Parallel full space SQP Lagrange-Newton-Krylov-Schwarz algorithms for PDE-constrained optimization problems

SIAM Journal on Scientific Computing

Volumn 27, Issue 4, 2006, Pages 1305-1328

  Prudenciq, Ernesto E ^a   Byrd, Richard ^b   Cai, Xiao Chuan ^b  

^a SLAC NATIONAL ACCELERATOR LABORATORY   (United States)

^b UCB 450   (United States)

Author keywords

Domain decomposition; Flow control; Incompressible Navier Stokes equations; Inexact Newton method; Nonlinear partial differential equations constrained optimization; Parallel computing; Schwarz preconditioners

Indexed keywords

ALGORITHMS; BOUNDARY ELEMENT METHOD; CONSTRAINT THEORY; FLOW CONTROL; ITERATIVE METHODS; NONLINEAR EQUATIONS; PROBLEM SOLVING; QUADRATIC PROGRAMMING; REYNOLDS NUMBER;

DOMAIN DECOMPOSITION; INCOMPRESSIBLE NAVIER-STOKES EQUATIONS; INEXACT NEWTON METHOD; NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS CONSTRAINED OPTIMIZATION; SCHWARZ PRECONDITIONERS;

PARALLEL PROCESSING SYSTEMS;

EID: 33646380214     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/040602997     Document Type: Article

References (45)

1. F. ABERGEL AND R. TEMAM, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dyn., 1 (1990), pp. 303-325.
2. S. BALAY, K. BUSCHELMAN, W. D. GROPP, D. KAUSHIK, M. KNEPLEY, L. C. MCINNES, B. F. SMITH, AND H. ZHANG, Portable, Extensible Toolkit for Scientific Computation (PETSc), http://www.mcs.anl.gov/petsc (2004).
3. H. T. BANKS AND K. KUNISCH, Estimation Techniques for Distributed Parameter Systems, Systems Control Found. Appl. 1, Birkhäuser Boston, Boston, 1989.
4. A. BATTERMANN AND M. HEINKENSCHLOSS, Preconditioned for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems, in Control and Estimation of Distributed Parameter Systems, Internat. Ser. Numer. Math. 126, W. Desch, F. Kappel, and K. Kunisch, eds., Birkhäuser-Verlag, Basel, Switzerland, 1998, pp. 15-32.
5. M. BERGOUNIOUX AND K. KUNISH, Augmented Lagrangian algorithms for state constrained optimal control problems, in Control and Estimation of Distributed Parameter Systems, Internat. Ser. Numer. Math. 126, W. Desch, F. Kappel, and K. Kunisch, eds., Birkhäuser-Verlag, Basel, Switzerland, 1998, pp. 33-48.
6. M. BERGOUNIOUX AND K. KUNISH, On the structure of the Lagrange multiplier for state-constrained optimal control problems, Systems Control Lett., 48 (2002), pp. 16-176.
7. T. R. BEWLEY, Flow control: New challenges for a new renaissance, Progr. Aerospace Sci., 37 (2001), pp. 21-58.
8. G. BIRO AND O. GHATTAS, Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: The Krylov-Schur solver, SIAM J. Sci. Comput., 27 (2005), pp. 687-713.
9. G. BIROS AND O. GHATTAS, 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 (2005), pp. 714-739.
10. A. BORZI AND K. KUNISH, A multigrid method for optimal control of time-dependent reaction diffusion processes, in Fast Solution of Discretized Optimization Problems, Internat. Ser. Numer. Math. 138, K.-H. Hoffman, R. H. W. Hoppe, and V. Schulz, eds., BirkhäuserVerlag, Basel, Switzerland, 2000, pp. 50-57.
11. X.-C. CAI, M. DRYJA, AND M. SARKIS, Restricted additive Schwarz preconditioned with harmonic overlap for symmetric positive definite linear systems, SIAM J. Numer. Anal., 41 (2003), pp. 1209-1231.
12. X.-C. CAI, W. D. GROPP, D. E. KEYES, R. G. MELVIN, AND D. P. YOUNG, Parallel NewtonKrylov-Schwarz algorithms for the transonic full potential equation, SIAM J. Sci. Comput., 19 (1998), pp. 246-265.
13. X.-C. CAI AND D. E. KEYES, Nonlinearly preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 24 (2002), pp. 183-200.
14. X.-C. CAI, D. E. KEYES, AND L. MARCINKOWSKI, Nonlinear additive Schwarz preconditioned and applications in computational fluid dynamics, Int. J. Numer. Meth. Fluids, 40 (2002), pp. 1463-1470.
15. X.-C. CAI AND M. SARKIS, A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 21 (1999), pp. 792-797.
16. H. CHOI, M. HINZE, AND K. KUNISH, Instantaneous control of backward-facing step flows, Appl. Numer. Math., 31 (1999), pp. 133-158.
17. T. F. COLEMAN AND J. J. MORE, Estimation of sparse Jacobian matrices and graph coloring problems, SIAM J. Numer. Anal., 20 (1983), pp. 187-209.
18. O. DAUBE, Resolution of the 2D Navier-Stokes equations in velocity-vorticity form by means of an influence matrix technique, J. Comput. Phys., 103 (1992), pp. 402-414.
19. J. E. DENNIS, JR., AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics Appl. Math. 16, SIAM, Philadelphia, 1996.
20. M. DRYJA AND O. B. WIDLUND, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comp., 15 (1994), pp. 604-620.
21. S. C. EISENSTAT AND H. F. WALKER, Globally convergent inexact Newton methods, SIAM J. Optim., 4 (1994), pp. 393-422.
22. S. C. EISENSTAT AND H. F. WALKER, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), pp. 16-32.
23. A. V. FURSIKOV, Optimal Control of Distributed Systems. Theory and Applications, Trans. Math. Monogr. 187, AMS, Providence, RI, 2000.
24. A. GREENBAUM, Iterative Methods for Solving Linear Systems, Frontiers Appl. Math. 17, SIAM, Philadelphia, 1997.
25. G. GUJ AND F. STELLA, Numerical solutions of high-Re recirculating flows in vorticity-velocity form, Internat. J. Numer. Methods Fluids, 8 (1988), pp. 405-416.
26. M. D. GUNZBURGER, Perspectives in Flow Control and Optimization, Adv. Des. Control 5, SIAM, Philadelphia, 2002.
27. M. D. GUNZBURGER AND L. S. HOU, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control Optim., 34 (1996), pp. 1001-1043.
28. M. HEINKENSCHLOSS, Formulation and analysis of a sequential quadratic programming method for the optimal Dirichlet boundary control of Navier-Stokes flow, in Optimal Control: Theory, Algorithms and Applications, Appl. Optim. 15, W. W. Hager and P. M. Pardalos, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998, pp. 178-203.
29. M. HINZE AND K. KUNISH, Second order methods for optimal control of time-dependent fluid flow, SIAM J. Control Optim., 40 (2001), pp. 925-946.
30. L. S. HOU AND S. S. RAVINDRAN, Numerical approximation of optimal flow control problems by a penalty method: Error estimates and numerical results, SIAM J. Sci. Comput., 20 (1999), pp. 1753-1777.
31. F.-N. HWANG AND X.-C. CAI, A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations, J. Comput. Phys., 204 (2005), pp. 666-691.
32. A. D. IOFFE AND V. M. TIHOMIROV, Theory of Extremal Problems, Stud. Math. Appl. 6, North-Holland, Amsterdam, New York, 1979.
33. K. ITO AND K. KUNISH, The augmented Lagrangian method for equality and inequality constrained problems in Hubert spaces, Math. Programming, 46 (1990), pp. 341-360.
34. K. ITO AND K. KUNISH, The augmented Lagrangian method for parameter estimation in elliptic systems, SIAM J. Control Optim., 28 (1990), pp. 113-136.
35. B. MOHAMMADI AND O. PIRONNEAU, Applied Shape Optimization for Fluids, Numer. Math. Sci. Comput., The Clarendon Press, Oxford University Press, New York, 2001.
36. J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, Springer Ser. Oper. Res., SpringerVerlag, New York, 1999.
37. L. QUARTAPELLE, Numerical Solution of the Incompressible Navier-Stokes Equations, Internat. Ser. Numer. Math. 113, Birkhäuser-Verlag, Basel, 1993.
38. Y. SAAD, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003.
39. B. SMITH, P. BJØRSTAD, AND W. GROPP, Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
40. S. S. SRITHARAN, Optimal Control of Viscous Flow, SIAM, Philadelphia, 1998.
41. V. M. TIHOMIROV, Fundamental Principles of the Theory of Extremal Problems, John Wiley & Sons, New York, 1986.
42. A. TOSELLI AND O. WIDLUND, Domain Decomposition Methods - Algorithms and Theory, Springer Ser. Comput. Math. 34, Springer-Verlag, Berlin, 2005.
43. M. ULBRICH, Semismooth Newton methods for operator equations in function spaces, SIAM J. Optim., 13 (2003), pp. 805-841.
44. M. ULBRICH, S. ULBRICH, AND M. HEINKENSCHLOSS, Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to point-wise bounds, SIAM J. Control Optim., 37 (1999), pp. 731-764.
45. C. R. VOGEL, Computational Methods for Inverse Problems, Frontiers Appl. Math., SIAM, Philadelphia, 2002.