# Model Order 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 order model (ROM). For such a ROM, the time of computing the solution is significantly lowered for each parameter. In order to obtain a certified ROM, the desire is that the error w.r.t. the high-dimensional space is controlled by an error bound.

Among other things I have previous work on model order reduction (MOR) for variational inequalities, MOR of time-dependent problems utilizing a space-time variational formulation and reduced basis methods for the wave equation. Currently, my research focus is on structure-preserving MOR as well as on MOR on manifolds, which I will explain briefly in the following.

## Structure-preserving Model Order Reduction

In case that the high-dimensional state-space models obtain a specific structure, e.g., Hamiltonian systems, Poisson systems or port-Hamiltonian systems, the ROMs should share the same physical structure and/or conservations laws. Otherwise, the ROM could be unphysical and yield poor approximation quality. To this end, we develop MOR methods to ensure that the ROMs of such systems are again of the same form as the high-dimensional models.

## Model Order Reduction on Manifolds

The quality of linear-subspace ROMs can be quantified by the Kolmogorov-N-width, which is the best achievable error for linear approximations of fixed dimension N. For problems with slowly decaying Kolmogorov-N-widths such as, e.g., transport equations, wave equations, low-dimensional linear-subspace ROMs might yield inaccurate results. To overcome these limitations, we work on extending the concept of linear-subspace ROMs and consider ROMs on nonlinear manifolds.

## Selected Related Output

**2022**

##### P. Buchfink, S. Glas and B. Haasdonk, Optimal Bases for Symplectic Model Order Reduction of Canonizable Linear Hamiltonian Systems

###### MATHMOD 2022 proceedings.

**2021**

##### P. Buchfink, S. Glas and B. Haasdonk, Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds.

###### accepted in SIAM Journal on Scientific Computing, 2022.

**2020**