Model Reduction

Standard approximations of parametric complex physical systems often result in high-dimensional models with long computing times. If the parametric problem needs to be solved in real-time or for multiple parameter values, the aim is to reduce the dimensionality in order to derive a low-dimensional reduced model. For such a reduced model, the time of computing the solution is significantly lowered for each parameter. In order to obtain a certified reduced model, the desire is that the error w.r.t. the high-dimensional space is controlled by an error bound.

Model Reduction for Variational Inequatilies

In this work, variational inequalities with different trial and test spaces and possibly noncoercive bilinear form are considered. Well-posedness has been shown under general conditions that are e.g., valid for the space-time formulation of parabolic variational inequalities. Fine discretizations of such probelms resolve in large scale problems and thus long computing times. To reduce the size of this problems, we use the reduced basis method. When combining the reduced basis method with the space-time formulation, a residual-based error estimator could be derived.

Symplectic Model Reduction for Hamiltonian Systems

Kinetic turbulent transport in plasma physics is of great importance nowadays for understanding and optimizing experiments for fusion devices such as stellarators or tokamaks - and even more for designing future fusion reactors. Existing codes usually need to run on high performance clusters, hence infeasible for a multi-query parametric setting. Therefore, we need to build a reduced parametric model in order to compute turbulent transport more efficiently. As the underlying kinetic equations can be rewritten in terms of a Hamiltonian system, we aim to preserve this structure within our reduced model. Due to the transport structure of the kinetic equations, the decay of the Kolmogorov N-width is slow. In order to overcome the linear subspace assumption of the Kolmogorov N-width, we extend existing work by considering symplectic model reduction on manifolds.

paper 2020
S. Glas, Anthony T. Patera and K. Urban, A Reduced Basis Method for the Wave Equation
International Journal of Computational Fluid Dynamics, 34(2):139–146.
proceedings 2017
S. Glas, A. Mayerhofer and K. Urban, Two Ways to Treat Time in Reduced Basis Methods.
In: Benner P., Ohlberger M., Patera A., Rozza G., Urban K. (eds) Model Reduction of Parametrized Systems. MS&A (Modeling, Simulation and Applications), vol 17. Springer, Cham