Lukas Renelt: Thursday, 24th July 2025 – 10h30 to 12h00
Abstract:
The numerical solution of partial differential equations (PDEs) is one of the main research fields in computational mathematics where a vast variety of numerical methods have been developed. Applications include the simulation of reactive transport, groundwater flow, electromagnetism or even the computation of quantum states. These equations often additionally depend on physical parameters such as coefficients or boundary data which can significantly influence the solutions behaviour. This poses an additional challenge if solutions for many different parameter values are required such as in parameter studies, optimization tasks, inverse problems or uncertainty quantification. While methods such as the finite element method, finite volumes or discontinuous Galerkin approaches work well for given fixed parameters, they are prohibitively costly if solutions for thousands of different parameters are needed.
To address this challenge, model order reduction methods have been developed which aim at approximating the highly complex set of all possible solutions jointly by a small (linear) subspace. In this talk, we will give a general introduction to the methodology with particular focus on the Reduced Basis (RB) approach highlighting both the abstract analysis and also showing concrete realizations.
In the second part of the talk, we will present recent results when applying the method to parametrized Friedrichs’ systems – a large abstract class of linear PDE problems including for example convection-diffusion-reaction, linear transport, linear elasticity or the time-harmonic Maxwell equations. From a theoretical point of view, these problems are particularly interesting as their Friedrichs’ formulation involves parameter-dependent function spaces – a setting which has not been explored by the model order reduction community thus far. We present a novel theoretical framework and highlight the connections to the established theory with implications beyond Friedrichs’ systems. Additionally, a normal-equation-based discretization is introduced and used to numerically verify our theoretical findings in an application to the reactive transport of a pollutant through a catalytic filter.
