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POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds

  • * Corresponding author: Abdallah El Hamidi

    * Corresponding author: Abdallah El Hamidi 
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  • 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.

    Mathematics Subject Classification: 65K05, 65D18, 65F15.


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  • 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

  •   P. A. Absil , R. 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.
      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.
      N. Akkari , A. Hamdouni , E. 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.
      N. Akkari , A. 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.
      N. Akkari , A. Hamdouni , E. 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. 
      Y. AminovThe Geometry of Submanifolds, Gordon and Breach Science Publishers, 2001. 
      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. 
      M. Azaïez , F. 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.
      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.
      B. Denis de Senneville , A. 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. 
      B. Haasdonk , M. Ohlberger  and  G. Rozza , A reduced basis method for evolution schemes with parameter-dependent explicit operators, Electron. Trans. Numer. Anal., 32 (2008) , 145-161. 
      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.
      H. Karcher , Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977) , 509-541.  doi: 10.1002/cpa.3160300502.
      S. E. Kozlov , Geometry of real Grassmannian manifolds, Zap. Nauchn. Semin. POMI, 246 (1997) , 108-129. 
      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.
      H. Le , Estimation of Riemannian barycenters, LMS J. Comput. Math., 7 (2004) , 193-200.  doi: 10.1112/S1461157000001091.
      C. Leblond , C. 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.
      E. Longatte , E. Liberge , M. Pomarède , J. 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. 
      Y. Lu , N. 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. 
      W. Milnor and  J. D. StasheffCharacteristic Classes, Ann. Math. Studies, Princeton University Press, 1974. 
      A. T. Patera and G. Rozza, A Posteriori Error Estimation for Parametrized Partial Differential Equations, MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007.
      P. PetersenRiemannian Geometry, Springer-Verlag, 2006. 
      D. Pigoli , A. 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.
      S. Roujol , M. Ries , B. Quesson , C. 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. 
      L. Sirovich , Turbulence and the dynamics of coherent structures, parts Ⅰ-Ⅲ, Quart. Appl. Math., 45 (1987) , 561-571.  doi: 10.1090/qam/910462.
      A. Tallet , C. Allery , C. 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.
      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.
      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.
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