Abstract
The computation of derivatives for the optimization of time-dependent flow problems is based on the integration of the adjoint differential equation. For this purpose, the knowledge of the complete forward solution is required. Similar information is needed for a posteriori error estimation with respect to a given functional. In the area of flow control, especially for three dimensional problems, it is usually impossible to store the full forward solution due to the lack of memory capacities. Additionally, adaptive time-stepping procedures are needed for efficient integration schemes in time. Therefore, standard optimal offline checkpointing strategies are usually not well-suited in that framework.
We present a new online procedure for determining the checkpoint distribution on the fly. Complexity estimates and consequences for storing and retrieving the checkpoints using parallel I/O are discussed. The resulting checkpointing approach is integrated in HiFlow, a multipurpose parallel finite-element package with a strong emphasis in computational fluid dynamic, reactive flows and related subjects. Using an adjoint-based error control for prototypical three dimensional flow problems, numerical experiments demonstrate the effectiveness of the proposed approach.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Brenner, S.C., Scott, R.L.: The mathematical theory of finite element methods. Springer, Heidelberg (1994)
Brezzi, F., Falk, R.: Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28(3), 581–590 (1991)
Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numerica 4, 105–158 (1995)
Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems, I: A linear model problem. SIAM J. Numer. Anal. 28, 43–77 (1991)
Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems, II, IV, V. SIAM J. Numer. Anal. 32, 706–740 (1995)
Giles, M.B.: On adjoint equations for error analysis and optimal grid adaptation. In: Caughey, D.A., Hafez, M.M. (eds.) Frontiers of Computational Fluid Dynamics 1998, pp. 155–170. World Scientific, Singapore (1998)
Griewank, A., Walther, A.: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Trans. Math. Software 26, 19–45 (2000)
Gunzburger, M.D.: Perspectives in flow control and optimization. Advances in Design and Control 5. SIAM, Philadelphia (2003)
Heuveline, V.: On higher-order mixed FEM for low Mach number flows: Application to a natural convection benchmark problem. Int. J. Num. Meth. Fluids 41(12), 1339–1356 (2003)
Heuveline, V., Walther, A.: Towards the economical computation of adjoints in PDEs using optimal online checkpointing (in preparation, 2006)
Hinze, M., Sternberg, J.: A-revolve: An adaptive memory- and run-time-reduced procedure for calculating adjoints; with an application to the instationary Navier-Stokes system. Opti. Meth. Softw. 20, 645–663 (2005)
Hood, P., Taylor, C.: A numerical solution of the Navier-Stokes equations using the finite element techniques. Comp. and Fluids 1, 73–100 (1973)
Machiels, L., Patera, A.T., Peraire, J.: Output bound approximation for partial differential equations; application to the incompressible Navier-Stokes equations. In: Biringen, S. (ed.) Industrial and Environmental Applications of Direct and Large Eddy Numerical Simulation, Springer, Heidelberg (1998)
Oden, J.T., Prudhomme, S.: On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors. Comput. Methods Appl. Mech. Eng. 176, 313–331 (1999)
Saad, Y.: Iterative methods for sparse linear systems. Computer Science/Numerical Methods. PWS Publishing Company (1996)
Schäfer, M., Turek, S.: Benchmark computations of laminar flow around cylinder. Notes on numerical fluid mechanics 52, 856–869 (1996)
Serban, R., Hindmarsh, A.C.: CVODES: An ODE solver with sensitivity analysis capabilities. UCRL-JP-20039, LLNL (2003)
Walther, A., Griewank, A.: Advantages of binomial checkpointing for memory-reduced adjoint calculations. In: Feistauer, M., et al. (eds.) Numerical mathematics and advanced applications, pp. 834–843. Springer, Heidelberg (2004)
Wesseling, P.: An introduction to multigrid methods. Wiley, Chichester (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Heuveline, V., Walther, A. (2006). Online Checkpointing for Parallel Adjoint Computation in PDEs: Application to Goal-Oriented Adaptivity and Flow Control. In: Nagel, W.E., Walter, W.V., Lehner, W. (eds) Euro-Par 2006 Parallel Processing. Euro-Par 2006. Lecture Notes in Computer Science, vol 4128. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11823285_72
Download citation
DOI: https://doi.org/10.1007/11823285_72
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37783-2
Online ISBN: 978-3-540-37784-9
eBook Packages: Computer ScienceComputer Science (R0)