Skip to main content
SpringerLink
Log in

POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, a reduced-order model (ROM) based on the proper orthogonal decomposition and the discrete empirical interpolation method is proposed for efficiently simulating time-fractional partial differential equations (TFPDEs). Both linear and nonlinear equations are considered. We demonstrate the effectiveness of the ROM by several numerical examples, in which the ROM achieves the same accuracy of the full-order model (FOM) over a long-term simulation while greatly reducing the computational cost. The proposed ROM is then regarded as a surrogate of FOM and is applied to an inverse problem for identifying the order of the time-fractional derivative of the TFPDE model. Based on the Levenberg–Marquardt regularization iterative method with the Armijo rule, we develop a ROM-based algorithm for solving the inverse problem. For cases in which the observation data is either uncontaminated or contaminated by random noise, the proposed approach is able to achieve accurate parameter estimation efficiently.

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

Similar content being viewed by others

References

  1. Armijo, L.: Minimization of functions having Lipschitz continuous partial derivatives. Pac. J. Math. 16, 1–3 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems, vol. 6. Society for Industrial and Applied Mathematics, Philadelphia, PA (2005)

    Book  MATH  Google Scholar 

  3. Benson, D., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36(6), 1413–1423 (2000)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Burkardt, J.V., Gunzburger, M., Lee, H.C.: POD and CVT-based reduced-order modeling of Navier-Stokes flows. Comput. Methods Appl. Mech. Eng. 196(1–3), 337–355 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carlberg, K., Farhat, C.: A low-cost, goal-oriented ’compact proper orthogonal decomposition’ basis for model reduction of static systems. Int. J. Numer. Meth. Eng. 86(3), 381–402 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaturantabut, S., Sorensen, D.C.: Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media. Math. Comput. Model. Dyn. 17(4), 337–353 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chaturantabut, S., Sorensen, D.C.: A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal. 50, 46–63 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chaturantabut, S., Sorensen, D.C., Steven, J.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chavent, G.: Nonlinear Least Squares for Inverse Problems: Theoretical Foundations and Step-by-Step Guide for Applications. Springer, Netherlands (2009)

    MATH  Google Scholar 

  11. Chen, S., Liu, F., Jiang, X., Turner, I., Burrage, K.: Fast finite difference approximation for identifying parameters in a two-dimensional space-fractional nonlocal model with variable diffusivity coefficients. SIAM J. Numer. Anal. 54(2), 606–624 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng, J., Nakagawa, J., Yamamoto, M., Yamazaki, T.: Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl. 25(11), 115002 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Daescu, D.N., Navon, I.M.: A dual-weighted approach to order reduction in 4DVAR data assimilation. Mon. Weather Rev. 136(3), 1026–1041 (2008)

    Article  Google Scholar 

  14. del-Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Fractional diffusion in plasma turbulence. Phys. Plasmas 11(8), 3854 (2004)

  15. Gal, N., Weihs, D.: Experimental evidence of strong anomalous diffusion in living cells. Phys. Rev. E 81(2), 020903 (2010)

    Article  Google Scholar 

  16. Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68(1), 46–53 (1995)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  18. Iollo, A., Lanteri, S., Désidéri, J.A.: Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations. Theor. Comput. Fluid Dyn. 13(6), 377–396 (2000)

    Article  MATH  Google Scholar 

  19. Jin, B., Rundell, W.: An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Probl. 28(7), 075010 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ke, R., Ng, M.K., Sun, H.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations. J. Comput. Phys. 303, 203–211 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90(1), 117–148 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lu, X., Pang, H., Sun, H.: Fast approximate inversion of block triangular Toeplitz matrices with application to sub-diffusion equations. Numer. Linear Algebra Appl. 22(5), 866–882 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Maday, Y., Rønquist, E.M.: A reduced-basis element method. J. Sci. Comput. 17(1–4), 447–459 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Redding (2006)

    Google Scholar 

  26. Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics, vol. 43. Walter de Gruyter, Berlin (2012)

  27. Metler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Metler, R., Klafter, J.: The restaurant at the end of random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37(31), R161–R208 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006)

    MATH  Google Scholar 

  30. Patera A.T., Rozza, G.: Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Version 1.0, Copyright MIT 2006, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering

  31. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  32. Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: An empirical study. Phys. A: Stat. Mech. Appl. 314(1–4), 749–755 (2002)

    Article  MATH  Google Scholar 

  33. Stefănescu, R., Navon, I.M.: POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model. J. Comput. Phys. 237, 95–114 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sirisup, S., Karniadakis, G.E.: A spectral viscosity method for correcting the long-term behavior of POD models. J. Comput. Phys. 194(1), 92–116 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sun, W., Yuan, Y.: Optimization Theory and Methods: Nonlinear Programming. Springer, New York (2006)

    MATH  Google Scholar 

  36. 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, pp. 235–273. Springer, Switzerland (2014)

  37. Wang, J., Wei, T., Zhou, Y.: Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Appl. Math. Model. 37(18–19), 8518–8532 (2013)

    Article  MathSciNet  Google Scholar 

  38. Wei, H., Chen, W., Sun, H., Li, X.: A coupled method for inverse source problem of spatial fractional anomalous diffusion equations. Inverse Probl. Sci. Eng. 18(7), 945–956 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xu, Q., Hesthaven, J.S., Chen, F.: A parareal method for time-fractional differential equations. J. Comput. Phys. 293, 173–183 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhuang, Q., Yu, B., Jiang, X.: An inverse problem of parameter estimation for time fractional heat conduction in a composite medium using carbon-carbon experimental data. Phys. B: Condens. Matter. 456, 9–15 (2015)

    Article  Google Scholar 

  41. Wang, Z.: Nonlinear model reduction based on the finite element method with interpolated coefficients: semilinear parabolic equations. Numer. Methods Partial. Differ. Eqs. 31(6), 1713–1741 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zeng, F., Zhang, Z., Karniadakis, G.E.: Fast difference schemes for solving high-dimensional time-fractional subdiffusion equation. J. Comput. Phys. 307, 15–33 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author would like to thank the support of China Scholarship Council for visiting the Interdisciplinary Mathematics Institute at University of South Carolina during the year 2015–2016.

Author information

Authors and Affiliations

  1. College of Science, China University of Petroleum, Qingdao, 266580, China

    Hongfei Fu

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Hong Wang & Zhu Wang

Authors
  1. Hongfei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhu Wang.

Additional information

The authors are grateful for the supports from the National Natural Science Foundation of China through Grants 11201485, 11471194, 11571115 and 91630207, the Fundamental Research Funds for the Central Universities through Grants 14CX02217A and 16CX02050A, the National Science Foundation through Grants DMS-1522672, DMS-1216923 and DMS-1620194, the OSD/ARO MURI Grant W911NF-15-1-0562, and the office of the Vice President for Research at University of South Carolina from the ASPIRE-I program.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fu, H., Wang, H. & Wang, Z. POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification. J Sci Comput 74, 220–243 (2018). https://doi.org/10.1007/s10915-017-0433-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0433-8

Keywords

Mathematics Subject Classification

Search

Navigation