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A fast and stable preconditioned iterative method for optimal control problem of wave equations. (English) Zbl 1327.49052

Summary: In this paper, we develop a new central finite difference scheme in terms of both time and space for solving the first-order necessary optimality systems that characterize the optimal control of wave equations. The obtained new scheme is proved to be unconditionally convergent with a second-order accuracy, without the requirement of the Courant-Friedrichs-Lewy condition on the corresponding grid ratio. An efficient preconditioned iterative method is further developed for solving the discretized sparse linear system based on the relationship between the resultant matrix structure and the coupled PDE optimality system. Numerical examples are presented to verify the theoretical analysis and to demonstrate the high efficiency of the proposed preconditioned iterative solver.

MSC:

49M25 Discrete approximations in optimal control
49M05 Numerical methods based on necessary conditions
49K20 Optimality conditions for problems involving partial differential equations
35L05 Wave equation
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
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[1] M. Benzi, G. H. Golub, and J. Liesen, {\it Numerical solution of saddle point problems}, Acta Numer., 14 (2005), pp. 1-137. · Zbl 1115.65034
[2] M. Bergounioux, X. Bonnefond, T. Haberkorn, and Y. Privat, {\it An optimal control problem in photoacoustic tomography}, Math. Models Methods Appl. Sci., 24 (2014), pp. 2525-2548. · Zbl 1298.49054
[3] A. Borz\`\i, K. Kunisch, and D. Y. Kwak, {\it Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system}, SIAM J. Control Optim., 41 (2003), pp. 1477-1497. · Zbl 1031.49029
[4] A. Borz\`\i and V. Schulz, {\it Computational Optimization of Systems Governed by Partial Differential Equations}, SIAM, Philadelphia, 2012.
[5] F. Bucci, {\it A Dirichlet boundary control problem for the strongly damped wave equation}, SIAM J. Control Optim., 30 (1992), pp. 1092-1100. · Zbl 0771.49009
[6] A. Chertock, M. Herty, and A. Kurganov, {\it An Eulerian-Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs}, Comput. Optim. Appl., 59 (2014), pp. 689-724. · Zbl 1306.49047
[7] M. Gerdts, G. Greif, and H. J. Pesch, {\it Numerical optimal control of the wave equation: Optimal boundary control of a string to rest in finite time}, Math. Comput. Simulation, 79 (2008), pp. 1020-1032. · Zbl 1162.65358
[8] M. Gugat and V. Grimm, {\it Optimal boundary control of the wave equation with pointwise control constraints}, Comput. Optim. Appl., 49 (2011), pp. 123-147. · Zbl 1226.49023
[9] M. Gugat, A. Keimer, and G. Leugering, {\it Optimal distributed control of the wave equation subject to state constraints}, ZAMM Z. Angew. Math. Mech., 89 (2009), pp. 420-444. · Zbl 1176.35176
[10] M. D. Gunzburger, {\it Perspectives in Flow Control and Optimization}, SIAM, Philadelphia, 2003. · Zbl 1088.93001
[11] W. Hackbusch, {\it Elliptic Differential Equations: Theory and Numerical Treatment}, Springer-Verlag, Berlin, 2010. · Zbl 1205.35060
[12] M. Heinkenschloss, {\it A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems}, J. Comput. Appl. Math., 173 (2005), pp. 169-198. · Zbl 1075.65091
[13] M. Herty, A. Kurganov, and D. Kurochkin, {\it Numerical method for optimal control problems governed by nonlinear hyperbolic systems of PDEs}, Commun. Math. Sci., 13 (2015), pp. 15-48. · Zbl 1308.49020
[14] R. Herzog and E. Sachs, {\it Preconditioned conjugate gradient method for optimal control problems with control and state constraints}, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2291-2317. · Zbl 1209.49038
[15] J. G. Heywood and R. Rannacher, {\it Finite-element approximation of the nonstationary Navier-Stokes problem. {\it IV}. Error analysis for second-order time discretization}, SIAM J. Numer. Anal., 27 (1990), pp. 353-384. · Zbl 0694.76014
[16] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, {\it Optimization with PDE Constraints}, Springer, New York, 2009. · Zbl 1167.49001
[17] A. Jameson, {\it Aerodynamic design via control theory}, J. Sci. Comput., 3 (1988), pp. 233-260. · Zbl 0676.76055
[18] B. S. Jovanović and E. Süli, {\it Analysis of Finite Difference Schemes}, Springer Ser. Comput. Math. 46, Springer, London, 2014. · Zbl 1335.65071
[19] A. Kröner, {\it Adaptive finite element methods for optimal control of second order hyperbolic equations}, Comput. Methods Appl. Math., 11 (2011), pp. 214-240. · Zbl 1283.35054
[20] A. Kröner, {\it Numerical Methods for Control of Second Order Hyperbolic Equations}, Ph.D. thesis, Fakultät für Mathematik, Technische Universität München, 2011. · Zbl 1283.35054
[21] A. Kröner, {\it Semi-smooth Newton methods for optimal control of the dynamical Lamé system with control constraints}, Numer. Funct. Anal. Optim., 34 (2013), pp. 741-769. · Zbl 1278.49034
[22] A. Kröner and K. Kunisch, {\it A minimum effort optimal control problem for the wave equation}, Comput. Optim. Appl., 57 (2014), pp. 241-270. · Zbl 1283.49031
[23] A. Kröner, K. Kunisch, and B. Vexler, {\it Semismooth Newton methods for optimal control of the wave equation with control constraints}, SIAM J. Control Optim., 49 (2011), pp. 830-858. · Zbl 1218.49035
[24] K. Kunisch and D. Wachsmuth, {\it On time optimal control of the wave equation and its numerical realization as parametric optimization problem}, SIAM J. Control Optim., 51 (2013), pp. 1232-1262. · Zbl 1268.49035
[25] K. Kunisch and D. Wachsmuth, {\it On time optimal control of the wave equation, its regularization and optimality system}, ESAIM Control Optim. Calc. Var., 19 (2013), pp. 317-336. · Zbl 1268.49025
[26] R. LeVeque, {\it Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems}, SIAM, Philadelphia, 2007. · Zbl 1127.65080
[27] J.-L. Lions, {\it Optimal Control of Systems Governed by Partial Differential Equations}, Springer-Verlag, New York, 1971. · Zbl 0203.09001
[28] J. Liu, Y. Huang, H. Sun, and M. Xiao, {\it Numerical methods for weak solution of wave equation with Van Der Pol type nonlinear boundary conditions}, Numer. Methods Partial Differential Equations, DOI:10.1002/num.21997. · Zbl 1339.65131
[29] X. Luo, Y. Chen, and Y. Huang, {\it A priori error estimates of finite volume element method for hyperbolic optimal control problems}, Sci. China Math., 56 (2013), pp. 901-914. · Zbl 1264.65179
[30] J. W. Pearson and M. Stoll, {\it Fast iterative solution of reaction-diffusion control problems arising from chemical processes}, SIAM J. Sci. Comput., 35 (2013), pp. B987-B1009. · Zbl 1281.65095
[31] J. W. Pearson, M. Stoll, and A. J. Wathen, {\it Regularization-robust preconditioners for time-dependent PDE-constrained optimization problems}, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 1126-1152. · Zbl 1263.65035
[32] M. Porcelli, V. Simoncini, and M. Tani, {\it Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems}, arXiv:1407.1144, 2014. · Zbl 1325.65066
[33] T. Rees, H. S. Dollar, and A. J. Wathen, {\it Optimal solvers for PDE-constrained optimization}, SIAM J. Sci. Comput., 32 (2010), pp. 271-298. · Zbl 1208.49035
[34] A. Rincon and I.-S. Liu, {\it On numerical approximation of an optimal control problem in linear elasticity}, Divulg. Mat., 11 (2003), pp. 91-107. · Zbl 1058.49027
[35] Y. Saad, {\it Iterative Methods for Sparse Linear Systems}, SIAM, Philadelphia, 2003. · Zbl 1031.65046
[36] A. Schiela and S. Ulbrich, {\it Operator preconditioning for a class of inequality constrained optimal control problems}, SIAM J. Optim., 24 (2014), pp. 435-466. · Zbl 1291.65092
[37] J. Schöberl and W. Zulehner, {\it Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems}, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 752-773. · Zbl 1154.65029
[38] M. Stoll and A. Wathen, {\it Preconditioning for partial differential equation constrained optimization with control constraints}, Numer. Linear Algebra Appl., 19 (2012), pp. 53-71. · Zbl 1274.65189
[39] J. C. Strikwerda, {\it Finite Difference Schemes and Partial Differential Equations}, 2nd ed., SIAM, Philadelphia, 2004. · Zbl 1071.65118
[40] F. Tröltzsch, {\it Optimal Control of Partial Differential Equations}, AMS, Providence, RI, 2010. · Zbl 1195.49001
[41] E. Zuazua, {\it Propagation, observation, and control of waves approximated by finite difference methods}, SIAM Rev., 47 (2005), pp. 197-243. · Zbl 1077.65095
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