods for evaluating this posterior, such as Markov chain Monte Carlo methods and ensemble Kalman inversion. The theory for these methods does in general not apply unchanged when using particle-based Monte Carlo solvers to evaluate the likelihood. We study how these methods perform in combination with such solvers and develop new robust and efficient variants of these methods to deal with such stochastic solvers. We develop these methods on mathematical toy problems and then extend their application to practical problems within nuclear fusion research and other relevant domains.

• DFG Programme: WBP Position