Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with applications in all areas of mathematical modelling.
Overview Many modern mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and large size (dimension). Model order reduction aims to lower the computational complexity of such problems, for example, in simulations of large-scale dynamical systems and control systems. By a reduction of the model's associated state space dimension or degrees of freedom, an approximation to the original model is computed which is commonly referred to as a reduced order model. Reduced order models are useful in settings where it is often unfeasible to perform numerical simulations using the complete full order model. This can be due to limitations in computational resources or the requirements of the simulations setting, for instance real-time simulation settings or many-query settings in which a large number of simulations needs to be performed. Examples of Real-time simulation settings include control systems in electronics and visualization of model results while examples for a many-query setting can include optimization problems and design exploration. In order to be applicable to real-world problems, often the requirements of a reduced order model are:
A small approximation error compared to the full order model. Conservation of the properties and characteristics of the full order model (E.g. stability and passivity in electronics). Computationally efficient and robust reduced order modelling techniques. It is interesting to note that in some cases (e.g. constrained lumping of polynomial differential equations) it is possible to have a null approximation error, resulting in an exact model order reduction.
Methods Contemporary model order reduction techniques can be broadly classified into 5 classes:
Proper orthogonal decomposition methods. Reduced basis methods. Balancing methods Simplified physics or operational based reduction methods. Nonlinear and manifold model reduction methods. The simplified physics approach can be described to be analogous to the traditional mathematical modelling approach, in which a less complex description of a system is constructed based on assumptions and simplifications using physical insight or otherwise derived information. However, this approach is not often the topic of discussion in the context of model order reduction as it is a general method in science, engineering, and mathematics. Proper orthogonal decomposition, reduced basis, and balancing methods fall into the category of projection-based reduction. Projection-based reduction relies on the projection of either the model equations or the solution onto a basis of reduced dimensionality compared to the original solution space. Methods that also fall into this class but are perhaps less common are:
Proper generalized decomposition Matrix interpolation Transfer function interpolation Piecewise tangential interpolation Loewner framework (Empirical) cross Gramian Krylov subspace methods Nonlinear and manifold model reduction methods derive nonlinear approximations on manifolds and so can achieve higher accuracy with the same number of degrees of freedom than traditional methods that obtain linear approximations in subspaces. Building on nonlinear approximations is essential for efficiently reducing certain problem classes such as wave problems and advection-dominated problems in computational fluid dynamics. The nature and principles underlying nonlinear model reduction methods are broad and include template-based methods, the use of neural networks and online adaptive spaces. There are also nonintrusive model reduction methods that learn reduced models from data without requiring knowledge about the governing equations and internals of the full, high-fidelity model. Nonintrusive methods learn a low-dimensional approximation space or manifold and the reduced operators that represent the reduced dynamics from data. Methods that are non-intrusive include:
Dynamic mode decomposition Operator inference Loewner framework
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