Advertisement

Combined Parameter and Model Reduction of Cardiovascular Problems by Means of Active Subspaces and POD-Galerkin Methods

  • Marco Tezzele
  • Francesco Ballarin
  • Gianluigi Rozza
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 16)

Abstract

In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency.

Notes

Acknowledgements

This work was partially supported by the INDAM-GNCS 2017 project “Advanced numerical methods combined with computational reduction techniques for parameterised PDEs and applications”, and by European Union Funding for Research and Innovation—Horizon 2020 Program—in the framework of European Research Council Executive Agency: H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics” P.I. Gianluigi Rozza.

References

  1. 1.
    Agoshkov, V., Quarteroni, A., Rozza, G.: A mathematical approach in the design of arterial bypass using unsteady Stokes equations. J. Sci. Comput. 28, 139–165 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agoshkov, V., Quarteroni, A., Rozza, G.: Shape design in aorto-coronaric bypass anastomoses using perturbation theory. SIAM J. Numer. Anal. 44(1), 367–384 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ali, S., Ballarin, F., Rozza, G.: Stabilized reduced basis methods for parametrized Stokes and Navier-Stokes equations. (2018, in preparation)Google Scholar
  4. 4.
    Ambrosi, D., Quarteroni, A., Rozza, G.: Modeling of Physiological Flows. MS&A – Modeling, Simulation and Applications, vol. 5. Springer, Berlin (2012)Google Scholar
  5. 5.
    Ballarin, F., Manzoni, A., Rozza, G., Salsa, S.: Shape optimization by Free-Form Deformation: existence results and numerical solution for Stokes flows. J. Sci. Comput. 60(3), 537–563 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ballarin, F., Manzoni, A., Quarteroni, A., Rozza, G.: Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations. Int. J. Numer. Methods Eng. 102(5), 1136–1161 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ballarin, F., Faggiano, E., Ippolito, S., Manzoni, A., Quarteroni, A., Rozza, G., Scrofani, R.: Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD–Galerkin method and a vascular shape parametrization. J. Comput. Phys. 315, 609–628 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ballarin, F., Sartori, A., Rozza, G.: RBniCS – reduced order modelling in fenics (2016). http://mathlab.sissa.it/rbnics
  9. 9.
    Ballarin, F., D’Amario, A., Perotto, S., Rozza, G.: A POD-selective inverse distance weighting method for fast parametrized shape morphing (2017, submitted). arXiv preprint arXiv:1710.09243Google Scholar
  10. 10.
    Ballarin, F., Faggiano, E., Manzoni, A., Quarteroni, A., Rozza, G., Ippolito, S., Antona, C., Scrofani, R.: Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts. Biomech. Model. Mechanobiol. 16(4), 1373–1399 (2017)CrossRefGoogle Scholar
  11. 11.
    Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9), 667–672 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.): Model Reduction of Parametrized Systems. MS&A – Modeling, Simulation and Applications, vol. 17. Springer, Berlin (2017)Google Scholar
  13. 13.
    Berkooz, G., Holmes, P., Lumley, J.: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25(1), 539–575 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Box, G.E., Draper, N.R.: Empirical Model-Building and Response Surfaces, vol. 424. Wiley, New York (1987)zbMATHGoogle Scholar
  15. 15.
    Brown, S.A.: Building supermodels: emerging patient avatars for use in precision and systems medicine. Front. Physiol. 6, 318 (2015)CrossRefGoogle Scholar
  16. 16.
    Buhmann, M.D.: Radial Basis Functions: Theory and Implementations, vol. 12. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  17. 17.
    Caiazzo, A., Iliescu, T., John, V., Schyschlowa, S.: A numerical investigation of velocity-pressure reduced order models for incompressible flows. J. Comput. Phys. 259, 598–616 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chen, P., Quarteroni, A.: Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraint. SIAM/ASA J. Uncertain. Quantif. 2(1), 364–396 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chen, P., Quarteroni, A., Rozza, G.: Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 59(1), 187–216 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chinesta, F., Keunings, R., Leygue, A.: The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer. Springer Science & Business Media, Berlin (2013)zbMATHGoogle Scholar
  22. 22.
    Constantine, P.G.: Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies, vol. 2. SIAM, Philadelphia (2015)CrossRefGoogle Scholar
  23. 23.
    Constantine, P., Gleich, D.: Computing active subspaces with Monte Carlo. arXiv preprint arXiv:1408.0545 (2015)Google Scholar
  24. 24.
    Constantine, P.G., Emory, M., Larsson, J., Iaccarino, G.: Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet. J. Comput. Phys. 302, 1–20 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Constantine, P.G., Eftekhari, A., Ward, R.: A near-stationary subspace for ridge approximation (2016). arXiv preprint arXiv:1606.01929Google Scholar
  26. 26.
    Constantine, P., Howard, R., Glaws, A., Grey, Z., Diaz, P., Fletcher, L.: Python active-subspaces utility library. J. Open Source Softw. 1(5) (2016)CrossRefGoogle Scholar
  27. 27.
    Cook, R.D.: Regression Graphics: Ideas for Studying Regressions Through Graphics, vol. 482. Wiley, New York (2009)zbMATHGoogle Scholar
  28. 28.
    Cueto, E., Chinesta, F.: Real time simulation for computational surgery: a review. Adv. Model. Simul. Eng. Sci. 1(1), 11:1–11:18 (2014)CrossRefGoogle Scholar
  29. 29.
    Devore, J.L.: Probability and Statistics for Engineering and the Sciences. Cengage Learning, Boston (2015)Google Scholar
  30. 30.
    Doorly, D., Sherwin, S.: Geometry and flow. In: Formaggia, L., Quarteroni, A., Veneziani, A. (eds.) Cardiovascular Mathematics. MS&A – Modeling, Simulation and Applications, vol. 1. Springer Italia, Milano (2009)Google Scholar
  31. 31.
    Dryden, I., Mardia, K.: Statistical Analysis of Shape. Wiley, New York (1998)zbMATHGoogle Scholar
  32. 32.
    Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive Theory of Functions of Several Variables, pp. 85–100. Springer, Berlin (1977)CrossRefGoogle Scholar
  33. 33.
    Forti, D., Rozza, G.: Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid–structure interaction coupling problems. Int. J. Comput. Fluid Dyn. 28(3–4), 158–169 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Frey, P., George, P.: Mesh generation. Application to finite elements. Hermes Science Publishing, Paris, Oxford (2000)zbMATHGoogle Scholar
  35. 35.
    González, D., Cueto, E., Chinesta, F.: Computational patient avatars for surgery planning. Ann. Biomed. Eng. 44(1), 35–45 (2016)CrossRefGoogle Scholar
  36. 36.
    Guibert, R., Mcleod, K., Caiazzo, A., Mansi, T., Fernández, M.A., Sermesant, M., Pennec, X., Vignon-Clementel, I.E., Boudjemline, Y., Gerbeau, J.F.: Group-wise construction of reduced models for understanding and characterization of pulmonary blood flows from medical images. Med. Image Anal. 18(1), 63–82 (2014)CrossRefGoogle Scholar
  37. 37.
    Gunzburger, M.D.: Perspectives in Flow Control and Optimization, vol. 5. SIAM, Philadelphia (2003)zbMATHGoogle Scholar
  38. 38.
    Hesthaven, J.S., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics. Springer, Berlin (2015)Google Scholar
  39. 39.
    Hokanson, J.M., Constantine, P.G.: Data-driven polynomial ridge approximation using variable projection (2017). arXiv preprint arXiv:1702.05859Google Scholar
  40. 40.
    Hu, X., Parks, G.T., Chen, X., Seshadri, P.: Discovering a one-dimensional active subspace to quantify multidisciplinary uncertainty in satellite system design. Adv. Space Res. 57(5), 1268–1279 (2016)CrossRefGoogle Scholar
  41. 41.
    INRIA 3D Meshes Research Database. Available at: https://www.rocq.inria.fr/gamma/gamma/download/download.php
  42. 42.
    Jefferson, J.L., Gilbert, J.M., Constantine, P.G., Maxwell, R.M.: Reprint of: Active subspaces for sensitivity analysis and dimension reduction of an integrated hydrologic model. Comput. Geosci. 90, 78–89 (2016)CrossRefGoogle Scholar
  43. 43.
    Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems, vol. 160. Springer Science & Business Media, Berlin (2006)zbMATHGoogle Scholar
  44. 44.
    Keiper, S.: Analysis of generalized ridge functions in high dimensions. In: 2015 International Conference on Sampling Theory and Applications (SampTA), pp. 259–263. IEEE, New York (2015)Google Scholar
  45. 45.
    Lassila, T., Rozza, G.: Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Eng. 199(23–24), 1583–1592 (2010)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Logg, A., Mardal, K.A., Wells, G.N.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)CrossRefGoogle Scholar
  47. 47.
    Lukaczyk, T.W., Constantine, P., Palacios, F., Alonso, J.J.: Active subspaces for shape optimization. In: 10th AIAA Multidisciplinary Design Optimization Conference, p. 1171 (2014)Google Scholar
  48. 48.
    Manzoni, A., Quarteroni, A., Rozza, G.: Model reduction techniques for fast blood flow simulation in parametrized geometries. Int. J. Numer. Methods Biomed. Eng. 28(6–7), 604–625 (2012)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Marsden, A.L.: Optimization in cardiovascular modeling. Annu. Rev. Fluid Mech. 46(1), 519–546 (2014)MathSciNetCrossRefGoogle Scholar
  50. 50.
    McLeod, K., Caiazzo, A., Fernández, M., Mansi, T., Vignon-Clementel, I., Sermesant, M., Pennec, X., Boudjemline, Y., Gerbeau, J.F.: Atlas-based reduced models of blood flows for fast patient-specific simulations. In: Camara, O., Pop, M., Rhode, K., Sermesant, M., Smith, N., Young, A. (eds.) Statistical Atlases and Computational Models of the Heart. Lecture Notes in Computer Science, vol. 6364, pp. 95–104. Springer, Berlin/Heidelberg (2010)CrossRefGoogle Scholar
  51. 51.
    Metropolis, N., Ulam, S.: The monte carlo method. J. Am. Stat. Assoc. 44(247), 335–341 (1949)CrossRefGoogle Scholar
  52. 52.
    Morris, M.: Factorial sampling plans for preliminary computational experiments. Technometrics 33(2), 161–174 (1991)CrossRefGoogle Scholar
  53. 53.
    Morris, A., Allen, C., Rendall, T.: CFD-based optimization of aerofoils using radial basis functions for domain element parameterization and mesh deformation. Int. J. Numer. Methods Fluids 58(8), 827–860 (2008)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Pinkus, A.: Ridge Functions, vol. 205. Cambridge University Press, Cambridge (2015)CrossRefGoogle Scholar
  55. 55.
    PyGeM: Python Geometrical Morphing. Available at https://github.com/mathLab/PyGeM
  56. 56.
    Quarteroni, A., Rozza, G.: Reduced Order Methods for Modeling and Computational Reduction. MS&A – Modeling, Simulation and Applications, vol. 9. Springer, Berlin (2014)Google Scholar
  57. 57.
    Ravindran, S.: A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Meth. Fluids 34, 425–448 (2000)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196(7), 1244–1260 (2007)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Sederberg, T., Parry, S.: Free-Form Deformation of solid geometric models. In: Proceedings of SIGGRAPH - Special Interest Group on GRAPHics and Interactive Techniques, pp. 151–159. SIGGRAPH (1986)Google Scholar
  60. 60.
    Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings-1968 ACM National Conference, pp. 517–524. ACM, New York (1968)Google Scholar
  61. 61.
    Stabile, G., Rozza, G.: Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations. Comput. Fluids (2018). https://doi.org/10.1016/j.compfluid.2018.01.035 MathSciNetCrossRefGoogle Scholar
  62. 62.
    Tezzele, M., Salmoiraghi, F., Mola, A., Rozza, G.: Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems. Adv. Model. Simul. Eng. Sci. (2018, in press). Preprint, arXiv:1709.03298Google Scholar
  63. 63.
    Torlo, D., Ballarin, F., Rozza, G.: Stabilized reduced basis methods for advection dominated partial differential equations with random inputs (2017, submitted)Google Scholar
  64. 64.
    Venturi, L., Ballarin, F., Rozza, G.: Weighted POD–Galerkin methods for parametrized partial differential equations in uncertainty quantification problems (2017, submitted)Google Scholar
  65. 65.
    Wang, V.Y., Hoogendoorn, C., Frangi, A.F., Cowan, B.R., Hunter, P.J., Young, A.A., Nash, M.P.: Automated personalised human left ventricular FE models to investigate heart failure mechanics. In: Proceedings of the Third International Conference on Statistical Atlases and Computational Models of the Heart: Imaging and Modelling Challenges, STACOM’12, pp. 307–316. Springer, Berlin, Heidelberg (2013)CrossRefGoogle Scholar
  66. 66.
    Witteveen, J., Bijl, H.: Explicit mesh deformation using Inverse Distance Weighting interpolation. In: 19th AIAA Computational Fluid Dynamics. AIAA, Washington (2009)Google Scholar
  67. 67.
    Zarins, C.K., Giddens, D.P., Bharadvaj, B., Sottiurai, V.S., Mabon, R.F., Glagov, S.: Carotid bifurcation atherosclerosis: quantitative correlation of plaque localization with flow velocity profiles and wall shear stress. Circ. Res. 53(4), 502–514 (1983)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Marco Tezzele
    • 1
  • Francesco Ballarin
    • 1
  • Gianluigi Rozza
    • 1
  1. 1.Mathematics Area, mathLabSISSATriesteItaly

Personalised recommendations