Special Issue on “Model Order Reduction of Parameterized Problems”
The modelling and discretization of complex physical, chemical or technical systems leads to high-dimensional simulation models, making analysis, design and control difficult. This is even more dramatic in the case of parameterized problems. Here, a parameterized problem is understood as a partial differential equation (PDE) or ordinary differential equation (ODE) system that is depending on a finite-dimensional parameter vector, which characterizes a single system configuration. Such parameters can, for instance, be geometry, material or control parameters. Simulation scenarios frequently require repeated simulations under parameter variation. Examples comprise sensitivity analysis, optimization or statistical analysis. Other simulation requirements can be time restrictions or even the demand for real-time simulations. Methods of parametric model order reduction (MOR) facilitate the handling of such systems for multiple simulation requests or real-time requirements. These reduction techniques are in the focus of this special issue.
At least two large fields can be identified and are represented by growing largely disjoint scientific communities: MOR of state-space systems and reduced basis (RB) methods for parameterized PDEs.
Well-known approaches from the first field are traditional MOR techniques such as proper orthogonal decomposition (POD), balanced truncation or moment matching. These methods predominantly consider systems on a large- or infinite timescale, and the reduction process is frequently adopted to obtain certain behaviour in frequency domain.
The second field, RB methods, deals with low-dimensional approximation and an a posteriori error estimation for parameterized stationary and instationary, linear and non-linear PDEs. For instationary problems, the systems are mostly considered on finite time intervals and the approximating spaces are constructed by solution snapshots, to obtain good approximation in the finite time-domain. A predominant feature of the RB methods is rigorous and fast a posteriori error estimation and computational efficiency by full offline/online decomposition. In this special issue, we present some of the latest articles from these fields.
The first article by Ulrike Baur et al. deals with the reduction of parametric models occurring in Micro Electromechanical Systems (MEMS). A novel parameterization to numerically construct highly accurate parametric ODE systems from a small number of systems with different parameter settings is presented and applied to different problems, using two reduction techniques.
In the second contribution, Rudy Eid et al. present a stability-preserving MOR scheme for parametric models. It is based on achieving contractivity of the model and uses an interpolation of the system matrices in state space. Specially simple schemes result for systems in port-Hamiltonian and second-order form.
The third contribution by Saifon Chaturantabut and Danny C. Sorensen treats the effective reduction of general non-linear systems as exemplified by viscous flows in porous media. The main contribution is the so-called discrete empirical interpolation method for sparse approximation of the system's non-linearity. This can then be subject to arbitrary projection-based reduction methods such as POD.
The fourth article by Timo Tonn et al. considers reduction of linear-quadratic optimal control problems for the stationary Helmholtz equation. The RB and POD approaches are considered, a posteriori error estimators presented and the methods are experimentally compared.
The fifth article by Fabrizio Gelsomino and Gianluigi Rozza aims at model reduction of parameterized heat transfer problems. The methods in focus are the RB method and, for comparison, the POD. The article shows that RB methods can be beneficially applied to higher space dimensions, in particular three-dimensional instationary problems.
The sixth article by Jens Eftang et al. considers the effective basis generation for RB methods of parameterized parabolic PDEs. The method is based on an efficient consecutive bisection of the parametric domain and a construction of reduced models for the sub-domains. Rigorous a priori convergence analysis is performed.
The seventh contribution by Bernard Haasdonk et al. also addresses the goal of basis generation for parameterized problems. The two main approaches are a method for adaptive training-set extension and for adaptive parameter-domain partition. The methods can also be combined and are applicable to both parameterized state–space systems and RB methods.
The eighth article by David Knezevic considers an important example of a “many-query” scenario given by a multiscale model. The considered problem is a liquid crystal flow modelled by a Stokes Fokker–Planck system, where a micro-model is strongly coupled to a macro-model. The contribution presents a RB method and gives rigorous error analysis and experimental results.
By this special issue, we hope to stimulate further research in the area of MOR of parameterized systems. In particular, we expect that common problems and aspects in the different fields of parameterized state–space systems and parameterized PDEs will lead to fruitful exchange between the research communities. The development and verification of new methods of parametric order reduction and their algorithmic tools represent a theoretical challenge and are of great importance for many practical applications.
Guest Editors
Bernard Haasdonk
Institute of Applied Analysis and Numerical Simulation,
University of Stuttgart,
Pfaffenwaldring 57,
D-70569 Stuttgart,
Germany
Email: [email protected]
Boris Lohmann
Institute of Automatic Control,
Technische Universität München,
Boltzmannstr. 15,
D-85748 Garching/Munich,
Germany
Email: [email protected]