Dynamical low-rank approximation for computational plasma physics
Event details
| Date | 29.11.2022 |
| Hour | 16:15 › 17:15 |
| Speaker | Prof. Lukas Einkemmer, University of Innsbruck |
| Location | |
| Category | Conferences - Seminars |
| Event Language | English |
Computational Mathematics Seminar
Abstract: Solving high-dimensional kinetic equations (such as the Vlasov equation, gyrokinetic equations, or the Boltzmann equations) numerically is extremely challenging. Methods that discretize phase space suffer from the exponential growth of the number of degrees of freedom, the so-called curse of dimensionality, while Monte Carlo methods converge slowly and suffer from numerical noise. In addition, standard complexity reduction techniques (such as sparse grids) usually perform rather poorly due to the lack of smoothness for such problems.
Dynamical low-rank techniques approximate the dynamics by a set of lower-dimensional functions. For those low-rank factors, partial differential equations are derived that can then be solved numerically.
Finding a good separation between the variables is often driven by physical insight. For example, for kinetic Alfvén waves we separate the directions parallel and perpendicular to the magnetic field lines in order to obtain an efficient approximation. Due to this flexibility and their capacity to handle non-smooth solutions, dynamical low-rank approximations have been shown to work well for a range of kinetic equations. In this talk, we also discuss some recent advances in dynamical low-rank techniques that lead to structure-preserving algorithms and introduce our software framework (Ensign; see https://github.com/leinkemmer/Ensign) that facilitates the implementation of such methods on multi-core CPU and GPU based systems.
Abstract: Solving high-dimensional kinetic equations (such as the Vlasov equation, gyrokinetic equations, or the Boltzmann equations) numerically is extremely challenging. Methods that discretize phase space suffer from the exponential growth of the number of degrees of freedom, the so-called curse of dimensionality, while Monte Carlo methods converge slowly and suffer from numerical noise. In addition, standard complexity reduction techniques (such as sparse grids) usually perform rather poorly due to the lack of smoothness for such problems.
Dynamical low-rank techniques approximate the dynamics by a set of lower-dimensional functions. For those low-rank factors, partial differential equations are derived that can then be solved numerically.
Finding a good separation between the variables is often driven by physical insight. For example, for kinetic Alfvén waves we separate the directions parallel and perpendicular to the magnetic field lines in order to obtain an efficient approximation. Due to this flexibility and their capacity to handle non-smooth solutions, dynamical low-rank approximations have been shown to work well for a range of kinetic equations. In this talk, we also discuss some recent advances in dynamical low-rank techniques that lead to structure-preserving algorithms and introduce our software framework (Ensign; see https://github.com/leinkemmer/Ensign) that facilitates the implementation of such methods on multi-core CPU and GPU based systems.
Practical information
- General public
- Free
Organizer
- Prof. Daniel Kressner
Contact
- Prof. Daniel Kressner