Summary
Discrete adjoints are very popular in optimization and control since they can be constructed automatically by reverse mode automatic differentiation. In this paper we analyze the consistency and stability properties of discrete adjoints of linear multistep methods. The analysis reveals that the discrete linear multistep adjoints are, in general, inconsistent approximations of the adjoint ODE solution along the trajectory. The discrete adjoints at the initial time converge to the adjoint ODE solution with the same order as the original linear multistep method. Discrete adjoints inherit the zero-stability properties of the forward method. Numerical results confirm the theoretical findings.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baguer, M., Romisch, W.: Computing gradients in parametrization-discretization schemes for constrained optimal control problems. Approximation and Optimization in the Carribbean II. Peter Lang, Frankfurt am Main (1995)
Giles, M.: On the use of Runge-Kutta time-marching and multigrid for the solution of steady adjoint equations. Technical Report NA00/10, Oxford University Computing Laboratory (2000)
Hager, W.: Runge-Kutta methods in optimal control and the transformed adjoint system. Numerische Mathematik 87(2), 247–282 (2000)
Hairer, E., Norsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer-Verlag, Berlin (1993)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin (1991)
Sandu, A.: On the Properties of Runge-Kutta Discrete Adjoints. In: Lecture Notes in Computer Science, vol. LNCS 3994, Part IV, pp. 550–557. “International Conference on Computational Science” (2006)
Sandu, A.: On consistency properties of discrete adjoint linear multistep methods. Tech. Rep. CS–TR–07–40, Computer Science Department, Virginia Tech (2007)
Sandu, A., Daescu, D., Carmichael, G.: Direct and adjoint sensitivity analysis of chemical kinetic systems with KPP: I – Theory and software tools. Atmospheric Environment 37, 5083–5096 (2003)
Walther, A.: Automatic differentiation of explicit Runge-Kutta methods for optimal control. Computational Optimization and Applications 36(1), 83–108 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sandu, A. (2008). Reverse Automatic Differentiation of Linear Multistep Methods. In: Bischof, C.H., Bücker, H.M., Hovland, P., Naumann, U., Utke, J. (eds) Advances in Automatic Differentiation. Lecture Notes in Computational Science and Engineering, vol 64. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68942-3_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-68942-3_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68935-5
Online ISBN: 978-3-540-68942-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)