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Automatic differentiation of numerical integration algorithms

Authors: Peter Eberhard and Christian Bischof
Journal: Math. Comp. 68 (1999), 717-731
MSC (1991): Primary 34A12, 65L05, 65L06; Secondary 68N99
MathSciNet review: 1613703
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Abstract: Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations (ODEs), in particular, the ramifications of the fact that AD is applied not only to the solution of such an algorithm, but to the solution procedure itself. This subtle issue can lead to surprising results when AD tools are applied to variable-stepsize, variable-order ODE integrators. The computation of the final time step plays a special role in determining the computed derivatives. We investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives.

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  • Martin Berz, Christian Bischof, George Corliss, and Andreas Griewank (eds.), Computational differentiation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. Techniques, applications, and tools. MR 1431037
  • Dieter Bestle and Peter Eberhard, Analyzing and optimizing multibody systems, Mechanical Structures and Machinery 20 (1992), no. 1, 67–92.
  • Christian H. Bischof and Mohammad R. Haghighat, On hierarchical differentiation, Computational Differentiation: Techniques, Applications, and Tools (Martin Berz, Christian Bischof, George Corliss, and Andreas Griewank, eds.), SIAM, Philadelphia, 1996, pp. 83–94.
  • Christian Bischof, Alan Carle, Peyvand Khademi, and Andrew Mauer, ADIFOR 2.0: Automatic differentiation of Fortran 77 programs, IEEE Computational Science & Engineering 3 (1996), no. 3, 18–32.
  • Christian Bischof, Lucas Roh, and Andrew Mauer, ADIC — An extensible automatic differentiation tool for ANSI-C, Software–Practice and Experience 27 (1997), no. 12, 1427–1456.
  • J. C. Butcher, The numerical analysis of ordinary differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. MR 878564
  • J. Cash and A. Karp, A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Transactions on Mathematical Software 16 (1990), no. 3, 201–222.
  • Evin J. Cramer, J. E. Dennis Jr., Paul D. Frank, Robert Michael Lewis, and Gregory R. Shubin, Problem formulation for multidisciplinary optimization, SIAM J. Optim. 4 (1994), no. 4, 754–776. MR 1300850, DOI
  • Peter Eberhard, Zur Mehrkriterienoptimierung von Mehrkörpersystemen, VDI Fortschritt-Berichte 11 (1996), no. 227.
  • Andreas Griewank, Christian Bischof, George Corliss, Alan Carle, and Karen Williamson, Derivative convergence of iterative equation solvers, Optimization Methods and Software 2 (1993), 321–355.
  • Andreas Griewank and Shawn Reese, On the calculation of Jacobian matrices by the Markowitz rule, Automatic differentiation of algorithms (Breckenridge, CO, 1991) SIAM, Philadelphia, PA, 1991, pp. 126–135. MR 1143797
  • Andreas Griewank, On automatic differentiation, Mathematical programming (Tokyo, 1988) Math. Appl. (Japanese Ser.), vol. 6, SCIPRESS, Tokyo, 1989, pp. 83–107. MR 1114312
  • E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987. Nonstiff problems. MR 868663
  • E. Hairer and G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems. MR 1111480
  • Alan Hindmarsh, ODEPACK, a systematized collection of ODE solvers, pp. 55–64, North-Holland, Amsterdam, 1983.
  • William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical recipes in FORTRAN, 2nd ed., Cambridge University Press, Cambridge, 1992. The art of scientific computing; With a separately available computer disk. MR 1196230
  • Louis B. Rall, Automatic differentiation: Techniques and applications, Lecture Notes in Computer Science, vol. 120, Springer Verlag, Berlin, 1981.
  • A. Sandu, G. R. Carmichael, and F. A. Potra, Sensitivity analysis for atmospheric chemistry models via automatic differentiation, Atmospheric Environment 31 (1997), no. 3, 475–489.
  • Granville Sewell, The numerical solution of ordinary and partial differential equations, Academic Press, Inc., Boston, MA, 1988. MR 968668
  • L. F. Shampine and M. K. Gordon, Computer solution of ordinary differential equations, W. H. Freeman and Co., San Francisco, Calif., 1975. The initial value problem. MR 0478627
  • J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR 557543

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Additional Information

Peter Eberhard
Affiliation: Institute B of Mechanics, University of Stuttgart, 70550 Stuttgart, Germany

Christian Bischof
Affiliation: Rheinisch-Westfälische Technische Hochschule Aachen, Seffenter Weg 23, D-52056 Aachen, Germany

Received by editor(s): November 11, 1996
Received by editor(s) in revised form: July 24, 1997
Additional Notes: This work was partially completed while the first author was visiting the Department of Mechanical Engineering at the University of California at Berkeley supported by the German Research Council (DFG) under grant EB195/1-1.
The second author was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under contract W-31-109-Eng-38.
Article copyright: © Copyright 1999 American Mathematical Society