User:JFA DeWit/sandbox: Difference between revisions
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The simplified physics approach can be described to be analogous to the traditional [[Mathematical model]]ling 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 and is not the subject of the current article. |
The simplified physics approach can be described to be analogous to the traditional [[Mathematical model]]ling 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 and is not the subject of the current article. |
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The remaining listed methods fall into the category of projection-based reduction. |
The remaining listed methods fall into the category of projection-based reduction and rely on the projection of either the model equations or the solution onto a basis of reduced dimensionality compared to the solution space. Methods that also fall into the class of [[Projection (mathematics)|projection-based]] reduction but are perhaps less commonly found are: |
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The following methods also fall into the class of [[Projection (mathematics)|projection-based]] reduction: |
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* [[Proper generalized decomposition]]<ref name=PGD>{{cite journal |last1=Chinesta |first1=Francisco |last2=Ladeveze |first2=Pierre |last3=Cueto |first3=Elías |title=A Short Review on Model Order Reduction Based on Proper Generalized Decomposition |journal=Archives of Computational Methods in Engineering |date=11 October 2011 |volume=18 |issue=4 |pages=395–404 |doi=10.1007/s11831-011-9064-7}}</ref> |
* [[Proper generalized decomposition]]<ref name=PGD>{{cite journal |last1=Chinesta |first1=Francisco |last2=Ladeveze |first2=Pierre |last3=Cueto |first3=Elías |title=A Short Review on Model Order Reduction Based on Proper Generalized Decomposition |journal=Archives of Computational Methods in Engineering |date=11 October 2011 |volume=18 |issue=4 |pages=395–404 |doi=10.1007/s11831-011-9064-7}}</ref> |
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===Fluid mechanics=== |
===Fluid mechanics=== |
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Current Problems in fluid mechanics involve large [[dynamical system]]s representing many effects on many different scales. [[Computational fluid dynamics]] studies often involve models solving the navier-stokes with a number of [[degrees of freedom]] in the order of magnitude upwards of 10^6. The first usage of model order reduction techniques dates back to the work of Lumley in 1967<ref>{{cite book |last1=Lumley |first1=J.L. |title=The Structure of Inhomogeneous Turbulence,” In: A. M. Yaglom and V. I. Tatarski, Eds., Atmospheric Turbulence and Wave Propagation |date=1967 |publisher=Nauka |location=Moscow}}</ref> where it was used to gain insight into the mechanisms of [[turbulence]] and [[Coherent turbulent structure|large coherent structure]]s. |
Current Problems in fluid mechanics involve large [[dynamical system]]s representing many effects on many different scales. [[Computational fluid dynamics]] studies often involve models solving the navier-stokes with a number of [[degrees of freedom]] in the order of magnitude upwards of 10^6. The first usage of model order reduction techniques dates back to the work of Lumley in 1967<ref>{{cite book |last1=Lumley |first1=J.L. |title=The Structure of Inhomogeneous Turbulence,” In: A. M. Yaglom and V. I. Tatarski, Eds., Atmospheric Turbulence and Wave Propagation |date=1967 |publisher=Nauka |location=Moscow}}</ref> where it was used to gain insight into the mechanisms of [[turbulence]] and [[Coherent turbulent structure|large coherent structure]]s. Model order reduction also finds applications in aeornautics<ref>{{Cite journal|last=Walton|first=S.|last2=Hassan|first2=O.|last3=Morgan|first3=K.|date=2013-11|title=Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions|url=http://linkinghub.elsevier.com/retrieve/pii/S0307904X13002771|journal=Applied Mathematical Modelling|volume=37|issue=20-21|pages=8930–8945|doi=10.1016/j.apm.2013.04.025|issn=0307-904X}}</ref> to model the flow over the body of aircrafts as in the study of Lieu et al<ref>{{Cite journal|last=Lieu|first=T.|last2=Farhat|first2=C.|last3=Lesoinne|first3=M.|date=2006-08|title=Reduced-order fluid/structure modeling of a complete aircraft configuration|url=http://linkinghub.elsevier.com/retrieve/pii/S0045782505005153|journal=Computer Methods in Applied Mechanics and Engineering|volume=195|issue=41-43|pages=5730–5742|doi=10.1016/j.cma.2005.08.026|issn=0045-7825}}</ref> in which the full order model with over 2.1 million degrees of freedom was reduced to a model of just 90 degrees of freedom. Additionally reduced order modelling has been applied to study [[Hemodynamics]] and the fluid structure interaction between the blood flowing through the vascular system and the vascular walls |
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== See also == |
== See also == |
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Revision as of 12:09, 15 June 2018
Model Order Reduction
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 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][2] 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 optimisation problems and design exploration. In order to be applicable to real-world problems, often the requirements of a reduced order model are: [3][4]
- 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.
Methods
Model order reduction techniques used most commonly nowadays can be broadly classified into 4 classes: [1][5]
- Proper orthogonal decompostion methods.[6]
- Reduced basis methods.[7]
- Balancing methods
- simplified physics[8] or operational based reduction methods[3].
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 and is not the subject of the current article.
The remaining listed methods fall into the category of projection-based reduction and rely on the projection of either the model equations or the solution onto a basis of reduced dimensionality compared to the solution space. Methods that also fall into the class of projection-based reduction but are perhaps less commonly found are:
- Proper generalized decomposition[9]
- Matrix interpolation
- Transfer function interpolation
- Piecewise tangential interpolation
- Loewner framework
- (Empirical) cross Gramian
- Krylov subspace methods[10]
Applications
Model order reduction finds application within all fields involving mathematical modelling and many reviews exist for the topics of electronics, fluid and structural mechanics.[11][9]
Fluid mechanics
Current Problems in fluid mechanics involve large dynamical systems representing many effects on many different scales. Computational fluid dynamics studies often involve models solving the navier-stokes with a number of degrees of freedom in the order of magnitude upwards of 10^6. The first usage of model order reduction techniques dates back to the work of Lumley in 1967[12] where it was used to gain insight into the mechanisms of turbulence and large coherent structures. Model order reduction also finds applications in aeornautics[13] to model the flow over the body of aircrafts as in the study of Lieu et al[14] in which the full order model with over 2.1 million degrees of freedom was reduced to a model of just 90 degrees of freedom. Additionally reduced order modelling has been applied to study Hemodynamics and the fluid structure interaction between the blood flowing through the vascular system and the vascular walls
See also
- Dimension reduction
- System identification
- Principal component analysis
- Singular value decomposition
- Nonlinear dimensionality reduction
References
- ^ a b 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.
- ^ Rozza, G.; Huynh, D. B. P.; Patera, A. T. (2008-05-21). "Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations". Archives of Computational Methods in Engineering. 15 (3): 229–275. doi:10.1007/s11831-008-9019-9. ISSN 1134-3060.
- ^ 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= - ^ Silva, João M. S.; Villena, Jorge Fernández; Flores, Paulo; Silveira, L. Miguel (2007), "Outstanding Issues in Model Order Reduction", Scientific Computing in Electrical Engineering, Springer Berlin Heidelberg, pp. 139–152, doi:10.1007/978-3-540-71980-9_13, ISBN 9783540719793, retrieved 2018-06-15
- ^ Kerschen, Gaetan; Golinval, Jean-claude; VAKAKIS, ALEXANDER F.; BERGMAN, LAWRENCE A. (2005-08). "The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview". Nonlinear Dynamics. 41 (1–3): 147–169. doi:10.1007/s11071-005-2803-2. ISSN 0924-090X.
{{cite journal}}: Check date values in:|date=(help) - ^ Boyaval, S.; Le Bris, C.; Lelièvre, T.; Maday, Y.; Nguyen, N. C.; Patera, A. T. (16 October 2010). "Reduced Basis Techniques for Stochastic Problems". Archives of Computational Methods in Engineering. 17 (4): 435–454. doi:10.1007/s11831-010-9056-z.
{{cite journal}}: no-break space character in|last2=at position 3 (help) - ^ Benner, Peter; Gugercin, Serkan; Willcox, Karen (2015-01). "A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems". SIAM Review. 57 (4): 483–531. doi:10.1137/130932715. ISSN 0036-1445.
{{cite journal}}: Check date values in:|date=(help) - ^ a b Chinesta, Francisco; Ladeveze, Pierre; Cueto, Elías (11 October 2011). "A Short Review on Model Order Reduction Based on Proper Generalized Decomposition". Archives of Computational Methods in Engineering. 18 (4): 395–404. doi:10.1007/s11831-011-9064-7.
- ^ "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) - ^ Benner, Peter; Gugercin, Serkan; Willcox, Karen (2015-01). "A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems". SIAM Review. 57 (4): 483–531. doi:10.1137/130932715. ISSN 0036-1445.
{{cite journal}}: Check date values in:|date=(help) - ^ Lumley, J.L. (1967). The Structure of Inhomogeneous Turbulence,” In: A. M. Yaglom and V. I. Tatarski, Eds., Atmospheric Turbulence and Wave Propagation. Moscow: Nauka.
- ^ Walton, S.; Hassan, O.; Morgan, K. (2013-11). "Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions". Applied Mathematical Modelling. 37 (20–21): 8930–8945. doi:10.1016/j.apm.2013.04.025. ISSN 0307-904X.
{{cite journal}}: Check date values in:|date=(help) - ^ Lieu, T.; Farhat, C.; Lesoinne, M. (2006-08). "Reduced-order fluid/structure modeling of a complete aircraft configuration". Computer Methods in Applied Mechanics and Engineering. 195 (41–43): 5730–5742. doi:10.1016/j.cma.2005.08.026. ISSN 0045-7825.
{{cite journal}}: Check date values in:|date=(help)