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Model Order Reduction

Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations.

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 infeasible 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.^[1] Real-time simulation settings include, for example, controllers in electronics and visualization of model results while examples for a many-query setting can include optimisation problems and design exploration. In order to be applicable to real-world problems, often the requirements of reduced order models are: ^[2]^[3]

A small approximation error compared to the full order model.
Conservation of the properties and characteristics of the full order model.
Computationally efficient and robust reduced order modelling techniques.

Applications

Reduced order models find themselves within a wide range

Methods

The most commonly used model order reduction techniques can be broadly classified into 3 classes:

operational or physics based reduction^[2]
Projection based methods
Reduced basis methods

A common approach for model order reduction is projection-based reduction. The following methods fall into this class:

Proper orthogonal decomposition
Proper generalized decomposition
Balanced truncation
Approximate balancing
Reduced basis method
Matrix interpolation
Transfer function interpolation
Piecewise tangential interpolation
Loewner framework
(Empirical) cross Gramian
Krylov subspace methods^[4]

References

^ Lassila, Toni; Manzoni, Andrea; Quarteroni, Alfio; Rozza, Gianluigi (2014). "Model Order Reduction in Fluid Dynamics: Challenges and Perspectives". Reduced Order Methods for Modeling and Computational Reduction. Springer International Publishing: 235–273. doi:10.1007/978-3-319-02090-7_9.
^ ^a ^b Schilders, Wilhelmus; van der Vorst, Henk; Rommes, Joost (2008). Model Order Reduction: Theory, Research Aspects and Applications. Springer-Verlag. ISBN 978-3-540-78841-6. {{cite book}}: |access-date= requires |url= (help)
^ Antoulas, A.C. (July 2004). "Approximation of Large-Scale Dynamical Systems: An Overview". IFAC Proceedings Volumes. 37 (11): 19–28. doi:https://doi.org/10.1016/S1474-6670(17)31584-7. {{cite journal}}: Check |doi= value (help); External link in |doi= (help)
^ "Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems". Applied Numerical Mathematics. 43: 9–44. 2002. {{cite journal}}: Unknown parameter |authors= ignored (help)

External links

[1] Lassila, Toni; Manzoni, Andrea; Quarteroni, Alfio; Rozza, Gianluigi (2014). "Model Order Reduction in Fluid Dynamics: Challenges and Perspectives". Reduced Order Methods for Modeling and Computational Reduction. Springer International Publishing: 235–273. doi:10.1007/978-3-319-02090-7_9.

[Schilders-2] Schilders, Wilhelmus; van der Vorst, Henk; Rommes, Joost (2008). Model Order Reduction: Theory, Research Aspects and Applications. Springer-Verlag. ISBN 978-3-540-78841-6. {{cite book}}: |access-date= requires |url= (help)

[3] Antoulas, A.C. (July 2004). "Approximation of Large-Scale Dynamical Systems: An Overview". IFAC Proceedings Volumes. 37 (11): 19–28. doi:https://doi.org/10.1016/S1474-6670(17)31584-7. {{cite journal}}: Check |doi= value (help); External link in |doi= (help)

[Bai2002-4] "Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems". Applied Numerical Mathematics. 43: 9–44. 2002. {{cite journal}}: Unknown parameter |authors= ignored (help)

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@@ Line 8: / Line 8: @@
 Many modern [[mathematical model]]s of real-life processes pose challenges when used in [[numerical simulation]]s, 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 system]]s and [[control system]]s. 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 infeasible to perform [[numerical simulation]]s using the complete full order model.  This can be due to limitations in [[computational resource]]s or the requirements of the simulations setting, for instance [[Real-time computing|real-time simulation]] settings or many-query settings in which a large number of simulations needs to be performed.<ref>{{cite journal |last1=Lassila |first1=Toni |last2=Manzoni |first2=Andrea |last3=Quarteroni |first3=Alfio |last4=Rozza |first4=Gianluigi |title=Model Order Reduction in Fluid Dynamics: Challenges and Perspectives |journal=Reduced Order Methods for Modeling and Computational Reduction |date=2014 |pages=235–273 |doi=10.1007/978-3-319-02090-7_9 |url=https://link.springer.com/chapter/10.1007/978-3-319-02090-7_9 |publisher=Springer International Publishing |language=en}}</ref> Real-time simulation settings include, for example, [[controller (control theory)|controller]]s in electronics and [[Data visualization|visualization]] of model results while examples for a many-query setting can include [[optimisation]] problems and design exploration. In order to be applicable to real-world problems, often the requirements of reduced order models are: <ref>{{cite book |last1=Schilders |first1=Wilhelmus |last2=van der Vorst |first2=Henk |last3=Rommes |first3=Joost |title=Model Order Reduction: Theory, Research Aspects and Applications |date=2008 |publisher=Springer-Verlag |isbn=978-3-540-78841-6 |accessdate=14 June 2018}}</ref><ref>{{cite journal |last1=Antoulas |first1=A.C. |title=Approximation of Large-Scale Dynamical Systems: An Overview |journal=IFAC Proceedings Volumes |date=July 2004 |volume=37 |issue=11 |pages=19–28 |doi=https://doi.org/10.1016/S1474-6670(17)31584-7}}</ref>
+Reduced order models are useful in settings where it is often infeasible to perform [[numerical simulation]]s using the complete full order model.  This can be due to limitations in [[computational resource]]s or the requirements of the simulations setting, for instance [[Real-time computing|real-time simulation]] settings or many-query settings in which a large number of simulations needs to be performed.<ref>{{cite journal |last1=Lassila |first1=Toni |last2=Manzoni |first2=Andrea |last3=Quarteroni |first3=Alfio |last4=Rozza |first4=Gianluigi |title=Model Order Reduction in Fluid Dynamics: Challenges and Perspectives |journal=Reduced Order Methods for Modeling and Computational Reduction |date=2014 |pages=235–273 |doi=10.1007/978-3-319-02090-7_9 |url=https://link.springer.com/chapter/10.1007/978-3-319-02090-7_9 |publisher=Springer International Publishing |language=en}}</ref> Real-time simulation settings include, for example, [[controller (control theory)|controller]]s in electronics and [[Data visualization|visualization]] of model results while examples for a many-query setting can include [[optimisation]] problems and design exploration. In order to be applicable to real-world problems, often the requirements of reduced order models are: <ref name=Schilders>{{cite book |last1=Schilders |first1=Wilhelmus |last2=van der Vorst |first2=Henk |last3=Rommes |first3=Joost |title=Model Order Reduction: Theory, Research Aspects and Applications |date=2008 |publisher=Springer-Verlag |isbn=978-3-540-78841-6 |accessdate=14 June 2018}}</ref><ref>{{cite journal |last1=Antoulas |first1=A.C. |title=Approximation of Large-Scale Dynamical Systems: An Overview |journal=IFAC Proceedings Volumes |date=July 2004 |volume=37 |issue=11 |pages=19–28 |doi=https://doi.org/10.1016/S1474-6670(17)31584-7}}</ref>
 * A small approximation error compared to the full order model.
 * Conservation of the properties and characteristics of the full order model.

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