Skip to main content
Springer Nature Link
Log in

Optimized Schwarz Methods for the Optimal Control of Systems Governed by Elliptic Partial Differential Equations

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

Abstract

Optimal control of systems governed by elliptic partial differential equations (PDEs) without constraint on the set of controls can be equivalently reformulated as a coupled system of second order elliptic PDEs, which has been considered to solve by a non-overlapping Schwarz domain decomposition method with the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions by Benamou (SIAM J Numer Anal 33:2401–2416, 1996) where a convergence analysis had been performed. Towards fast convergence of the overlapping and non-overlapping Schwarz subdomain iterations, in this paper we firstly perform, for fixed Tikhonov parameter \(\mu \), rigorous analyses based on optimization of the convergence factor of subdomain iterations in Fourier frequency domain to give the optimized transmission parameters involved in the above mentioned transmission conditions, as well as those involved in a Ventcell-like and a two-sided Robin-like transmission condition that we propose to accelerate the Schwarz subdomain iterations, and meanwhile we obtain also the corresponding asymptotic convergence rate estimates. The results show that the Tikhonov parameter \(\mu \) occurs in both the optimized transmission parameters and the corresponding convergence rate estimates and affects the performance of the Schwarz domain decomposition methods significantly: when \(\mu \) is less than a certain threshold value, with the decreasing of the Tikhonov parameter \(\mu \), the subdomain iteration converges more and more fast, though the regularity of the system deteriorates in this process. We lastly investigate the case where the Tikhonov parameter \(\mu =h^4\) that is suggested by Benamou (where h is the mesh size). We obtain as well the optimized transmission parameters involved in the non-coupled, the partially-coupled and the fully-coupled Robin-like transmission conditions, and find that they lead to optimized Schwarz methods that are very robust in the mesh size. The analysis also sheds light on optimizing the Schwarz domain decomposition methods for biharmonic equations, since they can also be reformulated as a system of second order elliptic PDEs. We use various numerical experiments to illustrate the theoretical findings.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

References

  1. Benamou, J.D.: Domain decomposition methods with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations. C. R. Acad. Sci. Paris 317, 205–209 (1993)

    MathSciNet  MATH  Google Scholar 

  2. Benamou, J.D.: A domain decomposition method with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations. SIAM J. Numer. Anal. 33(6), 2401–2416 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benamou, J.D.: Domain decomposition, optimal control of systems governed by partial differential equations, and synthesis of feedback laws. J. Optim. Theory Appl. 102(1), 15–36 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benamou, J.D., Despres, B.: A domain decomposition method for the Helmholtz equation and related optimal control problems. J. Comput. Phys. 136(1), 68–82 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bennequin, D., Gander, M.J., Halpern, L.: A homographic best approximation problem with application to optimized Schwarz waveform relaxation. Math. Comp. 78(265), 185–223 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blayo, E., Cherel, D., Rousseau, A.: Towards optimized Schwarz methods for the Navier–Stokes equations. J. Sci. Comput. 66(1), 275–295 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bouajaji, M.E., Dolean, V., Gander, M.J., Lanteri, S.: Optimized Schwarz methods for the time-harmonic Maxwell equations with damping. SIAM J. Sci. Comput. 34(4), A2048–A2071 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chang, H., Yang, D.: A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations. J. Comput. Appl. Math. 235(17), 5078–5094 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Z., Gander, M.J., Zhang, H.: On the relation between optimized Schwarz methods and source transfer. In: Dickopf, T., Gander, M.J., Halpern, L., Krause, R., Pavarino, L.F. (eds.) Domain Decomposition Methods in Science and Engineering XXII, pp. 217–225. Springer International Publishing, Berlin (2016)

    Chapter  Google Scholar 

  10. Chen, Z., Xiang, X.: A source transfer domain decomposition method for Helmholtz equations in unbounded domain. SIAM J. Numer. Anal. 51(4), 2331–2356 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Discacciati, M., Gerardo-Giorda, L.: Optimized Schwarz methods for the Stokes-Darcy coupling. IMA J. Numer. Aanal. 38(4), 1959–1983 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dolean, V., Gander, M.J., Gerardo-Giorda, L.: Optimized Schwarz methods for Maxwell’s equations. SIAM J. Sci. Comput. 31(3), 2193–2213 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dubois, O., Gander, M.J., Loisel, S., St-Cyr, A., Szyld, D.B.: The optimized schwarz method with a coarse grid correction. SIAM J. Sci. Comput. 34(1), A421–A458 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Mode. Simul. 9(2), 686–710 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gander, M.J.: Schwarz methods over the course of time. Electron. Trans. Numer. Anal. 31, 228–255 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Gander, M.J., Halpern, L.: Absorbing boundary conditions for the wave equation and parallel computing. Math. Comp. 74(249), 153–176 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gander, M.J., Halpern, L.: Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal. 45(2), 666–697 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gander, M.J., Halpern, L., Nataf, F.: Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41(5), 1643–1681 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gander, M.J., Liu, Y.: On the definition of dirichlet and neumann conditions for the biharmonic equation and its impact on associated schwarz methods. In: Lee, C.O., Cai, X.C., Keyes, D.E., Kim, H.H., Klawonn, A., Park, E.J., Widlund, O.B. (eds.) Domain Decomposition Methods in Science and Engineering XXIII, pp. 303–311. Springer International Publishing, Cham (2017)

    Chapter  Google Scholar 

  21. Gander, M.J., Magoulès, F., Nataf, F.: Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24(1), 38–60 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gander, M.J., Xu, Y.: Optimized schwarz methods for circular domain decompositions with overlap. SIAM J. Numer. Anal. 52(4), 1981–2004 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gander, M.J., Xu, Y.: Optimized Schwarz methods for model problems with continuously variable coefficients. SIAM J. Sci. Comput. 38(5), A2964–A2986 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gander, M.J., Xu, Y.: Optimized Schwarz methods for domain decompositions with parabolic interfaces. In: Lee, C.O., et al. (eds.) Domain Decomposition Methods in Science and Engineering XXIII, pp. 323–332. Springer, Cham (2017)

    Chapter  Google Scholar 

  25. Gander, M.J., Xu, Y.: Optimized Schwarz methods with nonoverlapping circular domain decompositions. Math. Comp. 86(304), 637–660 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Heinkenschloss, M., Nguyen, H.: Balancing Neumann–Neumann methods for elliptic optimal control problems. Lecture Notes in Computational Science and Engineering, vol. 40, pp. 589–596. Springer, Berlin, Heidelberg (2005)

    Google Scholar 

  27. Heinkenschloss, M., Nguyen, H.: Neumann-Neumann domain decomposition preconditioners for linear-quadratic elliptic optimal control problems. SIAM J. Sci. Comput. 28(3), 1001–1028 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Herzog, R., Rheinbach, O.: FETI-DP methods for optimal control problems. In: Erhel, J., Gander, M.J., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineering XXI, pp. 387–395. Springer International Publishing, Cham (2014)

    Google Scholar 

  29. Hou, L.S., Lee, J.: A Robin-Robin non-overlapping domain decomposition method for an elliptic boundary control problem. Int. J. Numer. Anal. Model. 8(3), 443–465 (2011)

    MathSciNet  MATH  Google Scholar 

  30. Japhet, C.: Domain decomposition methods and artificial boundary conditions in fluid dynamics: Optimized Order 2 method. Ph.D. thesis, Université Paris 13 (1998)

  31. Lagnese, J.E., Leugering, G.: Domain Decomposition for Elliptic Optimal Control Problems, pp. 107–129. Birkhäuser, Basel, Basel (2004)

    MATH  Google Scholar 

  32. Lions, J.L.: Contrôle Optimal De Systemes Gouvernés Par Des Équations Aux Dérivées Partielles. Dunod (1968)

  33. Lions, P.L.: On the Schwarz alternating method. III: a variant for nonoverlapping subdomains. In: Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, vol. 6, pp. 202–223. SIAM, Philadelphia, PA (1990)

  34. Magouls, F., Ivnyi, P., Topping, B.: Non-overlapping Schwarz methods with optimized transmission conditions for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 193(45–47), 4797–4818 (2004)

    Article  MathSciNet  Google Scholar 

  35. Martin, V.: An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions. Appl. Numer. Math. 52(4), 401–428 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pearson, J.W., Stoll, M.: Fast iterative solution of reaction–diffusion control problems arising from chemical processes. SIAM J. Sci. Comput. 35(5), 987–1009 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Qin, L., Xu, X.: Optimized Schwarz methods with Robin transmission conditions for parabolic problems. SIAM J. Sci. Comput. 31(1), 608–623 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32(1), 271–298 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rees, T., Stoll, M.: Block-triangular preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 17(6), 977–996 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Schwarz, H.A.: Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 15, 272–286 (1870)

    Google Scholar 

  41. Shang, Y.Q., He, Y.N.: Fourier analysis of schwarz domain decomposition methods for the biharmonic equation. Appl. Math. Mech. 30(9), 1177–1182 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xu, Y.: The influence of domain truncation on the performance of optimized Schwarz methods. Electron. Trans. Numer. Anal. 49, 182–209 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  43. Xu, Y.: Optimized Schwarz methods with Ventcell transmission conditions for model problems with continuously variable coefficients. J. Comput. Appl. Math. 334, 97–110 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Yingxiang Xu & Xin Chen

Authors
  1. Yingxiang Xu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Xin Chen
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Yingxiang Xu.

Additional information

Supported by NSFC-11671074, 11471047 and the Fundamental Research Funds for the Central Universities (No. 2412018ZD001).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, Y., Chen, X. Optimized Schwarz Methods for the Optimal Control of Systems Governed by Elliptic Partial Differential Equations. J Sci Comput 79, 1182–1213 (2019). https://doi.org/10.1007/s10915-018-0886-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0886-4

Keywords

Mathematics Subject Classification