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October  2019, 12(6): 1743-1759. doi: 10.3934/dcdss.2019115

POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds

Laboratoire LaSIE, Université de La Rochelle, Avenue M. Crépeau, 17042 La Rochelle, France

* Corresponding author: Abdallah El Hamidi

Received  February 2018 Revised  April 2018 Published  November 2018

An adaptation of the Inverse Distance Weighting (IDW) method to the Grassmann manifold is carried out for interpolation of parametric POD bases. Our approach does not depend on the choice of a reference point on the Grassmann manifold to perform the interpolation, moreover our results are more accurate than those obtained in [7]. In return, our approach is not direct but iterative and its relevance depends on the choice of the weighting functions which are inversely proportional to the distance to the parameter. More judicious choices of such weighting functions can be carried out via kriging technics [23], this is the subject of a work in progress.

Citation: Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115
References:
[1]

P. A. AbsilR. Mahony and R. Sepulchre, Riemannian geometry of Grassmann manifolds with a view on algorithmic computation, Acta Applicandae Mathematicae, 80 (2004), 199-220. doi: 10.1023/B:ACAP.0000013855.14971.91. Google Scholar

[2]

B. Afsari, Riemannian Lp center of mass: Existence, uniqueness, and convexity, Proc. Amer. Math. Soc., 139 (2011), 655-673. doi: 10.1090/S0002-9939-2010-10541-5. Google Scholar

[3]

N. AkkariA. HamdouniE. Liberge and M. Jazar, A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM-POD), for a 2D incompressible fluid flow, Journal of Computational and Applied Mathematics, 270 (2014), 522-530. doi: 10.1016/j.cam.2013.11.025. Google Scholar

[4]

N. AkkariA. Hamdouni and M. Jazar, Mathematical and numerical results on the sensitivity of the POD approximation relative to the Burgers equation, Applied Mathematics and Computation, 247 (2014), 951-961. doi: 10.1016/j.amc.2014.09.005. Google Scholar

[5]

N. AkkariA. HamdouniE. Liberge and M. Jazar, On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation, Advanced Modeling and Simulation in Engineering Sciences, 2 (2014), 1-16. Google Scholar

[6] Y. Aminov, The Geometry of Submanifolds, Gordon and Breach Science Publishers, 2001. Google Scholar
[7]

D. Amsallem and C. Farhat, An interpolation method for adapting reduced order models and application to aeroelasticity, Amer. Inst. Aeronaut. Astronaut., 46 (2008), 1803-1813. Google Scholar

[8]

M. AzaïezF. Ben Belgacem and T. Chacón Rebollo, Error bounds for POD expansions of parameterized transient temperatures, Comput. Methods Appl. Mech. Engrg., 305 (2016), 501-511. doi: 10.1016/j.cma.2016.02.016. Google Scholar

[9]

M. Azaïez and F. Ben Belgacem, Karhunen-Loève's truncation error for bivariate functions, Comput. Methods Appl. Mech. Engrg., 290 (2015), 57-72. doi: 10.1016/j.cma.2015.02.019. Google Scholar

[10]

B. Denis de SennevilleA. El Hamidi and C. Moonen, A direct PCA-based approach for real-time description of physiological organ deformations, IEEE Transactions on Medical Imaging, 34 (2015), 974-982. Google Scholar

[11]

B. HaasdonkM. Ohlberger and G. Rozza, A reduced basis method for evolution schemes with parameter-dependent explicit operators, Electron. Trans. Numer. Anal., 32 (2008), 145-161. Google Scholar

[12]

D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach utilizing proper orthogonal decomposition, Math. Comput. Model., 38 (2003), 1003-1028. doi: 10.1016/S0895-7177(03)90102-6. Google Scholar

[13]

H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), 509-541. doi: 10.1002/cpa.3160300502. Google Scholar

[14]

S. E. Kozlov, Geometry of real Grassmannian manifolds, Zap. Nauchn. Semin. POMI, 246 (1997), 108-129. Google Scholar

[15]

K. Kunisch and S. Volkwein, Control of Burgers equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371. doi: 10.1023/A:1021732508059. Google Scholar

[16]

H. Le, Estimation of Riemannian barycenters, LMS J. Comput. Math., 7 (2004), 193-200. doi: 10.1112/S1461157000001091. Google Scholar

[17]

C. LeblondC. Allery and C. Inard, An optimal projection method for the reduce order modeling of incompressible flows, Comp. Meth, in Applied Mechanics and Engineering, 200 (2011), 2507-2527. doi: 10.1016/j.cma.2011.04.020. Google Scholar

[18]

E. LongatteE. LibergeM. PomarèdeJ. F. Sigrist and A. Hamdouni, Parametric study of flow-induced vibrations in cylinder arrays under single-phase fluid cross flows using POD-ROM, Journal of Fluids and Structures, 78 (2018), 314-330. Google Scholar

[19]

Y. LuN. Blal and A. Gravouil, Space time POD based computational vademecums for parametric studies: Application to thermo-mechanical problems, Advanced Modeling and Simulation in Engineering Sciences, 5 (2018), 1-27. Google Scholar

[20] W. Milnor and J. D. Stasheff, Characteristic Classes, Ann. Math. Studies, Princeton University Press, 1974. Google Scholar
[21]

A. T. Patera and G. Rozza, A Posteriori Error Estimation for Parametrized Partial Differential Equations, MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007.Google Scholar

[22] P. Petersen, Riemannian Geometry, Springer-Verlag, 2006. Google Scholar
[23]

D. PigoliA. Menafoglio and P. Secchi, Kriging prediction for manifold-valued random fields, Journal of Multivariate Analysis, 145 (2016), 117-131. doi: 10.1016/j.jmva.2015.12.006. Google Scholar

[24]

S. RoujolM. RiesB. QuessonC. Moonen and B. Denis de Senneville, Real-time MR-thermometry and dosimetry for interventional guidance on abdominal organs, Magnetic Resonance in Medicine, 63 (2010), 1080-7. Google Scholar

[25]

L. Sirovich, Turbulence and the dynamics of coherent structures, parts Ⅰ-Ⅲ, Quart. Appl. Math., 45 (1987), 561-571. doi: 10.1090/qam/910462. Google Scholar

[26]

A. TalletC. AlleryC. Leblond and E. Liberge, A minimum residual projection to build coupled velocity-pressure POD-ROM for incompressible Navier-Stokes equations, Comm. in Nonlin. Science and Num. Simulation, 22 (2015), 909-932. doi: 10.1016/j.cnsns.2014.09.009. Google Scholar

[27]

S. Volkwein, Optimal control of a phase-field model using the proper orthogonal decomposition, Z. Angew. Math. Mech., 81 (2001), 83-97. doi: 10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R. Google Scholar

[28]

Y. C. Wong, Differential geometry of Grassmann manifolds, Proc Natl Acad Sci U S A., 57 (1967), 589-594. doi: 10.1073/pnas.57.3.589. Google Scholar

show all references

References:
[1]

P. A. AbsilR. Mahony and R. Sepulchre, Riemannian geometry of Grassmann manifolds with a view on algorithmic computation, Acta Applicandae Mathematicae, 80 (2004), 199-220. doi: 10.1023/B:ACAP.0000013855.14971.91. Google Scholar

[2]

B. Afsari, Riemannian Lp center of mass: Existence, uniqueness, and convexity, Proc. Amer. Math. Soc., 139 (2011), 655-673. doi: 10.1090/S0002-9939-2010-10541-5. Google Scholar

[3]

N. AkkariA. HamdouniE. Liberge and M. Jazar, A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM-POD), for a 2D incompressible fluid flow, Journal of Computational and Applied Mathematics, 270 (2014), 522-530. doi: 10.1016/j.cam.2013.11.025. Google Scholar

[4]

N. AkkariA. Hamdouni and M. Jazar, Mathematical and numerical results on the sensitivity of the POD approximation relative to the Burgers equation, Applied Mathematics and Computation, 247 (2014), 951-961. doi: 10.1016/j.amc.2014.09.005. Google Scholar

[5]

N. AkkariA. HamdouniE. Liberge and M. Jazar, On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation, Advanced Modeling and Simulation in Engineering Sciences, 2 (2014), 1-16. Google Scholar

[6] Y. Aminov, The Geometry of Submanifolds, Gordon and Breach Science Publishers, 2001. Google Scholar
[7]

D. Amsallem and C. Farhat, An interpolation method for adapting reduced order models and application to aeroelasticity, Amer. Inst. Aeronaut. Astronaut., 46 (2008), 1803-1813. Google Scholar

[8]

M. AzaïezF. Ben Belgacem and T. Chacón Rebollo, Error bounds for POD expansions of parameterized transient temperatures, Comput. Methods Appl. Mech. Engrg., 305 (2016), 501-511. doi: 10.1016/j.cma.2016.02.016. Google Scholar

[9]

M. Azaïez and F. Ben Belgacem, Karhunen-Loève's truncation error for bivariate functions, Comput. Methods Appl. Mech. Engrg., 290 (2015), 57-72. doi: 10.1016/j.cma.2015.02.019. Google Scholar

[10]

B. Denis de SennevilleA. El Hamidi and C. Moonen, A direct PCA-based approach for real-time description of physiological organ deformations, IEEE Transactions on Medical Imaging, 34 (2015), 974-982. Google Scholar

[11]

B. HaasdonkM. Ohlberger and G. Rozza, A reduced basis method for evolution schemes with parameter-dependent explicit operators, Electron. Trans. Numer. Anal., 32 (2008), 145-161. Google Scholar

[12]

D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach utilizing proper orthogonal decomposition, Math. Comput. Model., 38 (2003), 1003-1028. doi: 10.1016/S0895-7177(03)90102-6. Google Scholar

[13]

H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), 509-541. doi: 10.1002/cpa.3160300502. Google Scholar

[14]

S. E. Kozlov, Geometry of real Grassmannian manifolds, Zap. Nauchn. Semin. POMI, 246 (1997), 108-129. Google Scholar

[15]

K. Kunisch and S. Volkwein, Control of Burgers equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371. doi: 10.1023/A:1021732508059. Google Scholar

[16]

H. Le, Estimation of Riemannian barycenters, LMS J. Comput. Math., 7 (2004), 193-200. doi: 10.1112/S1461157000001091. Google Scholar

[17]

C. LeblondC. Allery and C. Inard, An optimal projection method for the reduce order modeling of incompressible flows, Comp. Meth, in Applied Mechanics and Engineering, 200 (2011), 2507-2527. doi: 10.1016/j.cma.2011.04.020. Google Scholar

[18]

E. LongatteE. LibergeM. PomarèdeJ. F. Sigrist and A. Hamdouni, Parametric study of flow-induced vibrations in cylinder arrays under single-phase fluid cross flows using POD-ROM, Journal of Fluids and Structures, 78 (2018), 314-330. Google Scholar

[19]

Y. LuN. Blal and A. Gravouil, Space time POD based computational vademecums for parametric studies: Application to thermo-mechanical problems, Advanced Modeling and Simulation in Engineering Sciences, 5 (2018), 1-27. Google Scholar

[20] W. Milnor and J. D. Stasheff, Characteristic Classes, Ann. Math. Studies, Princeton University Press, 1974. Google Scholar
[21]

A. T. Patera and G. Rozza, A Posteriori Error Estimation for Parametrized Partial Differential Equations, MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007.Google Scholar

[22] P. Petersen, Riemannian Geometry, Springer-Verlag, 2006. Google Scholar
[23]

D. PigoliA. Menafoglio and P. Secchi, Kriging prediction for manifold-valued random fields, Journal of Multivariate Analysis, 145 (2016), 117-131. doi: 10.1016/j.jmva.2015.12.006. Google Scholar

[24]

S. RoujolM. RiesB. QuessonC. Moonen and B. Denis de Senneville, Real-time MR-thermometry and dosimetry for interventional guidance on abdominal organs, Magnetic Resonance in Medicine, 63 (2010), 1080-7. Google Scholar

[25]

L. Sirovich, Turbulence and the dynamics of coherent structures, parts Ⅰ-Ⅲ, Quart. Appl. Math., 45 (1987), 561-571. doi: 10.1090/qam/910462. Google Scholar

[26]

A. TalletC. AlleryC. Leblond and E. Liberge, A minimum residual projection to build coupled velocity-pressure POD-ROM for incompressible Navier-Stokes equations, Comm. in Nonlin. Science and Num. Simulation, 22 (2015), 909-932. doi: 10.1016/j.cnsns.2014.09.009. Google Scholar

[27]

S. Volkwein, Optimal control of a phase-field model using the proper orthogonal decomposition, Z. Angew. Math. Mech., 81 (2001), 83-97. doi: 10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R. Google Scholar

[28]

Y. C. Wong, Differential geometry of Grassmann manifolds, Proc Natl Acad Sci U S A., 57 (1967), 589-594. doi: 10.1073/pnas.57.3.589. Google Scholar

Figure 1.  Geometrical properties of the flow around a circular cylinder
Figure 2.  Isovalue of the velocity magnitude for Re = 180 at the first snapshot
Figure 3.  Influence of the power $p$ of the IDW-G interpolation on the prediction of the drag and lift for Reynolds number $Re = 160$ and $Re = 190$
Figure 4.  Comparison of the drag and lift coefficients obtained by the two interpolations methods IDW-G and Grassmann and those obtained with the full model for Re = 140 and 160
Figure 5.  Comparison of the drag and lift coefficients obtained by the two interpolations methods IDW-G and Grassmann and those obtained with the full model for Re = 170 and 190
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