Model order reduction

Model order reduction (MOR)is a technique for reducing thecomputational complexityofmathematical modelsinnumerical simulations.As such it is closely related to the concept ofmetamodeling,with applications in all areas ofmathematical modelling.

Overview

Many modernmathematical modelsof real-life processes pose challenges when used innumerical 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-scaledynamical systemsandcontrol systems.By a reduction of the model's associatedstate spacedimension ordegrees 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 performnumerical simulationsusing the complete full order model. This can be due to limitations incomputational resourcesor the requirements of the simulations setting, for instancereal-time simulationsettings or many-query settings in which a large number of simulations needs to be performed.^[1]^[2]Examples of Real-time simulation settings includecontrol systemsin electronics andvisualizationof model results while examples for a many-query setting can includeoptimizationproblems and design exploration. In order to be applicable to real-world problems, often the requirements of a reduced order model are:^[3]^[4]

A smallapproximation errorcompared to the full order model.
Conservation of the properties and characteristics of the full order model (E.g. stability andpassivityin 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.^[5]

Methods

Contemporary model order reduction techniques can be broadly classified into 5 classes:^[1]^[6]

Proper orthogonal decompositionmethods.^[7]^[8]
Reduced basis methods.^[9]
Balancing methods
Simplified physics^[10]or operational based reduction methods.^[3]
Nonlinear manifold methods.^[11]^[12]

The simplified physics approach can be described to be analogous to the traditionalmathematical modellingapproach, 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.

The remaining listed 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^[13]
Matrix interpolation^[14]
Transfer function interpolation
Piecewise tangential interpolation
Loewner framework
(Empirical)cross Gramian
Krylov subspace methods^[15]

Implementations

RBmatlab:A MATLAB library containing all reduced simulation approaches for linear and nonlinear, affine or arbitrarily parameter dependent evolution problems with finite element, finite volume or local discontinuous Galerkin discretizations. Further information can be found on thedownload and documentation page.

Model Reduction inside ANSYS:implements a Krylov-based model order reduction for multiphysical finite element models in Ansys. Model simplification via Model Reduction inside Ansys is suitable for optimization strategies in component development as well as for integrating compact models into an overall system simulation in the fields of electronics, automotive or microsystems. Despite reduction, the examination parameters are retained, which means fast results can be achieved with regards to designs and system simulations. For more information, visithttps:// cadfem.net/en/our-solutions/cadfem-ansys-extensions/model-reduction-inside-ansys.html

pyMOR:pyMOR is a software library for building model order reduction applications with the Python programming language. Its main focus lies on the application of reduced basis methods to parameterized partial differential equations. All algorithms in pyMOR are formulated in terms of abstract interfaces for seamless integration with external high-dimensional PDE solvers. Moreover, pure Python implementations of finite element and finite volume discretizations using the NumPy/SciPy scientific computing stack are provided for getting started quickly. For more information, visithttp://pymor.org
emgr:Empirical Gramian Framework. Empirical gramians can be computed for linear and nonlinear control systems for purposes of model order reduction, uncertainty quantification or system identification. The emgr framework is a compact open source toolbox for gramian-based model reduction and compatible with OCTAVE and MATLAB. More at:http://gramian.de
KerMor:An object-oriented MATLAB© library providing routines for model order reduction of nonlinear dynamical systems. Reduction can be achieved via subspace projection and approximation of nonlinearities via kernels methods or DEIM. Standard procedures like the POD-Greedy method are readily implemented as well as advanced a-posteriori error estimators for various system configurations. KerMor also includes several working examples and some demo files to quickly get familiarized with the provided functionality. More information can be found athttp:// morepas.org/software/kermor/
JaRMoS:JaRMoS stands for "Java Reduced Model Simulations" and aims to enable import and simulation of various reduced models from multiple sources on any java-capable platform. So far support for RBmatlab, KerMor and rbMIT reduced models is present, where we can only import the rbMIT models that have previously been published with the rbAppMIT Android application. Extensions so far are a desktop-version to run reduced models and initial support for KerMor kernel-based reduced models is on the way. More information can be found athttp:// morepas.org/software/jarmos/
MORLAB:Model Order Reduction Laboratory. This toolbox is a collection of MATLAB/OCTAVE routines for model order reduction of linear dynamical systems based on the solution of matrix equations. The implementation is based on spectral projection methods, e.g., methods based on the matrix sign function and the matrix disk function. For more details on this software, see:https:// mpi-magdeburg.mpg.de/projects/morlab
Dune-RB:A module for the Dune library (dune-project.org,http://dune.mathematik.uni-freiburg.de), which realizes C++ template classes for use in snapshot generation and RB offline phases for various discretizations. Apart from single-core algorithms, the package also aims at using parallelization techniques for efficient snapshot generation. More at:http://users.dune-project.org/projects/dune-rb/wiki
libROM:Collection of C++ classes that compute model order reduction and hyper-reduction for systems of partial and ordinary differential equations. libROM includes scalable and parallel, adaptive methods for proper orthogonal decomposition, parallel, non-adaptive methods for hyper-reduction, and randomized singular value decomposition. libROM also includes the dynamic mode decomposition capability. libROM has physics-informed greedy sampling capability. Source codes can be found at:https://github /LLNL/libROM.Webpage can be found at:https:// librom.net,where you can find many examples, e.g., reduced order models for Lagrangian hydrodynamics with shock-moving wave.^[16]
Pressio:Pressio is an open-source project aimed at alleviating the intrusive nature of projection-based reduced-order models for large-scale codes. The core of the project is a header-only C++ library that leverages generic programming to interface with shared or distributed memory applications using arbitrary data-types. Pressio provides numerous functionalities and solvers for performing model reduction, such as Galerkin and least-squares Petrov–Galerkin projections. The Pressio ecosystem also offers: (1)pressio4py,a Python binding library for ease of prototyping, (2)pressio-tutorials,a library also offering end-to-end demos that one can easily play with, which can be found athttps://pressio.github.io/pressio-tutorials/,(3)pressio-tools,a library for large-scale SVD, QR and sample mesh, and (4)pressio-demoapps,a suite of 1d, 2d and 3d demo applications for testing ROMs and hyper-reduction. The ecosystem main website can be found athttps://pressio.github.io/,the C++ library documentation athttps://pressio.github.io/pressio/.

Applications

Model order reduction finds application within all fields involving mathematical modelling and many reviews^[10]^[13]exist for the topics ofelectronics,^[17]fluid mechanics,^[18]hydrodynamics,^[16]structural mechanics,^[7]MEMS,^[19]Boltzmann equation,^[8]anddesign optimization.^[14]^[20]

Fluid mechanics

Current problems in fluid mechanics involve largedynamical systemsrepresenting many effects on many different scales.Computational fluid dynamicsstudies often involve models solving theNavier–Stokes equationswith a number ofdegrees of freedomin 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,^[21]where it was used to gain insight into the mechanisms and intensity ofturbulenceandlarge coherent structurespresent in fluid flow problems. Model order reduction also finds modern applications inaeronauticsto model the flow over the body of aircraft.^[22]An example can be found in Lieu et al^[23]in which the full order model of anF16fighter-aircraft with over 2.1 million degrees of freedom, was reduced to a model of just 90 degrees of freedom. Additionally reduced order modeling has been applied to studyrheologyinhemodynamicsand thefluid–structure interactionbetween the blood flowing through the vascular system and the vascular walls.^[24]^[25]

References

^^a ^bLassila, 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(PDF).pp. 235–273.doi:10.1007/978-3-319-02090-7_9.ISBN 978-3-319-02089-1.
^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.S2CID 13511413.
^^a ^bSchilders, Wilhelmus; van der Vorst, Henk; Rommes, Joost (2008).Model Order Reduction: Theory, Research Aspects and Applications.Springer-Verlag.ISBN 978-3-540-78841-6.
^Antoulas, A.C.(July 2004). "Approximation of Large-Scale Dynamical Systems: An Overview".IFAC Proceedings Volumes.37(11): 19–28.CiteSeerX10.1.1.29.3565.doi:10.1016/S1474-6670(17)31584-7.
^Ovchinnikov, Alexey; Pérez Verona, Isabel; Pogudin, Gleb; Tribastone, Mirco (2021-07-19). Valencia, Alfonso (ed.)."CLUE: exact maximal reduction of kinetic models by constrained lumping of differential equations".Bioinformatics.37(12): 1732–1738.arXiv:2004.11961.doi:10.1093/bioinformatics/btab010.ISSN 1367-4803.PMID 33532849.
^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 978-3-540-71979-3
^^a ^bKerschen, Gaetan; Golinval, Jean-claude; VAKAKIS, ALEXANDER F.; BERGMAN, LAWRENCE A. (2005). "The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview".Nonlinear Dynamics.41(1–3): 147–169.CiteSeerX10.1.1.530.8349.doi:10.1007/s11071-005-2803-2.ISSN 0924-090X.S2CID 17625377.
^^a ^bChoi, Youngsoo; Brown, Peter; Arrighi, William; Anderson, Robert; Huynh, Kevin (2021)."Space--time reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems".Journal of Computational Physics.424:109845.arXiv:1910.01260.Bibcode:2021JCoPh.42409845C.doi:10.1016/j.jcp.2020.109845.ISSN 0021-9991.S2CID 203641768.
^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.arXiv:1004.0357.doi:10.1007/s11831-010-9056-z.hdl:1721.1/63915.S2CID 446613.
^^a ^bBenner, Peter; Gugercin, Serkan; Willcox, Karen (2015)."A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems"(PDF).SIAM Review.57(4): 483–531.doi:10.1137/130932715.hdl:1721.1/100939.ISSN 0036-1445.S2CID 16186635.
^Kim, Youngkyu; Choi, Youngsoo; Widemann, David; Zohdi, Tarek (2021)."A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder".Journal of Computational Physics.451:110841.arXiv:2009.11990.doi:10.1016/j.jcp.2021.110841.S2CID 221949087.
^Mojgani, Rambod; Balajewicz, Maciej (2021)."Low-Rank Registration Based Manifolds for Convection-Dominated PDEs".Proceedings of the AAAI Conference on Artificial Intelligence.35:399-407.arXiv:2006.15655.doi:10.1609/aaai.v35i1.16116.S2CID 220249659.
^^a ^bChinesta, Francisco; Ladeveze, Pierre; Cueto, Elías (11 October 2011)."A Short Review on Model Order Reduction Based on Proper Generalized Decomposition"(PDF).Archives of Computational Methods in Engineering.18(4): 395–404.doi:10.1007/s11831-011-9064-7.S2CID 54512292.
^^a ^bChoi, Youngsoo; Boncoraglio, Gabriele; Spenser, Anderson; Amsallem, David; Farhat, Charbel (2020)."Gradient-based constrained optimization using a database of linear reduced-order models".Journal of Computational Physics.423:109787.arXiv:1506.07849.Bibcode:2020JCoPh.42309787C.doi:10.1016/j.jcp.2020.109787.S2CID 60788542.
^Bai, Zhaojun (2002). "Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems".Applied Numerical Mathematics.43(1–2): 9–44.CiteSeerX10.1.1.131.8251.doi:10.1016/S0168-9274(02)00116-2.
^^a ^bCopeland, Dylan; Cheung, Siu Wun; Huynh, Kevin; Choi, Youngsoo (2021)."Reduced order models for Lagrangian hydrodynamics".Computer Methods in Applied Mechanics and Engineering.388:114259.arXiv:2104.11404.doi:10.1016/j.cma.2021.114259.ISSN 0045-7825.S2CID 233388014.
^Umunnakwe, Chisom Bernhard; Zawra, Ibrahim; Niessner, Martin; Rudnyi, Evgenii; Hohlfeld, Dennis; Bechtold, Tamara (2023). "Compact modelling of a thermo-mechanical finite element model of a microelectronic package".Microelectronics Reliability.151(115238).doi:10.1016/j.microrel.2023.115238.
^Holmes, Philip; Lumley, John L.; Berkooz, Gal (1996).Turbulence, Coherent Structures, Dynamical Systems and Symmetry.Cambridge: Cambridge University Press.doi:10.1017/cbo9780511622700.ISBN 978-0-511-62270-0.
^Bechtold, Tamara; Schrag, Gabriela; Feng, Lihong (2013).System-Level Modeling of MEMS.Wiley-VCH Verlag GmbH & Co. KGaA.ISBN 978-3-527-31903-9.
^McBane, Sean; Choi, Youngsoo (1 August 2021)."Component-wise reduced order model lattice-type structure design".Computer Methods in Applied Mechanics and Engineering.381(113813): 113813.arXiv:2010.10770.Bibcode:2021CMAME.381k3813M.doi:10.1016/j.cma.2021.113813.S2CID 224818337.
^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)."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.
^Lieu, T.; Farhat, C.; Lesoinne, M. (2006). "Reduced-order fluid/structure modeling of a complete aircraft configuration".Computer Methods in Applied Mechanics and Engineering.195(41–43): 5730–5742.Bibcode:2006CMAME.195.5730L.doi:10.1016/j.cma.2005.08.026.ISSN 0045-7825.
^Xiao, D.; Yang, P.; Fang, F.; Xiang, J.; Pain, C.C.; Navon, I.M. (2016)."Non-intrusive reduced order modelling of fluid–structure interactions"(PDF).Computer Methods in Applied Mechanics and Engineering.303:35–54.Bibcode:2016CMAME.303...35X.doi:10.1016/j.cma.2015.12.029.ISSN 0045-7825.
^Colciago, C.M.; Deparis, S.;Quarteroni, A.(2014)."Comparisons between reduced order models and full 3D models for fluid–structure interaction problems in haemodynamics".Journal of Computational and Applied Mathematics.265:120–138.doi:10.1016/j.cam.2013.09.049.ISSN 0377-0427.

External links

[:0-1] 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(PDF).pp. 235–273.doi:10.1007/978-3-319-02090-7_9.ISBN 978-3-319-02089-1.

[2] 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.S2CID 13511413.

[Schilders-3] 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.

[4] Antoulas, A.C.(July 2004). "Approximation of Large-Scale Dynamical Systems: An Overview".IFAC Proceedings Volumes.37(11): 19–28.CiteSeerX10.1.1.29.3565.doi:10.1016/S1474-6670(17)31584-7.

[5] Ovchinnikov, Alexey; Pérez Verona, Isabel; Pogudin, Gleb; Tribastone, Mirco (2021-07-19). Valencia, Alfonso (ed.)."CLUE: exact maximal reduction of kinetic models by constrained lumping of differential equations".Bioinformatics.37(12): 1732–1738.arXiv:2004.11961.doi:10.1093/bioinformatics/btab010.ISSN 1367-4803.PMID 33532849.

[6] 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 978-3-540-71979-3

[:1-7] Kerschen, Gaetan; Golinval, Jean-claude; VAKAKIS, ALEXANDER F.; BERGMAN, LAWRENCE A. (2005). "The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview".Nonlinear Dynamics.41(1–3): 147–169.CiteSeerX10.1.1.530.8349.doi:10.1007/s11071-005-2803-2.ISSN 0924-090X.S2CID 17625377.

[boltzmann-8] Choi, Youngsoo; Brown, Peter; Arrighi, William; Anderson, Robert; Huynh, Kevin (2021)."Space--time reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems".Journal of Computational Physics.424:109845.arXiv:1910.01260.Bibcode:2021JCoPh.42409845C.doi:10.1016/j.jcp.2020.109845.ISSN 0021-9991.S2CID 203641768.

[9] 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.arXiv:1004.0357.doi:10.1007/s11831-010-9056-z.hdl:1721.1/63915.S2CID 446613.

[:2-10] Benner, Peter; Gugercin, Serkan; Willcox, Karen (2015)."A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems"(PDF).SIAM Review.57(4): 483–531.doi:10.1137/130932715.hdl:1721.1/100939.ISSN 0036-1445.S2CID 16186635.

[11] Kim, Youngkyu; Choi, Youngsoo; Widemann, David; Zohdi, Tarek (2021)."A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder".Journal of Computational Physics.451:110841.arXiv:2009.11990.doi:10.1016/j.jcp.2021.110841.S2CID 221949087.

[12] Mojgani, Rambod; Balajewicz, Maciej (2021)."Low-Rank Registration Based Manifolds for Convection-Dominated PDEs".Proceedings of the AAAI Conference on Artificial Intelligence.35:399-407.arXiv:2006.15655.doi:10.1609/aaai.v35i1.16116.S2CID 220249659.

[PGD-13] Chinesta, Francisco; Ladeveze, Pierre; Cueto, Elías (11 October 2011)."A Short Review on Model Order Reduction Based on Proper Generalized Decomposition"(PDF).Archives of Computational Methods in Engineering.18(4): 395–404.doi:10.1007/s11831-011-9064-7.S2CID 54512292.

[gradient-14] Choi, Youngsoo; Boncoraglio, Gabriele; Spenser, Anderson; Amsallem, David; Farhat, Charbel (2020)."Gradient-based constrained optimization using a database of linear reduced-order models".Journal of Computational Physics.423:109787.arXiv:1506.07849.Bibcode:2020JCoPh.42309787C.doi:10.1016/j.jcp.2020.109787.S2CID 60788542.

[Bai2002-15] Bai, Zhaojun (2002). "Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems".Applied Numerical Mathematics.43(1–2): 9–44.CiteSeerX10.1.1.131.8251.doi:10.1016/S0168-9274(02)00116-2.

[hydrodynamics-16] Copeland, Dylan; Cheung, Siu Wun; Huynh, Kevin; Choi, Youngsoo (2021)."Reduced order models for Lagrangian hydrodynamics".Computer Methods in Applied Mechanics and Engineering.388:114259.arXiv:2104.11404.doi:10.1016/j.cma.2021.114259.ISSN 0045-7825.S2CID 233388014.

[17] Umunnakwe, Chisom Bernhard; Zawra, Ibrahim; Niessner, Martin; Rudnyi, Evgenii; Hohlfeld, Dennis; Bechtold, Tamara (2023). "Compact modelling of a thermo-mechanical finite element model of a microelectronic package".Microelectronics Reliability.151(115238).doi:10.1016/j.microrel.2023.115238.

[18] Holmes, Philip; Lumley, John L.; Berkooz, Gal (1996).Turbulence, Coherent Structures, Dynamical Systems and Symmetry.Cambridge: Cambridge University Press.doi:10.1017/cbo9780511622700.ISBN 978-0-511-62270-0.

[19] Bechtold, Tamara; Schrag, Gabriela; Feng, Lihong (2013).System-Level Modeling of MEMS.Wiley-VCH Verlag GmbH & Co. KGaA.ISBN 978-3-527-31903-9.

[20] McBane, Sean; Choi, Youngsoo (1 August 2021)."Component-wise reduced order model lattice-type structure design".Computer Methods in Applied Mechanics and Engineering.381(113813): 113813.arXiv:2010.10770.Bibcode:2021CMAME.381k3813M.doi:10.1016/j.cma.2021.113813.S2CID 224818337.

[21] 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.

[22] Walton, S.; Hassan, O.; Morgan, K. (2013)."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.

[23] Lieu, T.; Farhat, C.; Lesoinne, M. (2006). "Reduced-order fluid/structure modeling of a complete aircraft configuration".Computer Methods in Applied Mechanics and Engineering.195(41–43): 5730–5742.Bibcode:2006CMAME.195.5730L.doi:10.1016/j.cma.2005.08.026.ISSN 0045-7825.

[24] Xiao, D.; Yang, P.; Fang, F.; Xiang, J.; Pain, C.C.; Navon, I.M. (2016)."Non-intrusive reduced order modelling of fluid–structure interactions"(PDF).Computer Methods in Applied Mechanics and Engineering.303:35–54.Bibcode:2016CMAME.303...35X.doi:10.1016/j.cma.2015.12.029.ISSN 0045-7825.

[25] Colciago, C.M.; Deparis, S.;Quarteroni, A.(2014)."Comparisons between reduced order models and full 3D models for fluid–structure interaction problems in haemodynamics".Journal of Computational and Applied Mathematics.265:120–138.doi:10.1016/j.cam.2013.09.049.ISSN 0377-0427.

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