zbMATH — the first resource for mathematics

Simultaneous single-step one-shot optimization with unsteady PDEs. (English) Zbl 1327.65123
Summary: The single-step one-shot method has proven to be very efficient for PDE-constrained optimization where the partial differential equation (PDE) is solved by an iterative fixed point solver. In this approach, the simulation and optimization tasks are performed simultaneously in a single iteration. If the PDE is unsteady, finding an appropriate fixed point iteration is non-trivial. In this paper, we provide a framework that makes the single-step one-shot method applicable for unsteady PDEs that are solved by classical time-marching schemes. The one-shot method is applied to an optimal control problem with unsteady incompressible Navier-Stokes equations that are solved by an industry standard simulation code. With the Van-der-Pol oscillator as a generic model problem, the modified simulation scheme is further improved using adaptive time scales. Finally, numerical results for the advection-diffusion equation are presented.

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M25 Discrete approximations in optimal control
35Q30 Navier-Stokes equations
Full Text: DOI arXiv
[1] Lee-Rausch, E.; Hammond, D.; Nielsen, E.; Pirzadeh, S.; Rumsey, C., Application of the FUN3D solver to the 4th AIAA drag prediction workshop, AIAA J. Aircr., 51, 1149-1160, (2014)
[2] S. Langer, A. Schwöppe, N. Kroll, The DLR flow solver TAU—Status and recent algorithmic developments, in: 52nd Aerospace Sciences Meeting, 13-17. January 2014, National Harbor, Maryland, USA, 2014.
[3] Nocedal, J.; Wright, S., Numerical optimization, (2006), Springer Science + Business Media · Zbl 1104.65059
[4] Lions, J., Optimal control of systems governed by partial differential equations, (1971), Springer · Zbl 0203.09001
[5] Pironneau, O., On optimum design in fluid mechanics, J. Fluid Mech., 64, 97-110, (1974) · Zbl 0281.76020
[6] Nadarajah, S.; Jameson, A., Optimum shape design for unsteady flows with time-accurate continuous and discrete adjoint methods, AIAA J., 45, 1478-1491, (2007)
[7] Griewank, A.; Walther, A., Evaluating derivatives: principles and techniques of algorithmic differentiation, (2008), SIAM · Zbl 1159.65026
[8] Akçelik, V.; Biros, G.; Ghattas, O.; Hill, J.; Keyes, D.; van Bloemen Waanders, B., Parallel algorithms for PDE-constrained optimization, (Heroux, M. A.; Raghaven, P.; Simon, H. D., Parallel Processing for Scientific Computing, (2006), SIAM), 291-322
[9] Biegler, L. T.; Ghattas, O.; Heinkenschloss, M.; van Bloemen Waanders, B., Large-scale PDE-constrained optimization, vol. 30, (2003), Springer · Zbl 1021.00015
[10] Ziems, J. C.; Ulbrich, S., Adaptive multilevel inexact SQP methods for PDE-constrained optimization, SIAM J. Optim., 21, 1, 1-40, (2011) · Zbl 1214.49027
[11] 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, 687-713, (2005) · Zbl 1091.65061
[12] 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
[13] Abbeloos, D.; Diehl, M.; Hinze, M.; Vandewalle, S., Nested multigrid methods for time-periodic, parabolic optimal control problems, Comput. Vis. Sci., 14, 1, 27-38, (2011) · Zbl 1241.65061
[14] Ta’asan, S., One-shot methods for optimal control of distributed parameter systems I: finite dimensional control, icase report no, tech. rep, (1991), Institute for Computer Applications in Science and Engineering, NASA Langley Research Center
[15] Gauger, N.; Özkaya, E., Single-step one-shot aerodynamic shape optimization, Internat. Ser. Numer. Math., 158, 191-204, (2009) · Zbl 1197.49046
[16] Hazra, S.; Schulz, V., Simultaneous pseudo-timestepping for PDE-model based optimization problems, BIT, 44, 457-472, (2004) · Zbl 1066.65071
[17] Gauger, N.; Griewank, A.; Hamdi, A.; Kratzenstein, C.; Özkaya, E.; Slawig, T., Automated extension of fixed point PDE solvers for optimal design with bounded retardation, (Leugering, G.; Engell, S.; Griewank, A.; Hinze, M.; Rannacher, R.; Schulz, V.; Ulbrich, M.; Ulbrich, S., Constrained Optimization and Optimal Control for Partial Differential Equations, (2012), Springer Basel), 99-122 · Zbl 1356.49060
[18] Griewank, A.; Hamdi, A., Properties of an augmented Lagrangian for design optimization, Optim. Methods Softw., 25, 645-664, (2010) · Zbl 1225.90124
[19] Hazra, S.; Schulz, V.; Brezillon, J.; Gauger, N., Aerodynamic shape optimization using simultaneous pseudo-timestepping, J. Comput. Phys., 204, 46-64, (2005) · Zbl 1143.76564
[20] Bosse, T.; Gauger, N.; Griewank, A.; Günther, S.; Kaland, L., Optimal design with bounded retardation for problems with non-separable adjoints, Internat. Ser. Numer. Math., 165, 67-84, (2014) · Zbl 1332.49034
[21] Griewank, A.; Hamdi, A., Reduced quasi-Newton method for simultaneous design and optimization, Comput. Optim. Appl., 49, 521-548, (2011) · Zbl 1251.90364
[22] A. Jameson, Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, in: Proc. 10th Comp. Fluid Dyn. Conf., Honolulu, USA, June 24-26, AIAA-Paper 91-1596, 1991.
[23] Günther, S.; Gauger, N.; Wang, Q., Extension of the one-shot method for optimal control with unsteady pdes, (Computational Methods in Applied Sciences Series, Vol. 36, (2015), Springer), 127-142
[24] Barner, M.; Flohr, F., Analysis II, (1995), Walter de Gruyter
[25] Naumann, U., The art of differentiating computer programs: an introduction to algorithmic differentiation, software, environments, and tools, (2012), Society for Industrial and Applied Mathematics
[26] Bosse, T.; Gauger, N.; Griewank, A.; Günther, S.; Schulz, V., One-shot approaches to design optimzation, Internat. Ser. Numer. Math., 165, 43-66, (2014) · Zbl 1327.90300
[27] Özkaya, E., One-shot methods for aerodynamic shape optimization, (2014), RWTH Aachen University, (Ph.D. thesis)
[28] Lambert, J. D., Numerical methods for ordinary differential systems: the initial value problem, (1991), Wiley Chichester · Zbl 0745.65049
[29] Breuer, M.; Hänel, D., A dual time-stepping method for 3D, viscous, incompressible vortex flows, Comput. & Fluids, 22, 4-5, 467-484, (1993) · Zbl 0779.76049
[30] Höll, T.; Vel Job, A. K.; Giacopinelli, P.; Thiele, F., Numerical study of active flow control on a high-lift configuration, J. Aircr., 49, 5, 1406-1422, (2012)
[31] Ferziger, J.; Peric, M., Computational methods for fluid dynamics, (2002), Springer Berlin, Heidelberg · Zbl 0998.76001
[32] TROPICS (Research Team of INRIA Sophia-Antipolis), Tapenade 3.8, online automatic differentiation engine, August 2014. http://www-sop.inria.fr/tropics/tapenade.html, tropics@sophia.inria.fr.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.