Fully implicit Lagrange-Newton-Krylov-Schwarz algorithms for boundary control of unsteady incompressible flows. (English) Zbl 1253.65145
Summary: We develop a parallel fully implicit domain decomposition algorithm for solving optimization problems constrained by time-dependent nonlinear partial differential equations. In particular, we study the boundary control of unsteady incompressible Navier-Stokes equations. After an implicit discretization in time, a fully coupled sparse nonlinear optimization problem needs to be solved at each time step. The class of full space Lagrange-Newton-Krylov-Schwarz algorithms is used to solve the sequence of optimization problems. Among optimization algorithms, the fully implicit full space approach is considered to be the easiest to formulate and the hardest to solve. We show that Lagrange-Newton-Krylov-Schwarz, with a one-level restricted additive Schwarz preconditioner, is an efficient class of methods for solving these hard problems. To demonstrate the scalability and robustness of the algorithm, we consider several problems with a wide range of Reynolds numbers and time step sizes, and we present numerical results for large-scale calculations involving several million unknowns obtained on machines with more than 1000 processors.
MSC:
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 76D55 | Flow control and optimization for incompressible viscous fluids |
| 49M25 | Discrete approximations in optimal control |
| 65Y05 | Parallel numerical computation |
Keywords:
Schwarz preconditioner; parallel computing; nonlinear constrained optimization; inexact Newton flow control; unsteady incompressible Navier-Stokes equationsSoftware:
PETScReferences:
| [1] | Gunzburger, Perspectives in Flow Control and Optimization (2003) |
| [2] | Prudencio, Parallel full space SQP Lagrange-Newton-Krylov-Schwarz algorithms for PDE-constrained optimization problems, SIAM Journal on Scientific Computing 27 pp 1305– (2006) · Zbl 1092.49024 · doi:10.1137/040602997 |
| [3] | Prudencio E Parallel fully coupled Lagrange-Newton-Krylov-Schwarz algorithms and software for optimization problems constrained by partial differential equations PhD Thesis 2005 |
| [4] | Prudencio, Parallel multilevel restricted Schwarz preconditioners with pollution removing for PDE-constrained optimization, SIAM Journal on Scientific Computing 29 pp 964– (2007) · Zbl 1357.76020 · doi:10.1137/050635663 |
| [5] | Bewley, Flow control: new challenges for a new renaissance, Progress in Aerospace Sciences 37 pp 21– (2001) · doi:10.1016/S0376-0421(00)00016-6 |
| [6] | Bewley, A general framework for robust control in fluid mechanics, Physica D 138 pp 360– (2000) · Zbl 0981.76026 · doi:10.1016/S0167-2789(99)00206-7 |
| [7] | Moin, Feedback control of turbulence, Applied Mechanics Reviews 47 pp 3– (1994) · doi:10.1115/1.3124438 |
| [8] | Biros, Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization, part I: the Krylov-Schur solver, SIAM Journal on Scientific Computing 27 pp 687– (2005) · Zbl 1091.65061 · doi:10.1137/S106482750241565X |
| [9] | Biros, 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 Journal on Scientific Computing 27 pp 714– (2005) · Zbl 1091.65062 · doi:10.1137/S1064827502415661 |
| [10] | Aamo, Control of mixing by boundary feedback in 2D channel flow, Automatica 39 pp 1597– (2003) · Zbl 1055.93039 · doi:10.1016/S0005-1098(03)00140-7 |
| [11] | Balogh, Stabililty enhancement by boundary control in 2-D channel flow, IEEE Transactions on Automatic Control 46 pp 1696– (2001) · Zbl 1011.93061 · doi:10.1109/9.964681 |
| [12] | Ravindran, Numerical approximation of optimal control of unsteady flows using SQP and time decomposition, International Journal for Numerical Methods in Fluids 45 pp 21– (2004) · Zbl 1085.76513 · doi:10.1002/fld.651 |
| [13] | Gunzburger, The velocity tracking problem for Navier-Stokes flow with boundary control, SIAM Journal on Numerical Analysis 39 pp 594– (2000) · Zbl 0991.49002 |
| [14] | Gunzburger, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control, SIAM Journal on Numerical Analysis 37 pp 481– (2000) · Zbl 0963.35150 · doi:10.1137/S0036142997329414 |
| [15] | Quartapelle, International Series of Numerical Mathematics pp 1– (1993) · Zbl 0784.76020 |
| [16] | Fursikov, Boundary value problems and optimal boundary control for the Navier-Stokes system: the two-dimensional case, SIAM Journal on Control and Optimization 36 pp 852– (1998) · Zbl 0910.76011 · doi:10.1137/S0363012994273374 |
| [17] | Girault, The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms (1986) · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5 |
| [18] | Hou, A penalized Neumann control approach for solving an optimal Dirichlet control problem for the Navier-Stokes equations, SIAM Journal on Control and Optimization 36 pp 1795– (1998) · Zbl 0917.49003 · doi:10.1137/S0363012996304870 |
| [19] | Ito, Optimal control of thermally convected fluid flows, SIAM Journal on Scientific Computing 199 pp 1847– (1998) · Zbl 0918.49004 · doi:10.1137/S1064827596299731 |
| [20] | Hairer, Solving Ordinary Differential Equations I (1993) |
| [21] | Nocedal, Numerical Optimization (2000) |
| [22] | Ghattas, Optimal control of two- and three-dimensional incompressible Navier-Stokes flows, Journal of Computational Physics 136 pp 231– (1997) · Zbl 0893.76067 · doi:10.1006/jcph.1997.5744 |
| [23] | Ghattas, Multidisciplinary Design Optimization: State of the Art (1997) |
| [24] | Kunisch, Reduced SQP methods for parameter identification problems, SIAM Journal on Numerical Analysis 29 pp 1793– (1992) · Zbl 0772.65085 · doi:10.1137/0729100 |
| [25] | Kupfer, Numerical solution of a nonlinear parabolic control problem by a reduced SQP method, Computational Optimization and Applications 1 pp 113– (1992) · Zbl 0771.49010 · doi:10.1007/BF00247656 |
| [26] | Schulz, Solving discretized optimization problems by partially reduced SQP methods, Computing and Visualization in Science 1 pp 83– (1998) · Zbl 0970.65066 · doi:10.1007/s007910050008 |
| [27] | Eisenstat, Globally convergent inexact Newton method, SIAM Journal on Optimization 4 pp 393– (1994) · Zbl 0814.65049 · doi:10.1137/0804022 |
| [28] | Eisenstat, Choosing the forcing terms in an inexact Newton method, SIAM Journal on Scientific Computing 17 pp 16– (1996) · Zbl 0845.65021 · doi:10.1137/0917003 |
| [29] | Smith, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (1996) · Zbl 0857.65126 |
| [30] | Toselli, Domain Decomposition Methods-Algorithms and Theory (2005) · Zbl 1069.65138 · doi:10.1007/b137868 |
| [31] | Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1996) · Zbl 0847.65038 · doi:10.1137/1.9781611971200 |
| [32] | Cai, A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM Journal on Scientific Computing 21 pp 92– (1999) · Zbl 0944.65031 · doi:10.1137/S106482759732678X |
| [33] | Cai, Additive Schwarz algorithms for parabolic convection-diffusion equations, Numerische Mathematik 60 pp 41– (1990) · Zbl 0737.65078 · doi:10.1007/BF01385713 |
| [34] | Balay, PETSc Users Manual (2009) |
| [35] | Saad, Iterative Methods for Sparse Linear Systems (2003) · Zbl 1031.65046 · doi:10.1137/1.9780898718003 |
| [36] | Coleman, Estimation of sparse Jacobian matrices and graph coloring problems, SIAM Journal on Numerical Analysis 20 pp 187– (1983) · Zbl 0527.65033 · doi:10.1137/0720013 |
| [37] | Cai, Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation, SIAM Journal on Scientific Computing 19 pp 246– (1998) · Zbl 0917.76035 · doi:10.1137/S1064827596304046 |
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