Proper Orthogonal Decomposition and Radial Basis Functions for Fast Simulations

  • Vladimir BuljakEmail author
Part of the Computational Fluid and Solid Mechanics book series (COMPFLUID)


Proper Orthogonal Decomposition (POD) is a powerful method for low-order approximation of some high dimensional processes. It is widely used in the situations where model reduction is required. The most favorable feature of the method is its optimality: it provides the most efficient way of capturing the dominant components of high-dimensional processes with, sometimes surprisingly small number of “modes”. This chapter will present an algorithm that combines POD with Radial Basis Functions (RBF) used for the interpolation of the data with previously reduced dimensionality by the POD.


Radial Basis Function Singular Value Decomposition Proper Orthogonal Decomposition Nodal Displacement Proper Orthogonal Decomposition Mode 
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  1. 1.
    Pearson, K.: On lines planes of closes fit to system of points in space. The London, Edinburgh Dublin Philos. Mag. J. Sci. 2, 559–572 (1901)CrossRefGoogle Scholar
  2. 2.
    Hotelling, H.: Analyses of complex statistical variables intro principal components. J. Educ. Psychol. 24, 417–441 (1933)CrossRefGoogle Scholar
  3. 3.
    Karhunen, K.: Uber linear Methoden fur Wahrscheiniogkeitsrechnung. Ann. Acad. Sci. Fennicae Series Al Math. Phys. 37, 3–79 (1946)Google Scholar
  4. 4.
    Loeve, M.M.: Probabilty Theoiry. Van Nostrand, Princeton (1955)Google Scholar
  5. 5.
    Lumley, J.L.: Stochastic Tools in Turbulence. Academic, New York (1970)zbMATHGoogle Scholar
  6. 6.
    Liang, Y.C., Lee, H.P., Lim, S.P., Lin, W.Z., Lee, K.H., Wu, C.G.: Proper orthogonal decomposition and its applications: part I – theory. J. Sound Vib. 252(3), 527–544 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bialecki, R.A., Kassab, A.J., Fic, A.: Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis. Int. J. Numer. Meth. Eng. 62, 774–797 (2005)zbMATHCrossRefGoogle Scholar
  8. 8.
    Holmes, P., Lumley, J.L., Berkoz, G.: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575 (1993)CrossRefGoogle Scholar
  9. 9.
    Kerschen, G., Ponceletm, F., Golinval, J.C.: Physical interpretation of independent component analysis in structural dynamics. Mech. Syst. Signal Process 21, 1561–1575 (2007)CrossRefGoogle Scholar
  10. 10.
    Ly, H.V., Tran, H.T.: Modeling and control of physical processes using proper orthogonal decomposition. Math. Comput. Model 33, 223–236 (2001)zbMATHCrossRefGoogle Scholar
  11. 11.
    Ruotolo, R., Surace, C.: Using SVD to detect damage in structures with different operational conditions. J. Sound Vib. 226(3), 425–439 (1999)CrossRefGoogle Scholar
  12. 12.
    Sirovich, L., Kirby, M.: Low-dimensional procedure for the characterization of human faces. J. Opt. Soc. Am. 4, 519–524 (1987)CrossRefGoogle Scholar
  13. 13.
    Tang, D., Kholodar, D., Juang, J.N., Dowell, E.H.: System identification and proper orthogonal decomposition method applied to unsteady aerodynamics. AIAA J. 39(8), 1569–1575 (2001)CrossRefGoogle Scholar
  14. 14.
    Jolliffe, I.T.: Principal Component Analysis. Springer, New York (2002)zbMATHGoogle Scholar
  15. 15.
    Golub, G.H., Van Loan, C.F.: Matrix Computations. The Johns Hopkins University Press, Baltimore/London (1993)Google Scholar
  16. 16.
    Ostrowski, Z., Bialecki, R.A., Kassab, A.J.: Solving inverse heat conduction problems using trained POD-RBF network. Inverse Probl. Sci. Eng. 16(1), 705–714 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Buljak, V.: Assessment of material mechanical properties and residual stresses by indentation simulation and proper orthogonal decomposition. Ph.D. thesis, Politecnico di Milano, Milano (2009)Google Scholar
  18. 18.
    Buljak V., Maier G.: Proper orthogonal decomposition and radial basis functions in material characterization based on instrumented indentation. J. Eng. Struct. (2009, submitted)Google Scholar
  19. 19.
    Bolzon, G., Buljak, V.: An indentation-based technique to determine in-depth residual stress profiles by surface treatment of metal components. Fatigue Fract. Eng. Mater. Struct. (2010, in press)Google Scholar
  20. 20.
    Buhmann, M.D.: Radial Basis Functions. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  21. 21.
    Aoki, S., Amaya, K., Sahashi, M., Nakamura, T.: Identification of Gurson’s material constants by using Kalman filter. Comput. Mech. 19, 501–506 (2007)CrossRefGoogle Scholar
  22. 22.
    Kansa, E.J.: Motivations for using radial basis functions to solve PDEs., pp. 1–8 (2001)
  23. 23.
    Holmes, P., Lumley, J.L., Berkoz, D.: Turbulence, coherent structures, dynamical systems and symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, UK (1996)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria StrutturalePolitecnico di MilanoMilanoItaly

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