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Isogeometric analysis and proper orthogonal decomposition for parabolic problems

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Abstract

We investigate the combination of Isogeometric Analysis (IGA) and proper orthogonal decomposition (POD) based on the Galerkin method for model order reduction of linear parabolic partial differential equations. For the proposed fully discrete scheme, the associated numerical error features three components due to spatial discretization by IGA, time discretization with the \(\theta \)-scheme, and eigenvalue truncation by POD. First, we prove a priori error estimates of the spatial IGA semi-discrete scheme. Then, we show stability and prove a priori error estimates of the space-time discrete scheme and the fully discrete IGA-\(\theta \)-POD Galerkin scheme. Numerical tests are provided to show efficiency and accuracy of NURBS-based IGA for model order reduction in comparison with standard finite element-based POD techniques.

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References

  1. Bazilevs, Y., Beirão de Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16, 1031–1090 (2006)

  2. Bazilevs, Y., Calo, V.M., Hughes, T.J.R., Zhang, Y.: Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput. Mech. 43, 3–37 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beirão de Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for \(h\)-\(p\)-\(k\)-refinement in isogeometric analysis. Numer. Math. 118, 271–305 (2011)

  4. Bergmann, M., Bruneau, C.H., Iollo, A.: Enablers for robust POD models. J. Comput. Phys. 228, 516–538 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575 (1993)

    Article  MathSciNet  Google Scholar 

  6. Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buffa, A., Sangalli, G., Vázquez, R.: Isogeometric analysis in electromagnetics: B-splines approximation. J. Comput. Phys. 199, 1143–1152 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Carlberg, K., Charbel, B.M., Farhat, C.: Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. Int. J. Numer. Meth. Eng. 86, 155–181 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chapelle, D., Gariah, A., Sainte-Marie, J.: Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples. ESAIM Math. Model. Numer. Anal. 46, 731–757 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009)

    Book  Google Scholar 

  12. Cottrell, J.A., Hughes, T.J.R., Reali, A.: Studies of refinement and continuity in Isogeometric structural analysis. Comput. Methods Appl. Mech. Eng. 196, 4160–4183 (2007)

    Article  MATH  Google Scholar 

  13. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, in Evolution Problems I, vol. 5. Springer, Berlin (1992)

  14. Dedè, L., Borden, M.J., Hughes, T.J.R.: Isogeometric analysis for topology optimization with a phase field model. Arch. Comput. Methods Eng. 19, 427–465 (2012)

    Article  MathSciNet  Google Scholar 

  15. Dedè, L., Jäggli, C., Quarteroni, A.: Isogeometric numerical dispersion analysis for elastic wave propagation. Comput. Methods Appl. Mech. Eng. 284, 320–348 (2015)

    Article  MathSciNet  Google Scholar 

  16. Dedè, L., Santos, H.A.F.A.: B-spline goal-oriented error estimators for geometrically nonlinear rods. Comput. Mech. 49, 35–52 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Evans, J.A., Bazilevs, Y., Babuška, I., Hughes, T.J.R.: \(n\)-Widths, sup-infs, and optimality ratios for the \(k\)-version of the isogeometric finite element method. Comput. Methods Appl. Mech. Eng. 198, 1726–1741 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. de Falco, C., Reali, A., Vàzquez, R.: GeoPDEs: a research tool for isogeometric analysis of PDEs. Adv. Eng. Softw. 42, 1020–1034 (2011)

    Article  MATH  Google Scholar 

  19. Fang, F., Pain, C.C., Navon, I.M., Elsheikh, A.H., Du, J., Xiao, D.: Non-linear Petrov–Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods. J. Comput. Phys. 234, 540–559 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Galàn del Sastre, P., Bermejo, R.: Error estimates of proper orthogonal decomposition eigenvectors and Galerkin projection for a general dynamical system arising in fluid models. Numer. Math. 110, 49–81 (2008)

  21. Gunzburger, M.D., Peterson, J., Shadid, J.N.: Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput. Methods Appl. Mech. Eng. 196, 1030–1047 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Henri, T., Yvon, J.P.: Stability of the POD and convergence of the POD Galerkin method for parabolic problems. IRMAR No 02-40 (2002)

  23. Henri, T., Yvon, J.P.: Convergence estimates of POD-Galerkin methods for parabolic problems. In: System Modeling and Optimization, IFIP International Federation for Information Processing, vol. 166, pp. 295–306 (2005)

  24. Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, New York (1996)

    Book  MATH  Google Scholar 

  25. Hughes, T.J.R., Cottrell, J., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hughes, T.J.R., Evans, J.A., Reali, A.: Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Eng. 272, 290–320 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hughes, T.J.R., Feijoo, G.R., Mazzei, L., Quincy, J.B.: The variational multiscale method-a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166, 3–24 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Iliescu, T., Wang, Z.: Variational multiscale proper orthogonal decomposition: convection dominated convection-diffusion-reaction equations. Math. Comput. 82, 1357–1378 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Iliescu, T., Wang, Z.: Are the snapshot difference quotients needed in the proper orthogonal decomposition? SIAM J. Sci. Comput. 36, A1221–A1250 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Iliescu, T., Wang, Z.: Variational multiscale proper orthogonal decomposition: Navier–Stokes equations. Numer. Methods Part. Differ. Equ. 30, 641–663 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kunisch, K., Volkwein, S.: Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102, 345–371 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117–148 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: Model order reduction in fluid dynamics: challenges and perspectives. In: Reduced Order Methods for Modeling and Computational Reduction, vol. 9, pp. 235–273. Springer Milano, MS&A Series (2013)

  35. Lass, O., Volkwein, S.: POD Galerkin schemes for nonlinear-parabolic systems. SIAM J. Sci. Comput. 35, A1271–1298 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Luo, Z., Chen, J., Navon, I.M., Yang, X.: Mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations. SIAM. J. Numer. Anal. 47, 1–19 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Luo, Z., Xie, Z., Shang, Y., Chen, J.: A reduced finite volume element formulation and numerical simulations based on POD for parabolic equations. J. Comput. Appl. Math. 235, 2098–2111 (2011)

    Article  MATH  Google Scholar 

  38. Piegl, L., Tiller, W.: The NURBS Book. Springer, New York (1997)

    Book  MATH  Google Scholar 

  39. Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Indian 1, 1–44 (2011)

    MathSciNet  MATH  Google Scholar 

  40. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Problems. Springer, Berlin (1997)

    MATH  Google Scholar 

  41. Rathinam, M., Petzold, L.R.: A new look at proper orthogonal decomposition. SIAM J. Numer. Anal. 41, 1893–1925 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rowley, C.W.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurc. Chaos 15, 997–1013 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Schillinger, D., Dedè, L., Scott, M.A., Evans, J.A., Borden, M.J., Rank, E., Hughes, T.J.R.: An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput. Methods Appl. Mech. Eng. 249–252, 116–150 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Schmidt, A., Potschka, A., Krkel, S., Bock, H.G.: Derivative-extended POD reduced-order modeling for parameter estimation. SIAM J. Sci. Comput. 35, A2696–A2717 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Singler, J.R.: Balanced POD for model reduction of linear PDE systems: convergence theory. Numer. Math. 121, 127–164 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Singler, J.R.: New POD error expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs. SIAM J. Numer. Anal. 52, 852–876 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tagliabue, A., Dedè, L., Quarteroni, A.: Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics. Comput. Fluids 102, 277–303 (2014)

    Article  MathSciNet  Google Scholar 

  48. Volkwein, S.: Proper orthogonal decomposition: theory and reduced-order modelling. Universität Konstanz, Lecture Notes (2013)

  49. Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA 40, 2323–2330 (2002)

    Article  Google Scholar 

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Acknowledgments

The authors acknowledge Dr. Peng Chen and Dr. Toni Lassila for helpful discussions on model order reduction and Prof. Fabio Nobile for insights on error estimates. The help of Federico Negri with the library MLife developed by Prof. Fausto Saleri is acknowledged. The use of the IGA library GeoPDEs [18] is also acknowledged. S. Zhu greatly appreciates the warm hospitality of Prof. Quarteroni and other group members throughout his visit as a postdoctoral researcher at CMCS-MATHICSE-EPFL.

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Authors and Affiliations

  1. Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Department of Mathematics, East China Normal University, Shanghai, 200241, People’s Republic of China

    Shengfeng Zhu

  2. CMCS-Chair of Modeling and Scientific Computing, MATHICSE-Mathematics Institute of Computational Science and Engineering, EPFL-École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland

    Luca Dedè & Alfio Quarteroni

  3. MOX-Modeling and Scientific Computing, Department of Mathematics, Politecnico di Milano, Piazza L. da Vinci 32, 20133, Milan, Italy

    Alfio Quarteroni

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  1. Shengfeng Zhu
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Correspondence to Shengfeng Zhu.

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S. Zhu was financially supported in part by the National Natural Science Foundation of China under Grant 11201153 and a postdoctoral scholarship from China Scholarship Council.

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Zhu, S., Dedè, L. & Quarteroni, A. Isogeometric analysis and proper orthogonal decomposition for parabolic problems. Numer. Math. 135, 333–370 (2017). https://doi.org/10.1007/s00211-016-0802-5

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