Journal of Scientific Computing

, Volume 63, Issue 1, pp 138–162 | Cite as

Discretely Exact Derivatives for Hyperbolic PDE-Constrained Optimization Problems Discretized by the Discontinuous Galerkin Method

  • Lucas C. Wilcox
  • Georg Stadler
  • Tan Bui-Thanh
  • Omar Ghattas


This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. An efficient and accurate computation of these derivatives is important, for instance, in inverse problems and optimal control problems. This computation is usually based on an adjoint PDE system, and the question addressed in this paper is how the discretization of this adjoint system should relate to the dG discretization of the hyperbolic state equation. Adjoint-based derivatives can either be computed before or after discretization; these two options are often referred to as the optimize-then-discretize and discretize-then-optimize approaches. We discuss the relation between these two options for dG discretizations in space and Runge–Kutta time integration. The influence of different dG formulations and of numerical quadrature is discussed. Discretely exact discretizations for several hyperbolic optimization problems are derived, including the advection equation, Maxwell’s equations and the coupled elastic-acoustic wave equation. We find that the discrete adjoint equation inherits a natural dG discretization from the discretization of the state equation and that the expressions for the discretely exact gradient often have to take into account contributions from element faces. For the coupled elastic-acoustic wave equation, the correctness and accuracy of our derivative expressions are illustrated by comparisons with finite difference gradients. The results show that a straightforward discretization of the continuous gradient differs from the discretely exact gradient, and thus is not consistent with the discretized objective. This inconsistency may cause difficulties in the convergence of gradient based algorithms for solving optimization problems.


Discontinuous Galerkin PDE-constrained optimization  Discrete adjoints Elastic wave equation Maxwell’s equations 



We would like to thank Jeremy Kozdon and Gregor Gassner for fruitful discussions and helpful comments, and Carsten Burstedde for his help with the implementation of the numerical example presented in Sect. 4. Support for this work was provided through the U.S. National Science Foundation (NSF) grant CMMI-1028889, the Air Force Office of Scientific Research’s Computational Mathematics program under the grant FA9550-12-1-0484, and through the Mathematical Multifaceted Integrated Capability Centers (MMICCs) effort within the Applied Mathematics activity of the U.S. Department of Energy’s Advanced Scientific Computing Research program, under Award Number DE-SC0009286. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.


  1. 1.
    Alexe, M., Sandu, A.: Space-time adaptive solution of inverse problems with the discrete adjoint method. J. Comput. Phy. 270, 21–39 (2014). doi: 10.1016/
  2. 2.
    Becker, R., Meidner, D., Vexler, B.: Efficient numerical solution of parabolic optimization problems by finite element methods. Optim. Methods Softw. 22, 813–833 (2007). doi: 10.1080/10556780701228532 CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Borzì, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. SIAM, Philadelphia (2012). doi: 10.1137/1.9781611972054 zbMATHGoogle Scholar
  4. 4.
    Braack, M.: Optimal control in fluid mechanics by finite elements with symmetric stabilization. SIAM J. Control Optim. 48, 672–687 (2009). doi: 10.1137/060653494 CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bui-Thanh, T., Burstedde, C., Ghattas, O., Martin, J., Stadler, G., Wilcox, L.C.: Extreme-scale UQ for Bayesian inverse problems governed by PDEs. In: SC12 Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2012). doi: 10.1109/SC.2012.56
  6. 6.
    Bui-Thanh, T., Ghattas, O., Martin, J., Stadler, G.: A computational framework for infinite-dimensional Bayesian inverse problems part I: the linearized case, with application to global seismic inversion. SIAM J. Sci. Comput. 35, A2494–A2523 (2013). doi: 10.1137/12089586X CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Collis, S.S., Heinkenschloss, M.: Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. Technical Report TR02-01, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005–1892 (2002).
  8. 8.
    Collis, S.S., Ober, C.C., van Bloemen Waanders, B.G.: Unstructured discontinuous Galerkin for seismic inversion. In: Conference Paper, SEG International Exposition and 80th Annual Meeting (2010)Google Scholar
  9. 9.
    Epanomeritakis, I., Akçelik, V., Ghattas, O., Bielak, J.: A Newton-CG method for large-scale three-dimensional elastic full-waveform seismic inversion. Inverse Probl. 24 (2008), 034015 (26pp). doi: 10.1088/0266-5611/24/3/034015
  10. 10.
    Feng, K.-A., Teng, C.-H., Chen, M.-H.: A pseudospectral penalty scheme for 2D isotropic elastic wave computations. J. Sci. Comput. 33, 313–348 (2007). doi: 10.1007/s10915-007-9154-8 CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Fichtner, A.: Full Seismic Waveform Modelling and Inversion. Springer, Berlin (2011)CrossRefGoogle Scholar
  12. 12.
    Giles, M., Pierce, N.: Adjoint equations in CFD: duality, boundary conditions and solution behaviour. American Institute of Aeronautics and Astronautics (1997). doi: 10.2514/6.1997-1850
  13. 13.
    Giles, M., Ulbrich, S.: Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 1: linearized approximations and linearized output functionals. SIAM J. Numer. Anal. 48, 882–904 (2010). doi: 10.1137/080727464 CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Giles, M., Ulbrich, S.: Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions. SIAM J. Numer. Anal. 48, 905–921 (2010). doi: 10.1137/09078078X CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Griewank, A., Walther, A.: Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Trans. Math. Softw. 26, 19–45 (2000). doi: 10.1145/347837.347846 CrossRefzbMATHGoogle Scholar
  16. 16.
    Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd edn. SIAM, Philadelphia (2008). doi: 10.1137/1.9780898717761 CrossRefGoogle Scholar
  17. 17.
    Gunzburger, M.D.: Perspectives in Flow Control and Optimization. SIAM, Philadelphia (2003). doi: 10.1137/1.9780898718720 zbMATHGoogle Scholar
  18. 18.
    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, New York (1995)zbMATHGoogle Scholar
  19. 19.
    Hager, W.W.: Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87, 247–282 (2000). doi: 10.1007/s002110000178 CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Harriman, K., Gavaghan, D., Süli, E.: The importance of adjoint consistency in the approximation of linear functionals using the discontinuous Galerkin finite element method. Technical Report NA04/18, Oxford University Computing Laboratory (2004)Google Scholar
  21. 21.
    Hartmann, R.: Adjoint consistency analysis of discontinuous Galerkin discretizations. SIAM J. Numer. Anal. 45, 2671–2696 (2007). doi: 10.1137/060665117 CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 186–221 (2002). doi: 10.1006/jcph.2002.7118 CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, vol. 54 of Texts in Applied Mathematics. Springer, Berlin (2008)CrossRefGoogle Scholar
  24. 24.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Berlin (2009)zbMATHGoogle Scholar
  25. 25.
    Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kopriva, D.A., Gassner, G.: On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods. J. Sci. Comput. 44, 136–155 (2010). doi: 10.1007/s10915-010-9372-3 CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Lagnese, J.E.: Exact boundary controllability of Maxwell’s equations in a general region. SIAM J. Control Optim. 27, 374–388 (1989). doi: 10.1137/0327019 CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Lekić, V., Romanowicz, B.: Inferring upper-mantle structure by full waveform tomography with the spectral element method. Geophys. J. Int. 185, 799–831 (2011). doi: 10.1111/j.1365-246X.2011.04969.x CrossRefGoogle Scholar
  29. 29.
    Leykekhman, D.: Investigation of commutative properties of discontinuous Galerkin methods in PDE constrained optimal control problems. J. Sci. Comput. 53, 483–511 (2012). doi: 10.1007/s10915-012-9582-y
  30. 30.
    Lions, J.-L.: Control of Distributed Singular Systems. Gauthier-Villars, Paris (1985)Google Scholar
  31. 31.
    Mohammadian, A.H., Shankar, V., Hall, W.F.: Computational of electromagmetic scattering and radiation using a time-domain finite-volume discretization procedure. Comput. Phys. Commun. 68, 175–196 (1991). doi: 10.1016/0010-4655(91)90199-U CrossRefGoogle Scholar
  32. 32.
    Nicaise, S.: Exact boundary controllability of Maxwell’s equations in heterogeneous media and an application to an inverse source problem. SIAM J. Control Optim. 38, 1145–1170 (2000). doi: 10.1137/S0363012998344373 CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Oliver, T.A., Darmofal, D.L.: Analysis of dual consistency for discontinuous Galerkin discretizations of source terms. SIAM J. Numer. Anal. 47, 3507–3525 (2009). doi: 10.1137/080721467 CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Peter, D., Komatitsch, D., Luo, Y., Martin, R., Le Goff, N., Casarotti, E., Le Loher, P., Magnoni, F., Liu, Q., Blitz, C., Nisson-Meyer, T., Basini, P., Tromp, J.: Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes. Geophys. J. Int. 186, 721–739 (2011). doi: 10.1111/j.1365-246X.2011.05044.x CrossRefGoogle Scholar
  35. 35.
    Schütz, J., May, G.: An adjoint consistency analysis for a class of hybrid mixed methods. IMA J. Numer. Anal. 34(3), 863–878 (2013). doi: 10.1093/imanum/drt036
  36. 36.
    Teng, C.-H., Lin, B.-Y., Chang, H.-C., Hsu, H.-C., Lin, C.-N., Feng, K.-A.: A legendre pseudospectral penalty scheme for solving time-domain Maxwell’s equations. J. Sci. Comput. 36, 351–390 (2008). doi: 10.1007/s10915-008-9194-8 CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, vol. 112 of Graduate Studies in Mathematics. American Mathematical Society (2010)Google Scholar
  38. 38.
    Utke, J., Hascoët, L., Heimbach, P., Hill, C., Hovland, P., Naumann, U.: Toward adjoinable MPI. In: IEEE International Symposium on Parallel & Distributed Processing, 2009. IPDPS 2009, pp. 1–8 (2009). doi: 10.1109/IPDPS.2009.5161165
  39. 39.
    Walther, A.: Automatic differentiation of explicit Runge–Kutta methods for optimal control. Comput. Optim. Appl. 36, 83–108 (2007). doi: 10.1007/s10589-006-0397-3 CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Wilcox, L.C., Stadler, G., Burstedde, C., Ghattas, O.: A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media. J. Comput. Phys. 229, 9373–9396 (2010). doi: 10.1016/ CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Yousept, I.: Optimal control of Maxwell’s equations with regularized state constraints. Comput. Optim. Appl. 52, 559–581 (2012). doi: 10.1007/s10589-011-9422-2 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York (outside the USA) 2014

Authors and Affiliations

  • Lucas C. Wilcox
    • 1
  • Georg Stadler
    • 2
  • Tan Bui-Thanh
    • 2
    • 3
  • Omar Ghattas
    • 2
    • 4
  1. 1.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA
  2. 2.Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
  3. 3.Department of Aerospace Engineering and Engineering MechanicsThe University of Texas at AustinAustinUSA
  4. 4.Departments of Mechanical Engineering and Jackson School of GeosciencesThe University of Texas at AustinAustinUSA

Personalised recommendations