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.
We encounter simulations unconsciously in many everyday situations: the daily weather forecast, non-destructive crash tests for car approval, lightweight and material-saving plastic parts in household appliances, or investment strategies for funds and pension investors... An interdisciplinary team has set itself the task of bringing the topics of simulation, mathematical modelling, and artificial intelligence to schools. How do we recognise simulations? How are the results to be understood? And what do employees in data centres work on? How does AI work? Simulated Worlds answers these questions with many practical examples.
The goal of this project is to use deep neural networks as building blocks in a numerical method to solve the Boltzmann equation. This is a particularly challenging problem since the equation is a high-dimensional integro-differential equation, which at the same time possesses an intricate structure that a numerical method needs to preserve. Thus, artificial neural networks might be beneficial, but cannot be used out-of-the-box. We follow two main strategies to develop structure-preserving neural network-enhanced numerical methods for the Boltzmann equation. First, we target the moment approach, where a structure-preserving neural network will be employed to model the minimal entropy closure of the moment system. By enforcing convexity of the neural network, one can show, that the intrinsic structure of the moment system, such as hyperbolicity, entropy dissipation and positivity is preserved. Second, we develop a neural network approach to solve the Boltzmann equation directly at discrete particle velocity level. Here, a neural network is employed to model the difference between the full non-linear collision operator of the Boltzmann equation and the BGK model, which preserves the entropy dissipation principle. Furthermore, we will develop strategies to generate training data which fully sample the input space of the respective neural networks to ensure proper functioning models.
Waves are everywhere, and understanding their behavior leads us to understand nature. The goal of CRC 1173 »Wave Phenomena« is therefore to analytically understand, numerically simulate, and eventually manipulate wave propagation under realistic scenarios by intertwining analysis and numerics.
CAMMP stands for Computational and Mathematical Modeling Program. It is an extracurricular offer of KIT for students of different ages. We want to make the public aware of the social importance of mathematics and simulation sciences. For this purpose, students actively engage in problem solving with the help of mathematical modeling and computer use in various event formats together with teachers. In doing so, they explore real problems from everyday life, industry or research.
The Simulated Worlds project aims to provide students in Baden-Württemberg with a deeper critical understanding of the possibilities and limitations of computer simulations. The project is jointly supported by the Scientific Computing Center (SCC), the High Performance Computing Center Stuttgart (HLRS) and the University of Ulm and is already working with several schools in Baden-Württemberg.
Together with partners at Forschungszentrum Jülich and Fritz Haber Institute Berlin, our goal is to develop a novel intelligent management system for electric batteries that can make better decisions about battery charging cycles based on a detailed surrogate model ("digital twin") of the battery and artificial intelligence.
