Skip to main content
SpringerLink
Log in

kPCA-Based Parametric Solutions Within the PGD Framework

  • S.I.: Machine learning in computational mechanics
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

Parametric solutions make possible fast and reliable real-time simulations which, in turn allow real time optimization, simulation-based control and uncertainty propagation. This opens unprecedented possibilities for robust and efficient design and real-time decision making. The construction of such parametric solutions was addressed in our former works in the context of models whose parameters were easily identified and known in advance. In this work we address more complex scenarios in which the parameters do not appear explicitly in the model—complex microstructures, for instance. In these circumstances the parametric model solution requires combining a technique to find the relevant model parameters and a solution procedure able to cope with high-dimensional models, avoiding the well-known curse of dimensionality. In this work, kPCA (kernel Principal Component Analysis) is used for extracting the hidden model parameters, whereas the PGD (Proper Generalized Decomposition) is used for calculating the resulting parametric solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  1. Aghighi MS, Ammar A, Metivier C, Chinesta F (2015) Parametric solution of the Rayleigh-Bénard convection model by using the pgd: application to nanofluids. Int J Numer Methods Heat Fluid Flows 25(6):1252–1281

    Article  MATH  Google Scholar 

  2. Aguado JV, Huerta A, Chinesta F, Cueto E (2015) Real-time monitoring of thermal processes by reduced order modelling. Int J Numer Meth Eng 102(5):991–1017

    Article  MATH  Google Scholar 

  3. Ammar A, Mokdad B, Chinesta F, Keunings R (2006) A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J Non Newton Fluid Mech 139:153–176

    Article  MATH  Google Scholar 

  4. Ammar A, Mokdad B, Chinesta F, Keunings R (2007) A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. part ii. J Non Newton Fluid Mech 144:98–121

    Article  MATH  Google Scholar 

  5. Ammar A, Normandin M, Chinesta F (2010) Solving parametric complex fluids models in rheometric flows. J Non Newton Fluid Mech 165:1588–1601

    Article  MATH  Google Scholar 

  6. Ammar A, Chinesta F, Cueto E, Doblaré M (2012) Proper generalized decomposition of time-multiscale models. Int J Numer Meth Eng 90(5):569–596

    Article  MathSciNet  MATH  Google Scholar 

  7. Ammar A, Huerta A, Chinesta F, Cueto E, Leygue A (2014) Parametric solutions involving geometry: a step towards efficient shape optimization. Comput Methods Appl Mech Eng 268:178–193

    Article  MathSciNet  MATH  Google Scholar 

  8. Amsallem D, Cortial J, Farhat C (2010) Towards real-time cfd-based aeroelastic computations using a database of reduced-order information. AIAA J 48:2029–2037

    Article  Google Scholar 

  9. Amsallem D, Farhat C (2008) An interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J 46:1803–1813

    Article  Google Scholar 

  10. Athanasios C, Sorensen ADC, Gugercin S (2001) A survey of model reduction methods for large-scale systems. Contemp Math 280:193–220

    Article  MathSciNet  MATH  Google Scholar 

  11. Ballarin F, Manzoni A, Rozza G, Salsa S (2014) Shape optimization by free-form deformation: existence results and numerical solution for stokes flows. J Sci Comput 60(3):537–563

    Article  MathSciNet  MATH  Google Scholar 

  12. Barrault M, Maday Y, Nguyen NC, Patera AT (2004) An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. CR Math 339(9):667–672

    MathSciNet  MATH  Google Scholar 

  13. Bialecki RA, Kassab AJ, Fic A (2005) Proper orthogonal decomposition and modal analysis for acceleration of transient fem thermal analysis. Int J Numer Meth Engrg 62:774–797

    Article  MATH  Google Scholar 

  14. Bognet B, Bordeu F, Chinesta F, Leygue A, Poitou A (2012) Advanced simulation of models defined in plate geometries: 3d solutions with 2d computational complexity. Comput Methods Appl Mech Eng 201–204:1–12

    Article  MATH  Google Scholar 

  15. Bognet B, Leygue A, Chinesta F (2014) Separated representations of 3d elastic solutions in shell geometries. Adv Modell Simul Eng Sci 1(1):1–34

    Google Scholar 

  16. Bui-Thanh T, Willcox K, Ghattas O, Van Bloemen Waanders B (2007) Goal-oriented, model-constrained optimization for reduction of large-scale systems. J Comput Phys 224(2):880–896

    Article  MathSciNet  MATH  Google Scholar 

  17. Cancès E, Defranceschi M, Kutzelnigg W, Le Bris C, Maday Y (2003) Computational quantum chemistry: a primer. In: Handbook of numerical analysis, vol X, pp 3–270

  18. Chaturantabut S, Sorensen DC (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32:2737–2764

    Article  MathSciNet  MATH  Google Scholar 

  19. Chinesta F, Ammar A, Cueto E (2010) Proper generalized decomposition of multiscale models. Int J Numer Meth Eng 83(8–9):1114–1132

    Article  MATH  Google Scholar 

  20. Chinesta F, Ammar A, Cueto E (2010) Recent advances in the use of the proper generalized decomposition for solving multidimensional models. Arch Comput Methods Eng 17(4):327–350

    Article  MathSciNet  MATH  Google Scholar 

  21. Chinesta F, Ammar A, Leygue A, Keunings R (2011) An overview of the proper generalized decomposition with applications in computational rheology. J Non Newton Fluid Mech 166(11):578–592

    Article  MATH  Google Scholar 

  22. Chinesta F, Leygue A, Bordeu F, Aguado JV, Cueto E, Gonzalez D, Alfaro I, Ammar A, Huerta A (2013) PGD-based computational vademecum for efficient design, optimization and control. Arch Comput Methods Eng 20(1):31–59

    Article  MathSciNet  MATH  Google Scholar 

  23. Chinesta F, Keunings R, Leygue A (2014) The proper generalized decomposition for advanced numerical simulations. Springer, Switzerland

    Book  MATH  Google Scholar 

  24. Chinesta F, Ladeveze P, Cueto E (2011) A short review on model order reduction based on proper generalized decomposition. Arch Comput Methods Eng 18:395–404

    Article  Google Scholar 

  25. Dowell E, Hall K (2001) Modeling of fluid-structure interaction. Annu Rev Fluid Mech 33:445–490

    Article  MATH  Google Scholar 

  26. Ghnatios C, Chinesta F, Cueto E, Leygue A, Poitou A, Breitkopf P, Villon P (2011) Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: Application to pultrusion. Compos A Appl Sci Manuf 42(9):1169–1178

    Article  Google Scholar 

  27. Ghnatios C, Masson F, Huerta A, Leygue A, Cueto E, Chinesta F (2012) Proper generalized decomposition based dynamic data-driven control of thermal processes. Comput Methods Appl Mech Eng 213–216:29–41

    Article  Google Scholar 

  28. Girault M, Videcoq E, Petit D (2010) Estimation of time-varying heat sources through inversion of a low order model built with the modal identification method from in-situ temperature measurements. Int J Heat Mass Transf 53:206–219

    Article  MATH  Google Scholar 

  29. Gonzalez D, Masson F, Poulhaon F, Cueto E, Chinesta F (2012) Proper generalized decomposition based dynamic data driven inverse identification. Math Comput Simul 82:1677–1695

    Article  MathSciNet  Google Scholar 

  30. González D, Alfaro I, Quesada C, Cueto E, Chinesta F (2015) Computational vademecums for the real-time simulation of haptic collision between nonlinear solids. Comput Methods Appl Mech Eng 283:210–223

    Article  Google Scholar 

  31. Gonzalez D, Cueto E, Chinesta F (2014) Real-time direct integration of reduced solid dynamics equations. Int J Numer Methods Eng 99(9):633–653

    Article  MathSciNet  MATH  Google Scholar 

  32. González D, Cueto E, Chinesta F (2015) Computational patient avatars for surgery planning. Ann Biomed Eng 44(1):35–45

    Article  Google Scholar 

  33. El Halabi F, González D, Chico A, Doblaré M (2013) FE2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition. Comput Methods Appl Mech Eng 257:183–202

    Article  MATH  Google Scholar 

  34. Hesthaven J, Rozza G, Stamm B (2015) Certified reduced basis methods for parametrized partial differential equations. Springer, New York

    MATH  Google Scholar 

  35. Heyberger C, Boucard P-A, Neron D (2013) A rational strategy for the resolution of parametrized problems in the PGD framework. Comput Methods Appl Mech Eng 259:40–49

    Article  MathSciNet  MATH  Google Scholar 

  36. Karhunen K (1946) Uber lineare methoden in der wahrscheinlichkeitsrechnung. Ann Acad Sci Fennicae Ser Al Math Phys 37:1–79

    Google Scholar 

  37. Ladeveze P (1985) On a family of algorithms for structural mechanics (in french). Comptes Rendus Académie Des Sci Paris 300(2):41–44

    MathSciNet  MATH  Google Scholar 

  38. Ladeveze P (1989) The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables. Comptes Rendus Académie Des Sci Paris 309:1095–1099

    MATH  Google Scholar 

  39. Lamari H, Ammar A, Cartraud P, Legrain G, Jacquemin F, Chinesta F (2010) Routes for efficient computational homogenization of non-linear materials using the proper generalized decomposition. Arch Comput Methods Eng 17(4):373–391

    Article  MathSciNet  MATH  Google Scholar 

  40. Lassila T, Manzoni A, Quarteroni A, Rozza G (2013) A reduced computational and geometrical framework for inverse problems in hemodynamics. Int J Numer Methods Biomed Eng 29(7):741–776

    Article  MathSciNet  Google Scholar 

  41. Lassila T, Rozza G (2010) Parametric free-form shape design with pde models and reduced basis method. Comput Methods Appl Mech Eng 199(23):1583–1592

    Article  MathSciNet  MATH  Google Scholar 

  42. Laughlin RB, Pines D (2000) The theory of everything. Proc Nat Acad Sci 97(1):28–31

    Article  MathSciNet  Google Scholar 

  43. Lee JA, Verleysen M (2007) Nonlinear dimensionality reduction. Springer, New York

    Book  MATH  Google Scholar 

  44. Loève MM (1963) Probability theory. The university series in higher mathematics, 3rd edn. Van Nostrand, Princeton, NJ

    MATH  Google Scholar 

  45. Lopez E, Gonzalez D, Aguado JV, Abisset-Chavanne E, Lebel F, Upadhyay R, Cueto E, Binetruy C, Chinesta F (2016) A manifold learning approach for integrated computational materials engineering (in press)

  46. Lorenz EN (1956) Empirical orthogonal functions and statistical weather prediction. MIT, Departement of Meteorology, Scientific Report Number 1, Statistical Forecasting Project

  47. Maday Y, Patera AT, Turinici G (2002) A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J Sci Comput 17(1-4):437–446

    Article  MathSciNet  MATH  Google Scholar 

  48. Maday Y, Ronquist EM (2004) The reduced basis element method: application to a thermal fin problem. SIAM J Sci Comput 26(1):240–258

    Article  MathSciNet  MATH  Google Scholar 

  49. Manzoni A, Quarteroni A, Rozza G (2012) Computational reduction for parametrized pdes: strategies and applications. Milan J Math 80:283–309

    Article  MathSciNet  MATH  Google Scholar 

  50. Manzoni A, Quarteroni A, Rozza G (2012) Model reduction techniques for fast blood flow simulation in parametrized geometries. Int J Numer Methods Biomed Eng 28(6–7):604–625

    Article  MathSciNet  Google Scholar 

  51. Manzoni A, Quarteroni A, Rozza G (2012) Shape optimization for viscous flows by reduced basis methods and free-form deformation. Int J Numer Meth Fluids 70(5):646–670

    Article  MathSciNet  Google Scholar 

  52. Mena A, Bel D, Alfaro I, Gonzalez D, Cueto E, Chinesta F (2015) Towards a pancreatic surgery simulator based on model order reduction. Adv Model Simul Eng Sci 2(1):31

    Article  Google Scholar 

  53. Millán D, Arroyo M (2013) Nonlinear manifold learning for model reduction in finite elastodynamics. Comput Methods Appl Mech Eng 261–262:118–131

    Article  MathSciNet  MATH  Google Scholar 

  54. Niroomandi S, González D, Alfaro I, Bordeu F, Leygue A, Cueto E, Chinesta F (2013) Real-time simulation of biological soft tissues: a PGD approach. Int J Numer Methods Biomed Eng 29(5):586–600

    Article  MathSciNet  Google Scholar 

  55. Niroomandi S, Gonzalez D, Alfaro I, Cueto E, Chinesta F (2013) Model order reduction in hyperelasticity: a proper generalized decomposition approach. Int J Numer Meth Eng 96(3):129–149

    MathSciNet  MATH  Google Scholar 

  56. Nouy A (2007) A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput Methods Appl Mech Eng 196:4521–4537

    Article  MathSciNet  MATH  Google Scholar 

  57. Willcox K, Benner P, Gugercin S (2016) A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev 57(4):483–531

    MathSciNet  MATH  Google Scholar 

  58. Park HM, Cho DH (1996) The use of the Karhunen-Loève decomposition for the modeling of distributed parameter systems. Chem Eng Sci 51(1):81–98

    Article  Google Scholar 

  59. Patera AT, Rozza G (2007) Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Technical report, MIT Pappalardo Monographs in Mechanical Engineering

  60. Pruliere E, Chinesta F, Ammar A (2010) On the deterministic solution of multidimensional parametric models using the proper generalized decomposition. Math Comput Simul 81(4):791–810

    Article  MathSciNet  MATH  Google Scholar 

  61. Quarteroni A, Rozza G, Manzoni A (2011) Certified reduced basis approximation for parametrized pde and applications. J Math Ind 1(1):1–49

    Article  MATH  Google Scholar 

  62. Quarteroni A, Rozza G (2003) Optimal control and shape optimization of aorto-coronaric bypass anastomoses. Math Models Methods Appl Sci 13(12):1801–1823

    Article  MathSciNet  MATH  Google Scholar 

  63. Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500):2323–2326

    Article  Google Scholar 

  64. Rozza G (2014) Fundamentals of reduced basis method for problems governed by parametrized pdes and applications. In: Ladeveze P, Chinesta F (eds) CISM lectures notes “Separated Representation and PGD based model reduction: fundamentals and applications”. Springer, New York

    Google Scholar 

  65. Ronquist EM, Maday Y (2002) A reduced-basis element method. C R Acad Sci Paris Ser I 335:195–200

    Article  MathSciNet  MATH  Google Scholar 

  66. Rozza G, Huynh DBP, Patera AT (2008) Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations - application to transport and continuum mechanics. Arch Comput Methods Eng 15(3):229–275

    Article  MathSciNet  MATH  Google Scholar 

  67. Ryckelynck D (2003) A priori model reduction method for the optimization of complex problems. In: Workshop on optimal design of materials and structures, ecole polytechnique, Palaiseau, Paris (France)

  68. Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202(1):346–366

    Article  MathSciNet  MATH  Google Scholar 

  69. Ryckelynck D, Chinesta F, Cueto E, Ammar A (2006) On the a priori model reduction: overview and recent developments. Arch Comput Methods Eng 12(1):91–128

    Article  MathSciNet  MATH  Google Scholar 

  70. Ryckelynck D, Hermanns L, Chinesta F, Alarcon E (2005) An efficient ‘a priori’ model reduction for boundary element models. Eng Anal Bound Elem 29(8):796–801

    Article  MATH  Google Scholar 

  71. Scholkopf B, Smola A, Muller KR (1999) Kernel principal component analysis. In: Advances in kernel methods—suport vector learning. MIT Press, New York, pp 327–352

  72. Schölkopf B, Smola A, Müller K-R (1998) Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput 10(5):1299–1319

    Article  Google Scholar 

  73. Videcoq E, Quemener O, Lazard M, Neveu A (2008) Heat source identification and on-line temperature control by a branch eigenmodes reduced model. Int J Heat Mass Transf 51:4743–4752

    Article  MATH  Google Scholar 

  74. Vitse M, Neron D, Boucard P-A (2014) Virtual charts of solutions for parametrized nonlinear equations. Comput Mech 54(6):1529–1539

    Article  MathSciNet  MATH  Google Scholar 

  75. Volkwein S (2001) Model reduction using proper orthogonal decomposition. Technical report, lecture notes. Institute of Mathematics and Scientific Computing, University of Graz

  76. Wang Q (2012) Kernel principal component analysis and its applications in face recognition and active shape models. ArXiv preprint arXiv:1207.3538

  77. Zimmer VA, Lekadir K, Hoogendoorn C, Frangi AF, Piella G (2015) A framework for optimal kernel-based manifold embedding of medical image data. Comput Med Imaging Graph 41:93–107 Machine Learning in Medical Imaging

    Article  Google Scholar 

Download references

Acknowledgments

This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grant number CICYT DPI2014-51844-C2-1-R and by the Regional Government of Aragon and the European Social Fund, research group T88. Professor Chinesta is also supported by the Institut Universitaire de France.

Author information

Authors and Affiliations

  1. I3A, Universidad de Zaragoza, Maria de Luna s/n, 50018, Zaragoza, Spain

    D. González & E. Cueto

  2. ESI Chair “Advanced Computational Manufacturing Processes”, ICI - High Performance Computing Institute @ Ecole centrale de Nantes, Institut Universtaire de France, 1 rue de la Noe, BP 92101, 44321, Nantes Cedex 3, France

    J. V. Aguado & F. Chinesta

  3. GEM, UMR CNRS - Centrale Nantes, 1 rue de la Noe, BP 92101, 44321, Nantes Cedex 3, France

    E. Abisset-Chavanne

Authors
  1. D. González
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. V. Aguado
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Cueto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Abisset-Chavanne
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. F. Chinesta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. González.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

González, D., Aguado, J.V., Cueto, E. et al. kPCA-Based Parametric Solutions Within the PGD Framework. Arch Computat Methods Eng 25, 69–86 (2018). https://doi.org/10.1007/s11831-016-9173-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-016-9173-4

Search

Navigation