Advertisement

Journal of Scientific Computing

, Volume 67, Issue 3, pp 860–882 | Cite as

Two-Level Space–Time Domain Decomposition Methods for Three-Dimensional Unsteady Inverse Source Problems

  • Xiaomao Deng
  • Xiao-Chuan CaiEmail author
  • Jun Zou
Article

Abstract

As the number of processor cores on supercomputers becomes larger and larger, algorithms with high degree of parallelism attract more attention. In this work, we propose a two-level space–time domain decomposition method for solving an inverse source problem associated with the time-dependent convection–diffusion equation in three dimensions. We introduce a mixed finite element/finite difference method and a one-level and a two-level space–time parallel domain decomposition preconditioner for the Karush–Kuhn–Tucker system induced from reformulating the inverse problem as an output least-squares optimization problem in the entire space-time domain. The new full space–time approach eliminates the sequential steps in the optimization outer loop and the inner forward and backward time marching processes, thus achieves high degree of parallelism. Numerical experiments validate that this approach is effective and robust for recovering unsteady moving sources. We will present strong scalability results obtained on a supercomputer with more than 1000 processors.

Keywords

Space–time method Multilevel method Domain decomposition preconditioner Unsteady inverse source problem  Parallel computing 

Mathematics Subject Classification

49K20 65F22 65F08 65F10 65M32 65M55 65Y05 90C06 

Notes

Acknowledgments

The authors would like to thank the anonymous referees for their insightful comments and suggestions that helped us improve the quality of the paper. The work was partly supported by NSFC 11501545, 91330111, Shenzhen Program JCYJ20140901003939012, KQCX20130628112914303, 201506303000093 and 863 Program 2015AA01A302. The second author was partly support by NSF CCF-1216314. The third author was substantially supported by Hong Kong RGC Grants 404611 and 405513.

References

  1. 1.
    Aitbayev, R., Cai, X.-C., Paraschivoiu, M.: Parallel two-level methods for three-dimensional transonic compressible flow simulations on unstructured meshes. In: Proceedings of Parallel CFD’99 (1999)Google Scholar
  2. 2.
    Akcelik, V., Biros, G., Draganescu, A., Ghattas, O., Hill, J., Waanders, B.: Dynamic data-driven inversion for terascale simulations: real-time identification of airborne contaminants. In: Proceedings of Supercomputing, Seattle, WA (2005)Google Scholar
  3. 3.
    Akcelik, V., Biros, G., Ghattas, O., Long, K.R., Waanders, B.: A variational finite element method for source inversion for convective–diffusive transport. Finite Elem. Anal. Des. 39, 683–705 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Atmadja, J., Bagtzoglou, A.C.: State of the art report on mathematical methods for groundwater pollution source identification. Environ. Forensics 2, 205–214 (2001)CrossRefGoogle Scholar
  5. 5.
    Baflico, L., Bernard, S., Maday, Y., Turinici, G., Zerah, G.: Parallel-in-time molecular-dynamics simulations. Phys. Rev. E. 66, 2–5 (2002)Google Scholar
  6. 6.
    Balay, S., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc users manual. In: Technical Report, Argonne National Laboratory (2014)Google Scholar
  7. 7.
    Battermann, A.: Preconditioners for Karush–Kuhn–Tucker Systems Arising in Optimal Control. In: Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia (1996)Google Scholar
  8. 8.
    Biros, G., Ghattas, O.: Parallel preconditioners for KKT systems arising in optimal control of viscous incompressible flows. In: Proceedings of Parallel CFD’99, Williamsburg, Virginia, USA (1999)Google Scholar
  9. 9.
    Cai, X.-C., Liu, S., Zou, J.: Parallel overlapping domain decomposition methods for coupled inverse elliptic problems. Commun. Appl. Math. Comput. Sci. 4, 1–26 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cai, X.-C., Sarkis, M.: A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 792–797 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, R.L., Cai, X.-C.: Parallel one-shot Lagrange–Newton–Krylov–Schwarz algorithms for shape optimization of steady incompressible flows. SIAM J. Sci. Comput. 34, 584–605 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deng, X.M., Cai, X.-C., Zou, J.: A parallel space–time domain decomposition method for unsteady source inversion problems. Inverse Probl. Imag. (2015)Google Scholar
  13. 13.
    Deng, X.M., Zhao, Y.B., Zou, J.: On linear finite elements for simultaneously recovering source location and intensity. Int. J. Numer. Anal. Model. 10, 588–602 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1998)zbMATHGoogle Scholar
  15. 15.
    Farhat, C., Chandesris, M.: Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications. Int. J. Numer. Methods Eng. 58, 1397–1434 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gander, M.J., Hairer, E.: Nonlinear convergence analysis for the parareal algorithm. In: Proceedings of the 17th International Conference on Domain Decomposition Methods, vol. 60, pp. 45–56 (2008)Google Scholar
  17. 17.
    Gander, M.J., Petcu, M.: Analysis of a Krylov subspace enhanced parareal algorithm for linear problems. In: Cances E. et al. (eds.) Paris-Sud Working Group on Modeling and Scientific Computing 2007–2008. ESAIM Proceedings of EDP Science, LesUlis, vol. 25, pp. 114–129 (2008)Google Scholar
  18. 18.
    Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29, 556–578 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gorelick, S., Evans, B., Remson, I.: Identifying sources of groundwater pollution: an optimization approach. Water Resour. Res. 19, 779–790 (1983)CrossRefGoogle Scholar
  20. 20.
    Hamdi, A.: The recovery of a time-dependent point source in a linear transport equation: application to surface water pollution. Inverse Probl. 24, 1–18 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Karalashvili, M., Groß, S., Marquardt, W., Mhamdi, A., Reusken, A.: Identification of transport coefficient models in convection–diffusion equations. SIAM J. Sci. Comput. 33, 303–327 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Keung, Y.L., Zou, J.: Numerical identifications of parameters in parabolic systems. Inverse Probl. 14, 83–100 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of 2nd Berkeley Symposium. University of California Press, Berkeley, pp. 481–492 (1951)Google Scholar
  24. 24.
    Lions, J.-L., Maday, Y., Turinici, G.: A parareal in time discretization of PDE’s. C. R. Acad. Sci. Ser. I Math. 332, 661–668 (2001)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Liu, X., Zhai, Z.: Inverse modeling methods for indoor airborne pollutant tracking literature review and fundamentals. Indoor Air 17, 419–438 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Maday, Y., Turinici, G.: The parareal in time iterative solver: a further direction to parallel implementation. Domain Decompos. Methods Sci. Eng. 40, 441–448 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nilssen, T.K., Karlsen, K.H., Mannseth, T., Tai, X.-C.: Identification of diffusion parameters in a nonlinear convection–diffusion equation using the augmented Lagrangian method. Comput. Geosci. 13, 317–329 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Prudencio, E., Byrd, R., Cai, X.-C.: Parallel full space SQP Lagrange–Newton–Krylov–Schwarz algorithms for PDE-constrained optimization problems. SIAM J. Sci. Comput. 27, 1305–1328 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Revelli, R., Ridolfi, L.: Nonlinear convection–dispersion models with a localized pollutant source II—-a class of inverse problems. Math. Comput. Model. 42, 601–612 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Saad, Y.: A flexible inner–outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. Walter de Gruyter, Berlin (2007)CrossRefzbMATHGoogle Scholar
  32. 32.
    Skaggs, T., Kabala, Z.: Recovering the release history of a groundwater contaminant. Water Resour. Res. 30, 71–80 (1994)CrossRefGoogle Scholar
  33. 33.
    Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  34. 34.
    Wong, J., Yuan, P.: A FE-based algorithm for the inverse natural convection problem. Int. J. Numer. Methods. Fluid 68, 48–82 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Woodbury, K.A.: Inverse Engineering Handbook. CRC Press, Boca Raton (2003)zbMATHGoogle Scholar
  36. 36.
    Yang, X.-H., She, D.-X., Li, J.-Q.: Numerical approach to the inverse convection-diffusion problem. In: 2007 International Symposium on Nonlinear Dynamics (2007 ISND), Journal of Physics: Conference Series, vol. 96,p. 012156 (2008)Google Scholar
  37. 37.
    Yang, L., Deng, Z.-C., Yu, J.-N., Luo, G.-W.: Optimization method for the inverse problem of reconstructing the source term in a parabolic equation. Math. Comput. Simul. 80, 314–326 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Yang, H., Prudencio, E., Cai, X.-C.: Fully implicit Lagrange–Newton–Krylov–Schwarz algorithms for boundary control of unsteady incompressible flows. Int. J. Numer. Methods Eng. 91, 644–665 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zhang, J., Delichatsios, M.A.: Determination of the convective heat transfer coefficient in three-dimensional inverse heat conduction problems. Fire Saf. J. 44, 681–690 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Laboratory for Engineering and Scientific Computing, Shenzhen Institutes of Advanced TechnologyChinese Academy of SciencesShenzhenPeople’s Republic of China
  2. 2.Department of Computer ScienceUniversity of Colorado BoulderBoulderUSA
  3. 3.Department of MathematicsThe Chinese University of Hong KongShatinPeople’s Republic of China

Personalised recommendations