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Generalized fast automatic differentiation technique

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Abstract

A new efficient technique intended for the numerical solution of a broad class of optimal control problems for complicated dynamical systems described by ordinary and/or partial differential equations is investigated. In this approach, canonical formulas are derived to precisely calculate the objective function gradient for a chosen finite-dimensional approximation of the objective functional.

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References

  1. A. F. Albu and V. I. Zubov, “Determination of functional gradient in an optimal control problem related to metal solidification,” Comput. Math. Math. Phys. 49 (1), 47–70 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  2. F. P. Vasil’ev, Optimization Methods (Faktorial, Moscow, 2002) [in Russian].

    Google Scholar 

  3. J.-M. Lellouche, J.-L. Devenon, and I. Dekeyser, “Boundary control of Burgers’ equation: A numerical approach,” Comput. Math. Appl. 28 (5), 33–44 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  4. Yu. G. Evtushenko, E. S. Zasukhina, and V. I. Zubov, “Numerical optimization of solutions to Burgers’ problems by means of boundary conditions,” Comput. Math. Math. Phys. 37 (12), 1406–1414 (1997).

    MathSciNet  MATH  Google Scholar 

  5. K. R. Aida-Zade and Yu. G. Evtushenko, “Fast automatic differentiation,” Mat. Model. 1, 121–139 (1989).

    MathSciNet  MATH  Google Scholar 

  6. Yu. G. Evtushenko and V. P. Mazurik, Optimization Software (Znanie, Moscow, 1989) [in Russian].

    MATH  Google Scholar 

  7. Y. G. Evtushenko, “Automatic differentiation viewed from optimal control theory,” Proceedings of the Workshop on Automatic Differentiation of Algorithms: Theory, Implementation, and Application (Philadelphia, SIAM, 1991).

    Google Scholar 

  8. M. Iri, “Simultaneous computation of functions, partial derivatives and estimates of rounding errors,” Jpn. J. Appl. Math. 1, 223–252 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  9. Automatic Differentiation of Algorithms: Theory, Implementation, and Application, Ed. by A. Griewank and G. F. Corliss (Philadelphia, SIAM, 1991).

  10. Yu. Evtushenko, E. Zasuhina, and V. Zubov, “Fast AD technique and inverse problems,” Abstracts of Third International Conference on Automatic Differentiation: From Simulation to Optimization, June 19–23, 2000 (INRIA, Sophia, Antipolis, France, 2000),pp. 23–24.

    Google Scholar 

  11. A. Griewank, Evaluating derivatives. Philadelphia: SIAM (2000).

    MATH  Google Scholar 

  12. U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation (SIAM, Philadelphia, 2012).

    MATH  Google Scholar 

  13. Y. G. Evtushenko, “Computation of exact gradients in distributed dynamic systems,” Optim. Methods Software 9, 45–75 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  14. Yu. G. Evtushenko, Optimization and Fast Automatic Differentiation (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2013) [in Russian].

    Google Scholar 

  15. A. F. Albu and V. I. Zubov, “Optimal control of the process of solidification,” Comput. Math. Math. Phys. 44 (1), 34–44 (2004).

    MathSciNet  Google Scholar 

  16. A. V. Albu, A. F. Albu, and V. I. Zubov, “Functional gradient evaluation in the optimal control of a complex dynamical system,” Comput. Math. Math. Phys. 51 (5), 762–780 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. V. Albu, A. F. Albu, and V. I. Zubov, “Control of substance solidification in a complex-geometry mold,” Comput. Math. Math. Phys. 52 (12), 1612–1623 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. F. Albu and V. I. Zubov, “Investigation of the optimal control problem for metal solidification in a new formulation,” Comput. Math. Math. Phys. 54 (5), 756–766 (2014).

    Article  MATH  Google Scholar 

  19. A. F. Albu and V. I. Zubov, “Investigation of the optimal control of metal solidification for a complex-geometry object in a new formulation,” Comput. Math. Math. Phys. 54 (12), 1804–1816 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  20. A. F. Albu and V. I. Zubov, “On the efficiency of solving optimal control problems by means of Fast Automatic Differentiation technique,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 21 (4), 20–29 (2015).

    MathSciNet  Google Scholar 

  21. S. L. Sobolev, Partial Differential Equations of Mathematical Physics, (Pergamon, Oxford, 1964; Nauka, Moscow, 1966).

    MATH  Google Scholar 

  22. G. I. Marchuk, Adjoint Equations and Analysis of Complex Systems (Nauka, Moscow, 1992; Kluwer Academic, Dordrecht, 1995).

    MATH  Google Scholar 

  23. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).

    MATH  Google Scholar 

  24. M. A. Lavrent’ev and L. A. Lusternik, A Course in Calculus of Variations (Gostekhizdat, Moscow, 1950) [in Russian].

    Google Scholar 

  25. I. M. Gelfand and S. V. Fomin, Calculus of Variations (Fizmatlit, Moscow, 1961; Prentice Hall, Englewood Cliffs, N.J., 1963).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control,”, Russian Academy of Sciences, Moscow, 119333, Russia

    Yu. G. Evtushenko & V. I. Zubov

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow oblast, 141700, Russia

    V. I. Zubov

Authors
  1. Yu. G. Evtushenko
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  2. V. I. Zubov
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Correspondence to V. I. Zubov.

Additional information

Original Russian Text © Yu.G. Evtushenko, V.I. Zubov, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 11, pp. 1847–1862.

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Evtushenko, Y.G., Zubov, V.I. Generalized fast automatic differentiation technique. Comput. Math. and Math. Phys. 56, 1819–1833 (2016). https://doi.org/10.1134/S0965542516110075

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  • DOI: https://doi.org/10.1134/S0965542516110075

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