Skip to main content
SpringerLink
Log in

An efficient alternating direction method of multipliers for optimal control problems constrained by random Helmholtz equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Based on the alternating direction method of multipliers (ADMM), we develop three numerical algorithms incrementally for solving the optimal control problems constrained by random Helmholtz equations. First, we apply the standard Monte Carlo technique and finite element method for the random and spatial discretization, respectively, and then ADMM is used to solve the resulting system. Next, combining the multi-modes expansion, Monte Carlo technique, finite element method, and ADMM, we propose the second algorithm. In the third algorithm, we preprocess certain quantities before the ADMM iteration, so that nearly no random variable is in the inner iteration. This algorithm is the most efficient one and is easy to implement. The error estimates of these three algorithms are established. The numerical experiments verify the efficiency of our algorithms.

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

Similar content being viewed by others

References

  1. Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52(2), 317–355 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDE’s with stochastic coefficients. Numer. Math. 1, 123–161 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2011)

    Article  MATH  Google Scholar 

  4. Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta. Numer. 7, 1–49 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, X., Chen, Y., Han, D.: Nonnegative tensor factorizations using an alternating direction method. Front. Math. China 8, 3–18 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, X.J., Gu, G.Y., He, B.S.: On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. 57, 339–363 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai, X.J., Han, D.R.: O(1/t) Complexity analysis of the alternating direction method of multipliers. Revision under review (2014)

  8. Cao, Y., Hussaini, M.Y., Zang, T.A.: An efficient monte carlo method for optimal control problems with uncertainty. Comput. Optim. Appl. 26, 219–230 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, C., Hong, J., Ji, L., Kong, L.: A compact scheme for coupled stochastic nonlinear Schrodinger equations. Commun. Comput. Phys. 21, 93–125 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, P., Quarteroni, A., Rozza, G.: Stochastic optimal Robin boundary control problems of advection-dominated elliptic equations. SIAM J. Numer. Anal. 51, 2700–2722 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, P., Quarteroni, A.: Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraints. SIAM/ASA. J. Uncert. Quantif. 2, 364–396 (2014)

    Article  MATH  Google Scholar 

  12. Chen, P., Quarteroni, A., Rozza, G.: Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations. Numer. Math. 133, 67–102 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cutland, N.J., Grzesiak, K.: Optimal control for two-dimensional stochastic Navier-Stokes equations. Appl. Math. Optim. 55, 61–91 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. De los Reyes, J.C.: Numerical PDE-constrained Optimization. Springer (2015)

  15. Debussche, A., Fuhrman, M., Tessitore, G.: Optimal control of a stochastic heat equation with boundary-noise and boundary control. ESAIM Control Optim. Calc. Var. 13, 178–205 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eckstein, J., Fukushima, M.: Some reformulations and applications of the alternating direction method of multipliers. Springer-Verlag, US (1994)

  17. Feng, X.B., Lin, J.S., Lorton, C.: An efficient numerical method for acoustic wave scattering in random media. SIAM/ASA J. Uncert. Quantif. 3, 790–822 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Glowinski, R., Marrocco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par penalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires. ESAIM Math. Model. Num. 2, 41–76 (1975)

    MATH  Google Scholar 

  19. Gunzburger, M.D., Lee, H.C., Lee, J.: Error estimates of stochastic optimal neumann boundary control problems. SIAM J. Numer. Anal. 49, 1532–1552 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. He, B.S.: Contraction methods for convex optimization and monotone variational inequalities (2014)

  21. He, B.S., Yuan, X.M.: On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, B.S., Yuan, X.M.: On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers. Numer. Math. 130, 567–577 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer Verlag (2010)

  24. Ito, K., Kunisch, K.: Lagrange multiplier approach to variational problems and applications. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  25. Kelley, C.T., Sachs, E.W.: Quasi-newton methods and unconstrained optimal control problems. SIAM J. Control. Optim. 25, 1503–1517 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kouri, D.P., Heinkenschloos, D., Ridzal, M., Van Bloemen Waanders, B.G.: A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty. SIAM J. Sci. Comput. 35, 1847–1879 (2012)

    Article  MathSciNet  Google Scholar 

  27. Kunisch, K., Wachsmuth, D.: Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities. ESAIM Contr. Optim. CA. 180, 520–547 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kuo, F.Y., Schwab, C., Sloan, I.H.: Quasi-Monte Carlo finite element methods for a class of elliptic partial diffirential equations with random coefficients. SIAM J. Numer. Anal. 50, 3351–3374 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Meyer, C., Yousept, I.: State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions. SIAM J. Control Optim. 48, 734–755 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Narayan, A., Zhou, T.: Stochastic collocation methods on unstructured meshes. Commun. Comput. Phys. 18, 1–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Naseri, R., Malek, A.: Numerical optimal control for problems with random forced SPDE constraints. Isrn Appl. Math. 2014, 1–11 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ng, M.K., Weiss, P.A., Yuan, X.M.: Solving constrained total-variation image reconstruction problems via alternating direction methods. SIAM J. Sci. Comput. 32, 2710–2736 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rosseel, E., Wells, G.N.: Optimal control with stochastic PDE constraints and uncertain controls. Comput. Method Appl. M. 213, 286–295 (2011)

    MATH  Google Scholar 

  34. Sun, W.: Optimal control of impressed cathodic protection systems in ship building. Appl. Math. Model. 20, 823–828 (1996)

    Article  MATH  Google Scholar 

  35. Tang, T., Zhao, W., Zhou, T.: Deferred correction methods for forward backward stochastic differential equations. Numer. Math. Theory Methods Appl. 10, 222–242 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tiesler, H., Kirby, R.M., Xiu, D., Preusser, T.: Stochastic collocation for optimal control problems with stochastic PDE constraints. SIAM. J. Control Optim. 50, 2659–2682 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wahlberg, B., Boyd, S., Annergren, M., Wang, Y.: An ADMM algorithm for a class of total variation regularized estimation problems. IFAC Proc. Vol. 45, 83–88 (2012)

    Article  Google Scholar 

  38. Xiu, D., Karniadakis, G.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yang, J., Zhang, Y.: Alternating direction algorithms for l 1-problems in compressive sensing. SIAM J. Sci. Comput. 33, 250–278 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zamani, N.G., Chuang, J.M.: Optimal control of current in a cathodic protection system: a numerical investigation. Optim. Contr. Appl. Met. 8, 339–350 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, K., Li, J.S., Song, Y.C., Wang, X.S.: An alternating direction method of multipliers for elliptic equation constrained optimization problem. Sci. China Math. 60, 361–378 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, Z., Hu, X., Hou, Y., Lin, G., Yan, M.: An adaptive ANOVA-based data-driven stochastic method for elliptic PDEs with random coefficient. Commun. Comput. Phys. 16, 571–598 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhao, W., Zhang, W., Ju, L.: A multistep scheme for decoupled forward-backward stochastic differential equations. Numer. Math. Theory Methods Appl. 9, 262–288 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11471141), the basic research of the Science and Technology Development Program of Jilin province (No. 20150101058JC). The authors also wish to thank the High-Performance Computing Center of Jilin University, Computing Center of Jilin Province, and Key Laboratory of Symbolic Computation and Knowledge Engineering of the Ministry of Education for essential computing support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jilin University, Changchun, Jilin, 130023, People’s Republic of China

    Jingshi Li & Kai Zhang

  2. Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiaoshen Wang

Authors
  1. Jingshi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoshen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kai Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, J., Wang, X. & Zhang, K. An efficient alternating direction method of multipliers for optimal control problems constrained by random Helmholtz equations. Numer Algor 78, 161–191 (2018). https://doi.org/10.1007/s11075-017-0371-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0371-4

Keywords

Mathematics subject classification (2010)

Search

Navigation