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Adjoint concepts for the optimal control of Burgers equation. (English) Zbl 1278.49028

Summary: Adjoint techniques are a common tool in the numerical treatment of optimal control problems. They are used for efficient evaluations of the gradient of the objective in gradient-based optimization algorithms. Different adjoint techniques for the optimal control of Burgers equation with Neumann boundary control are studied. The methods differ in the point in the numerical algorithm at which the adjoints are incorporated. Discretization methods for the continuous adjoint are discussed and compared with methods applying the application of the discrete adjoint. At the example of the implicit Euler method and the Crank Nicolson method it is shown that a discretization for the adjoint problem that is adjoint to the discretized optimal control problem avoids additional errors in gradient-based optimization algorithms. The approach of discrete adjoints coincides with that of automatic differentiation tools (AD) which provide exact gradient calculations on the discrete level.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
35Q53 KdV equations (Korteweg-de Vries equations)
76D55 Flow control and optimization for incompressible viscous fluids
49M25 Discrete approximations in optimal control
65K10 Numerical optimization and variational techniques
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[1] W. Alt, ”The Lagrange-Newton method for infinite-dimensional optimization problems,” Numer. Func. Anal. and Optim., vol. 11, pp. 201–224, 1990. · Zbl 0694.49022 · doi:10.1080/01630569008816371
[2] C. Bendtsen and O. Stauning, ”FADBAD, a flexible C++ package for automatic differentiation,” Technical University of Denmark, IMM, Technical Report IMM-REP-1996-17 (1996).
[3] G. Biros and O. Ghattas, ”Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: the Krylov-Schur solver,” SIAM Journal on Scientific Computing, to appear. · Zbl 1091.65061
[4] C. Bischof, A. Carle, P. Khademi, A. Maurer, and P. Hovland, ADIFOR 2.0 User’s Guide, ANL. ANL/MCS-TM-192, 1994.
[5] P.G. Ciarlet, Introduction to Numerical Linear Algebra and Optimization, Cambridge Univers. Press, 1989.
[6] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I, Springer Verlag, 1992. · Zbl 0755.35001
[7] R. Giering and T. Kaminski, ”Recies for adjoint code construction,” ACM Trans. Math. Software, vol. 24, pp. 437–474, 1998. · Zbl 0934.65027 · doi:10.1145/293686.293695
[8] R. Griesse and A. Walther, ”Evaluating gradients in optimal control–continuous adjoints versus automatic differentiation,” J. Optim. Theory and Appl.(JOTA), vol. 122, no. 1, pp. 63–86, 2004. · Zbl 1130.49308 · doi:10.1023/B:JOTA.0000041731.71309.f1
[9] A. Griewank, ”On automatic differentiation,” in Mathematical Programming: Recent Developments and Applications, (Eds.), M. Iri and K. Tanabe, Kluwer Academic Publishers, 1989, pp 83–107. · Zbl 0696.65015
[10] A. Griewank, ”Evaluating derivatives: Principles and techniques of algorithmic differentiation,” Frontiers in appl. math vol. 19, 2000. · Zbl 0958.65028
[11] A. Griewank, D. Juedes, and J. Utke, ”ADOL–C, a package for the automatic differentiation of algorithms written in C/C++,” ACM Trans. Math. Soft., vol. 22, pp. 131–167, 1996. · Zbl 0884.65015 · doi:10.1145/229473.229474
[12] A. Griewank and A. Walther, ”Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation,” TOMS, vol. 26, no. 1, 2000. · Zbl 1137.65330
[13] A. Griewank and A. Walther, ”Applying the checkpointing routine treeverse to discretizations of Burgers’ equation,” Lect. Notes Comput. Sci.and Engin. 8: in High Performance Scientific and Engineering Computing, H.-J. Bungartz, F. Durst, and C. Zenger (Eds.) Springer Berlin Heidelberg, 1999.
[14] Ch. Grossmann and A. Noack, ”Linearizations and adjoints of operator equations – constructions and selected applications,” Preprint MATH-NM-08-01, TU Dresden, 2001.
[15] Ch. Grossmann and H.-G. Roos, ”Numerik partieller Differentialgleichungen,” Teubner, 1992.
[16] M. Hinze, ”Optimal and instantaneous control of the instationary Navier-Stokes equations,” Habilitationsschrift, Fachbereich Mathematik, Technische Universität Berlin, 1999.
[17] M. Hinze and S. Volkwein, ”Analysis of instantaneous control for the Burgers equation,” Nonlinear Anal. TMA 50A, no. 1, pp. 1–26, 2002. · Zbl 1022.49001
[18] D. Keyes, P. Hovland, L. McInnes, and W. Samyono, ”Using automatic differentiation for second-order matrix-free methods in pde-constrained optimization,” in Automatic Differentiation of Algorithms: From Simulation to Optimization, G. Corliss, C. Faure, A. Griewank, L. Hascoet, and U. Naumann (Eds.), Springer, New York, 2001.
[19] K. Kunisch and S. Volkwein, ”Control of Burgers’ equation by a reduced order approach using proper orthogonal decomposition,” J. Optim. Theory and Appl, vol. 102, pp. 345–371, 1999. · Zbl 0949.93039 · doi:10.1023/A:1021732508059
[20] J.-M. Lellouche, J.-L. Devenon, and I. Dekeyser, ”Boundary control of Burgers’ equation–a numerical approach,” Comput Math. Applic., vol. 28, no. 5, pp. 33–44, 1994. · Zbl 0807.76065 · doi:10.1016/0898-1221(94)00138-3
[21] Y. Leredde, J.-M. Lellouche, J.-L. Devenon, and I. Dekeyser, ”On initial, boundary conditions and viscosity coefficient control for Burgers’ equation,” Int. J. Numer. Meth. Fluids, vol. 28, pp. 113–128, 1998. · Zbl 0928.76035 · doi:10.1002/(SICI)1097-0363(19980715)28:1<113::AID-FLD702>3.0.CO;2-1
[22] R. Temam, Navier-Stokes Equations,” Studies in Mathematics and its Applications. North-Holland, 1979. · Zbl 0426.35003
[23] F. Tröltzsch, ”On the Lagrange-Newton-SQP method for the optimal control of semilinear parabolic equations,” SIAM J. Control Optimization, vol. 38, no. 1, pp. 294–312, 1999. · Zbl 0954.49018 · doi:10.1137/S0363012998341423
[24] F. Tröltzsch, and S. Volkwein, ”The SQP method for control constrained optimal control of the Burgers equation,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 6, pp. 649–674, 2001. · Zbl 1001.49035 · doi:10.1051/cocv:2001127
[25] A. Verma, ”ADMAT: An automatic differentiation and MATLAB interface toolbox,” in Object-Oriented Methods for Interoperable Scientific and Engineering Computing, M. Henderson, C. Anderson, and S. Lyons (Eds.), SIAM, 1999, pp. 174–183.
[26] S. Volkwein, ”Second order conditions for boundary control problems of the Burgers equation,” Control and Cybernatics, vol. 30, pp. 249–278, 2001. · Zbl 1001.93033
[27] S. Volkwein, ”Mesh-independence of an augmented Lagrangian-SQP method in Hilbert spaces and control problems for the Burgers equation,” Diss. at the University of Technology Berlin, 1997.
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