Coupling Tangent-Linear and Adjoint Models

  • Uwe Naumann
  • Patrick Heimbach
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2668)


We consider the solution of a (generalized) eigenvalue problem arising in physical oceanography that involves the evaluation of both the tangent-linear and adjoint versions of the underlying numerical model. Two different approaches are discussed. First, tangent-linear and adjoint models are generated by the software tool TAF and used separately. Second, the two models are combined into a single derivative model based on optimally preaccumulated local gradients of all scalar assignments. The coupled tangent-linear / adjoint model promises to be a good solution for small or medium sized problems. However, the simplicity of the example code at hand prevents us from observing considerable run time differences between the two approaches.


Argonne National Laboratory Ocean General Circulation Model Local Gradient Thermohaline Circulation Computational Graph 
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  1. [1]
    Clark, P., Pisias, N., Stocker, T., Weaver, A.: The role of the thermohaline circulation in abrubt climate change. Nature 415 (2002) 863–869Google Scholar
  2. [2]
    Wunsch, C.: What is the thermohaline circulation? Science 298 (2002) 1179–1180Google Scholar
  3. [3]
    Farrell, B., Moore, A.: An adjoint method for obtaining the most rapidly growing perturbation to oceanic flows. J. Phys. Oceanogr. 22 (1992) 338–349Google Scholar
  4. [4]
    Farrell, B., Ioannou, P.: Perturbation growth and structure in uncertain flows. Part I. J. Atmos. Sci. 59 (2002) 2629–2646CrossRefMathSciNetGoogle Scholar
  5. [5]
    Tziperman, E., Ioannou, P.: Transient growth and optimal excitation of thermohaline variability. J. Phys. Oceanogr. 32 (2002) 3427–3435CrossRefGoogle Scholar
  6. [6]
    Stommel, H.: Thermohaline convection with two stable regimes of flow. Tellus 13 (1961) 224–230Google Scholar
  7. [7]
    Tziperman, E.: Inherently unstable climate behaviour due to weak thermohaline ocean circulation. Nature 386 (1997) 592–595Google Scholar
  8. [8]
    Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia (2000)zbMATHGoogle Scholar
  9. [9]
    Giering, R., Kaminski, T.: Recipes for adjoint code construction. ACM Transactions on Mathematical Software 24 (1998) 437–474zbMATHCrossRefGoogle Scholar
  10. [10]
    Giering, R.: Transformation of algorithms in Fortran (TAF). User manual version 1.3. Technical report, FastOpt (2001)
  11. [11]
    Marotzke, J., Giering, R., Zhang, K., Stammer, D., Hill, C., Lee, T.: Construction of the adjoint MIT ocean general circulation model and application to Atlantic heat transport variability. J. Geophys. Res. 104, C12 (1999) 29,529–29,547CrossRefGoogle Scholar
  12. [12]
    Stammer, D., Wunsch, C., Giering, R., Eckert, C., Heimbach, P., Marotzke, J., Adcroft, A., Hill, C., Marshall, J.: The global ocean circulation and transports during 1992–1997, estimated from ocean observations and a general circulation model. J. Geophys. Res. 107(C9) (2002) 3118–3144CrossRefGoogle Scholar
  13. [13]
    Stammer, D., Wunsch, C., Giering, R., Eckert, C., Heimbach, P., Marotzke, J., Adcroft, A., Hill, C., Marshall, J.: Volume, heat and freshwater transports of the global ocean circulation 1993–2000, estimated from a general circulation model constrained by WOCE data. J. Geophys. Res. (2002) in press.Google Scholar
  14. [14]
    Heimbach, P., Hill, C., Giering, R.: Automatic generation of e.cient adjoint code for a parallel Navier-Stokes solver. In Dongarra, J.J., Sloot, P.M.A., Tan, C.J.K., eds.: Computational Science-ICCS 2002. Volume 2331 of Lecture Notes in Computer Science. Springer-Verlag, Berlin (Germany) (2002) 1019–1028Google Scholar
  15. [15]
    Heimbach, P., Hill, C., Giering, R.: An efficient exact adjoint of the parallel MIT general circulation model, generated via automatic differentiation. Future Generation Computer Systems (FGCS) (2002) submitted.Google Scholar
  16. [16]
    Naumann, U.: On optimal Jacobian accumulation for single expression use programs. Preprint ANL-MCS/P944-0402, Argonne National Laboratory (2002)Google Scholar
  17. [17]
    Naumann, U.: Automatic generation of optimal gradient code for scalar assignments. Preprint ANL-MCS/P1020-0103, Argonne National Laboratory (2003)Google Scholar
  18. [18]
    Rivin, I., Tziperman, E.: Linear versus self-sustained interdecadal thermohaline variability in a coupled box model. J. Phys. Oceanogr. 27 (1997) 1216–1232CrossRefGoogle Scholar
  19. [19]
    Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK users’ guide: Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia (1998)Google Scholar
  20. [20]
    Restrepo, J., Leaf, G., Griewank, A.: Circumvening storage limitations in variational data assimilation studies. SIAM J. Sci. Comput. 19 (1998) 1586–1605zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Hovland, P., Naumann, U., Norris, B.: An XML-based platform for semantic transformation of numerical programs. In: M. Hamza, ed., Software Engineering and Applications, Proceedings of the Sixth IASTED International Conference, ACTA Press (2002) 530–538Google Scholar
  22. [22]
    Griewank, A., Reese, S.: On the calculation of Jacobian matrices by the Markovitz rule. In: [27]. (1991) 126–135Google Scholar
  23. [23]
    Hascoët, L., Naumann, U., Pascual, V.: TBR analysis in reverse-mode automatic differentiation. Elsevier Science (2002) under review.Google Scholar
  24. [24]
    Aho, A., Sethi, R., Ullman, J.: Compilers. Principles, Techniques, and Tools. Addison-Wesley, Reading, MA (1986)Google Scholar
  25. [25]
    Naumann, U.: Optimal accumulation of Jacobian matrices by elimination methods on the dual computational graph. Preprint ANL-MCS/P943-0402, Argonne National Laboratory (2002) To appear in Math. Prog.Google Scholar
  26. [26]
    Naumann, U.: Statement-level optimality of tangent-linear and adjoint models. Preprint ANL-MCS/P1021-0103, Argonne National Laboratory (2002)Google Scholar
  27. [27]
    Corliss, G., Griewank, A., eds.: Automatic Differentiation: Theory, Implementation, and Application. Proceedings Series, Philadelphia, SIAM (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Uwe Naumann
    • 1
  • Patrick Heimbach
    • 2
  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryUSA
  2. 2.Earth, Atmospheric and Planetary SciencesMassachusetts Institute of TechnologyUSA

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