Model order reduction (MOR) has become an established tool for the fast execution of computational tasks in simulation, control and optimization of various problems in the sciences and engineering. Modelling complex real-world problems usually leads to large sets of coupled differential and possibly algebraic equations. Often their sheer number makes them difficult to store on standard hardware and demanding to treat in reasonable time frames. MOR can drastically reduce both storage and time demands for many problems. As long as the differential equations are of first order and linear as well as time invariant, the MOR problem can be considered solved today. When certain design or material parameters need to be preserved, a whole class of methods from both classic systems’ theory-based approaches as well as reduced-basis-driven work exist and are constantly improved.
Today, the challenges are given when the system of differential (algebraic) equations is nonlinear, parameters vary over time or even the entire system is changing in the course of operation, e.g. due to moving loads. While the above approaches using reduced basis methods or driven by systems theoretic motivations are adapted to these fields, also a new class of methods is becoming increasingly important. The data-driven MOR approaches, which are closely related to system identification, exploit measured system data to derive the reduced order model, rather than setting up huge models that are reduced afterwards, or relying on simulation snapshots.
In this special issue we present recent work from all the three fields:
N. Lang, J. Saak and T. Stykel present some implicit numerical integrators for differential Lyapunov equations with large and sparse coefficient matrices and the application of these methods in the balanced truncation framework for linear time-varying systems.
B. Kramer and S. Gugercin present a multi-input-multi-output extension of the data-driven so-called Eigensystem Realization Algorithm, suitable for identification and reduced-order modelling of systems with even large numbers of inputs and outputs.
The contribution by M. Geuß and B. Lohmann deals with reducing parametric models by interpolating the system matrices of several strictly dissipative reduced models, thereby preserving stability.
M. Redeker and B. Haasdonk consider a two-scale model for precipitation in porous media. In this setting, only one of the equations on the microscale is reduced using proper orthogonal decomposition. The authors subsequently apply the Empirical Interpolation Method to deal with the parameter dependence and multiple basis sets due to the convective nature of the problem.
The paper by E. Bader, Z. Zhang, and K. Veroy considers variational inequalities such as the obstacle problem. The authors first reformulate the problem using a barrier method resulting in a nonlinear partial differential equation. They subsequently propose efficiently evaluable reduced basis approximations and associated error estimators which make use of the Empirical Interpolation Method.
L. Iapichino, S. Volkwein, and A. Wesche present reduced basis approximations and associated a posteriori error estimators for a nonlinear parabolic partial differential equation arising in the modelling of lithium-ion batteries. They present numerical results for the model reduction as well as a parameter estimation problem, thus showing the efficiency of the proposed approach.
The work by J. Fehr, P. Holzwarth and P. Eberhard presents an approach to save simulation time in explicit crash simulations of vehicles by splitting the model of a kart frame into a linear and a nonlinear part and by applying advanced interface reduction techniques.