Domain decomposition methods for advection-dominated linear-quadratic elliptic optimal control problems. (English) Zbl 1121.49031
Summary: We present an optimization-level domain decomposition (DD) preconditioner for the solution of advection dominated elliptic linear-quadratic optimal control problems, which arise in many science and engineering applications. The DD preconditioner is based on a decomposition of the optimality conditions for the elliptic linear-quadratic optimal control problem into smaller subdomain optimality conditions with Dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces. These subdomain optimality conditions are coupled through Robin transmission conditions for the states and the adjoints. The parameters in the Robin transmission condition depend on the advection. This decomposition leads to a Schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces. The Schur complement operator is the sum of subdomain Schur complement operators, the application of which is shown to correspond to the solution of subdomain optimal control problems, which are essentially smaller copies of the original optimal control problem. We show that, under suitable conditions, the application of the inverse of the subdomain Schur complement operators requires the solution of a subdomain elliptic linear-quadratic optimal control problem with Robin boundary conditions for the state.
Numerical tests for problems with distributed and with boundary control show that the dependence of the preconditioners on mesh size and subdomain size is comparable to its counterpart applied to a single advection dominated equation. These tests also show that the preconditioners are insensitive to the size of the control regularization parameter.
Numerical tests for problems with distributed and with boundary control show that the dependence of the preconditioners on mesh size and subdomain size is comparable to its counterpart applied to a single advection dominated equation. These tests also show that the preconditioners are insensitive to the size of the control regularization parameter.
MSC:
| 49M27 | Decomposition methods |
| 49M05 | Numerical methods based on necessary conditions |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 90C06 | Large-scale problems in mathematical programming |
| 90C20 | Quadratic programming |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76D55 | Flow control and optimization for incompressible viscous fluids |
| 76R99 | Diffusion and convection |
| 49N10 | Linear-quadratic optimal control problems |
Keywords:
Optimal control; Domain decomposition; Advection-diffusion equations; Preconditioning; Robin-Robin methods; Parallel computation; Stabilized finite elementsReferences:
| [1] | Abraham, F.; Behr, M.; Heinkenschloss, M., The effect of stabilization in finite element methods for the optimal boundary control of the Oseen equations, Finite Elem. Anal. Des., 41, 229-251 (2004) |
| [2] | Abraham, F.; Behr, M.; Heinkenschloss, M., Shape optimization in stationary blood flow: a numerical study of non-Newtonian effects, Comput. Methods Biomech. Biomed. Engrg., 8, 127-137 (2005) |
| [3] | Biros, G.; Ghattas, O., Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: The Krylov-Schur solver, SIAM J. Sci. Comput., 27, 2, 587-713 (2005) · Zbl 1091.65061 |
| [4] | Biros, G.; Ghattas, O., Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part II: The Lagrange-Newton solver and its application to optimal control of steady viscous flows, SIAM J. Sci. Comput., 27, 2, 714-739 (2005) · Zbl 1091.65062 |
| [5] | Gunzburger, M. D., Perspectives in Flow Control and Optimization (2003), SIAM: SIAM Philadelphia · Zbl 1088.93001 |
| [6] | Gunzburger, M. D.; Hou, L. S.; Svobotny, T. P., Heating and cooling control of temperature distributions along boundaries of flow domains, J. Math. Syst. Estimat. Control, 3, 147-172 (1993) |
| [7] | Gunzburger, M. D.; Lee, H. C., Analysis, approximation, and computation of a coupled solid/fluid temperature control problems, Comput. Methods Appl. Mech. Engrg., 118, 133-152 (1994) · Zbl 0847.76074 |
| [8] | He, J. W.; Glowinski, R.; Metcalfe, R.; Periaux, J., A numerical approach to the control and stabilization of advection-diffusion systems: application to viscous drag reduction, Int. J. Comput. Fluid Dyn., 11, 1-2, 131-156 (1998) · Zbl 0938.76024 |
| [9] | Ito, K.; Ravindran, S. S., Optimal control of thermally convected fluid flows, SIAM J. Sci. Comput., 19, 1847-1869 (1998) · Zbl 0918.49004 |
| [10] | Salinger, A. G.; Pawlowski, R. P.; Shadid, J. N.; van Bloemen Waanders, B. G., Computational analysis and optimization of a chemical vapor deposition reactor with large-scale computing, Ind. Eng. Chem. Res., 43, 16, 4612-4623 (2004) |
| [11] | Yang, G. Z.; Zabaras, N., An adjoint method for the inverse design of solidification processes with natural convection, Int. J. Numer. Methods Engrg., 42, 1121-1144 (1998) · Zbl 0912.76036 |
| [12] | Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differential Equations (1999), Oxford University Press: Oxford University Press Oxford · Zbl 0931.65118 |
| [13] | Smith, B.; Bjørstad, P.; Gropp, W., Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations (1996), Cambridge University Press: Cambridge University Press Cambridge, London, New York · Zbl 0857.65126 |
| [14] | Toselli, A.; Widlund, O., Domain Decomposition Methods—Algorithms and Theory. Domain Decomposition Methods—Algorithms and Theory, Computational Mathematics, vol. 34 (2004), Springer-Verlag: Springer-Verlag Berlin |
| [15] | Heinkenschloss, M.; Nguyen, H., Balancing Neumann-Neumann methods for elliptic optimal control problems, (Kornhuber, R.; Hoppe, R. H.W.; Periaux, J.; Pironneau, O.; Widlund, O. B.; Xu, J., Domain Decomposition methods in Science and Engineering. Domain Decomposition methods in Science and Engineering, Lecture Notes in Computational Science and Engineering, vol. 40 (2004), Springer-Verlag: Springer-Verlag Heidelberg), 589-596 · Zbl 1066.65072 |
| [16] | M. Heinkenschloss, H. Nguyen, Neumann-Neumann domain decomposition preconditioners for linear-quadratic elliptic optimal control problems, Tech. Rep. TR04-01, Department of Computational and Applied Mathematics, Rice University, SIAM J. Sci. Comput., in press.; M. Heinkenschloss, H. Nguyen, Neumann-Neumann domain decomposition preconditioners for linear-quadratic elliptic optimal control problems, Tech. Rep. TR04-01, Department of Computational and Applied Mathematics, Rice University, SIAM J. Sci. Comput., in press. · Zbl 1120.49025 |
| [17] | Achdou, Y.; Nataf, F., A Robin-Robin preconditioner for an advection-diffusion problem, C. R. Acad. Sci. Paris Sér. I Math., 325, 11, 1211-1216 (1997) · Zbl 0893.65061 |
| [18] | Achdou, Y.; Tallec, P. L.; Nataf, F.; Vidrascu, M., A domain decomposition preconditioner for an advection-diffusion problem, Comput. Methods Appl. Mech. Engrg., 184, 2-4, 145-170 (2000) · Zbl 0979.76043 |
| [19] | Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1994), Springer: Springer Berlin, Heidelberg, New York · Zbl 0852.76051 |
| [20] | Roos, H. G.; Stynes, M.; Tobiska, L., Numerical methods for singularly perturbed differential equations, Computational Mathematics, vol. 24 (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0844.65075 |
| [21] | Morton, K. W., Numerical Solution of Convection-Diffusion Problems (1996), Chapman & Hall: Chapman & Hall London, Glasgow, New York · Zbl 0861.65070 |
| [22] | Lions, J.-L., Optimal Control of Systems Governed by Partial Differential Equations (1971), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0203.09001 |
| [23] | Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041 |
| [24] | S.S. Collis, M. Heinkenschloss, Analysis of the streamline upwind/Petrov-Galerkin method applied to the solution of optimal control problems, Tech. Rep. TR02-01, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892. Available from: <http://www.caam.rice.edu/ heinken; S.S. Collis, M. Heinkenschloss, Analysis of the streamline upwind/Petrov-Galerkin method applied to the solution of optimal control problems, Tech. Rep. TR02-01, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892. Available from: <http://www.caam.rice.edu/ heinken |
| [25] | Knabner, P.; Angermann, L., Numerical Methods for Partial Differential Equations. Numerical Methods for Partial Differential Equations, Texts in Applied Mathematics, vol. 44 (2003), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0953.65075 |
| [26] | Gould, N. I.M., On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem, Math. Program., 32, 1, 90-99 (1985) · Zbl 0591.90068 |
| [27] | Keyes, D. E.; Gropp, W. D., A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM J. Sci. Statist. Comput., 8, S166-S202 (1987) · Zbl 0619.65088 |
| [28] | Haase, G.; Langer, U.; Meyer, A., Domain decomposition preconditioners with inexact subdomain solvers, J. Numer. Linear Alg. Appl., 1, 1, 27-42 (1992) |
| [29] | Davis, T. A., Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30, 2, 196-199 (2004) · Zbl 1072.65037 |
| [30] | Saad, Y.; Schultz, M. H., GMRES a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7, 856-869 (1986) · Zbl 0599.65018 |
| [31] | R.W. Freund, N.M. Nachtigal, A new Krylov-subspace method for symmetric indefinite linear systems, In: W.F. Ames (Ed.), Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, IMACS, 1994, pp. 1253-1256.; R.W. Freund, N.M. Nachtigal, A new Krylov-subspace method for symmetric indefinite linear systems, In: W.F. Ames (Ed.), Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, IMACS, 1994, pp. 1253-1256. |
| [32] | Freund, R. W.; Nachtigal, N. M., Software for simplified Lanczos and QMR algorithms, Appl. Numer. Math., 19, 319-341 (1995) · Zbl 0853.65041 |
| [33] | R. Bartlett, M. Heinkenschloss, D. Ridzal, B. van Bloemen Waanders, Domain decomposition methods for advection dominated linear-quadratic elliptic optimal control problems, Tech. Rep. SAND 2005-2895, Sandia National Laboratories, 2005.; R. Bartlett, M. Heinkenschloss, D. Ridzal, B. van Bloemen Waanders, Domain decomposition methods for advection dominated linear-quadratic elliptic optimal control problems, Tech. Rep. SAND 2005-2895, Sandia National Laboratories, 2005. · Zbl 1121.49031 |
| [34] | Babuška, I., The finite element method with penalty, Math. Comput., 27, 221-228 (1973) · Zbl 0299.65057 |
| [35] | Hou, L. S.; Ravindran, S. S., Numerical approximation of optimal control problems by a penalty method: error estimates and numerical results, SIAM J. Sci. Comput., 20, 1753-1777 (1999) · Zbl 0952.93036 |
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