Advertisement

Computational Optimization and Applications

, Volume 62, Issue 1, pp 131–155 | Cite as

Lossy compression for PDE-constrained optimization: adaptive error control

  • Sebastian Götschel
  • Martin Weiser
Article

Abstract

For the solution of optimal control problems governed by nonlinear parabolic PDEs, methods working on the reduced objective functional are often employed to avoid solving large systems in the dimension of the full spatio-temporal discretization. The evaluation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, requiring the storage of a full 4D data set. If Newton-CG methods are used, two additional trajectories have to be stored. To get numerical results which are accurate enough, in many cases very fine discretizations in time and space are necessary, which leads to a significant amount of data to be stored and transmitted to mass storage. Lossy compression methods were developed to overcome the storage problem by reducing the accuracy of the stored trajectories. The inexact data induces errors in the reduced gradient and reduced Hessian. In this paper, we analyze the influence of such a lossy trajectory compression method on Newton-CG methods for optimal control of parabolic PDEs and design an adaptive strategy for choosing appropriate quantization tolerances.

Keywords

Optimal control Semi-linear parabolic PDEs Newton-CG Trajectory storage Lossy compression 

Mathematics Subject Classification

35K58 49M15 65M60 68P30 94A29 

Notes

Acknowledgments

The authors gratefully acknowledge support by the DFG Research Center Matheon, project F9.

References

  1. 1.
    Bouras, A., Frayssé, V.: Inexact matrix-vector products in Krylov methods for solving linear systems: a relaxation strategy. SIAM J. Matrix Anal. Appl. 26(3), 660–678 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Britton, N.F.: Reaction-Diffusion Equations and Their Application to Biology. Academic Press, Waltham (1986)Google Scholar
  3. 3.
    Deuflhard, P., Weiser, M.: Local inexact Newton multilevel FEM for nonlinear elliptic problems. In: Bristeau, M.O., Etgen, G., Fitzgibbon, W., Lions, J.L., Periaux, J., Wheeler, M. (eds.) Computational Science for the 21st Century, pp. 129–138. Wiley, Hoboken (1997)Google Scholar
  4. 4.
    Du, X., Haber, E., Karampataki, M., Szyld, D.B.: Varying iteration accuracy using inexact conjugate gradients in control problems governed by PDEs. In: R. Chen (ed.) Proceedings of the 2nd Annual Conference on Computational Mathematics, Computational Geometry & Statistics (CMCGS 2013), pp. 29–38 (2013)Google Scholar
  5. 5.
    Fife, P.C., Tang, M.M.: Comparison principles for reaction-diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation of disturbances. J. Differ. Equ. 40(2), 168–185 (1981)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Götschel, S., Chamakuri, N., Kunisch, K., Weiser, M.: Lossy compression in optimal control of cardiac defibrillation. J. Sci. Comput. 60(1), 35–59 (2014)Google Scholar
  7. 7.
    Götschel, S., von Tycowicz, C., Polthier, K., Weiser, M.: Reducing memory requirements in scientific computing and optimal control. In: Carraro, T., Geiger, M., Körkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition Methods. Springer (2014, to appear)Google Scholar
  8. 8.
    Götschel, S., Weiser, M., Schiela, A.: Solving optimal control problems with the Kaskade 7 finite element toolbox. In: Dedner, A., Flemisch, B., Klöfkorn, R. (eds.) Advances in DUNE, pp. 101–112. Springer, Berlin (2012)CrossRefGoogle Scholar
  9. 9.
    Greenbaum, A.: Estimating the attainable accuracy of recursively computed residual methods. SIAM J. Matrix Anal. Appl. 18(3), 535–551 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Griewank, A.: Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation. Optim. Methods Softw. 1(1), 35–54 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Griewank, A., Walther, A.: Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Trans. Math. Softw. (TOMS) 26(1), 19–45 (2000)CrossRefMATHGoogle Scholar
  12. 12.
    Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia (2008)CrossRefGoogle Scholar
  13. 13.
    Gutknecht, M.H., Strakoš, Z.: Accuracy of two three-term and three two-term recurrences for Krylov space solvers. SIAM J. Matrix Anal. Appl. 22(1), 213–229 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kunisch, K., Wagner, M.: Optimal control of the bidomain system (I): the monodomain approximation with the Rogers–McCulloch model. Nonlinear Anal. Real World Appl. 13(4), 1525–1550 (2012).Google Scholar
  15. 15.
    Meurant, G.A.: The Lanczos and Conjugate Gadient Algorithms: From Theory to Finite Precision Computations, vol. 19. SIAM, Philadelphia (2006)Google Scholar
  16. 16.
    Meurant, G.A., Strakoš, Z.: The Lanczos and conjugate gradient algorithms in finite precision arithmetic. Acta Numerica 15(1), 471–542 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nagaiah, C., Kunisch, K., Plank, G.: Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology. Comput. Optim. Appl. 49, 149–178 (2011). doi: 10.1007/s10589-009-9280-3.
  18. 18.
    Nielsen, B.F., Ruud, T.S., Lines, G.T., Tveito, A.: Optimal monodomain approximations of the bidomain equations. Appl. Math. Comput. 184(2), 276–290 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)MATHGoogle Scholar
  20. 20.
    Rogers, J.M., McCulloch, A.D.: A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41, 743–757 (1994)CrossRefGoogle Scholar
  21. 21.
    Simoncini, V., Szyld, D.B.: Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM J. Sci. Comput. 25(2), 454–477 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sleijpen, G.L.G., van der Vorst, H.A.: Reliable updated residuals in hybrid Bi-CG methods. Computing 56(2), 141–163 (1996)Google Scholar
  23. 23.
    Strakoš, Z., Tichỳ, P.: On error estimation in the conjugate gradient method and why it works in finite precision computations. Electron. Trans. Numer. Anal. 13, 56–80 (2002)MathSciNetMATHGoogle Scholar
  24. 24.
    van den Eshof, J., Sleijpen, G.L.G.: Inexact Krylov subspace methods for linear systems. SIAM J. Matrix Anal. Appl. 26(1), 125–153 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    von Tycowicz, C., Kälberer, F., Polthier, K.: Context-based coding of adaptive multiresolution meshes. Comput. Graph. Forum 30(8), 2231–2245 (2011)CrossRefGoogle Scholar
  26. 26.
    van der Vorst, H.A., Ye, Q.: Residual replacement strategies for Krylov subspace iterative methods for the convergence of true residuals. SIAM J. Sci. Comput. 22(3), 835–852 (2000)Google Scholar
  27. 27.
    Weiser, M., Götschel, S.: State trajectory compression for optimal control with parabolic PDEs. SIAM J. Sci. Comput. 34(1), A161–A184 (2012)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Zuse Institute BerlinBerlinGermany

Personalised recommendations