Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: The Krylov-Schur solver

SIAM Journal on Scientific Computing

Volumn 27, Issue 2, 2006, Pages 687-713

  Biros, George ^a   Ghattas, Omar ^b  

^a NEW YORK UNIVERSITY   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Adjoint methods; Finite elements; Indefinite systems; Lagrange Newton Krylov Schur methods; Navier Stokes; Nonlinear equations; Optimal control; Parallel algorithms; PDE constrained optimization; Preconditioners; Sequential quadratic programming

Indexed keywords

ITERATIVE METHODS; LAGRANGE MULTIPLIERS; LINEAR SYSTEMS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; QUADRATIC PROGRAMMING;

ADJOINT METHODS; INDEFINITE SYSTEMS; LAGRANGE-NEWTON-KRYLOV-SCHUR METHODS; PDE-CONSTRAINED OPTIMIZATION; PRECONDITIONERS; SEQUENTIAL QUADRATIC PROGRAMMING;

PARALLEL ALGORITHMS;

EID: 33644934079     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S106482750241565X     Document Type: Article

References (41)

1. E. L. ALLGOWER, K. BÖHMER, F. A. POTRA, AND W. C. RHEINBOLDT, A mesh-independence principle for operator equations and their discretizations, SIAM J. Numer. Anal., 23 (1986), pp. 160-169.
2. E. ARIAN AND S. TA'ASAN, Analysis of the Hessian for Aerodynamic Optimization, Technical report 96-28, Institute for Computer Applications in Science and Engineering, Hampton, VA, 1996.
3. O. AXELSSON, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.
4. S. BALAY, K. BUSCHELMAN, W. D. GROPP, D. KAUSHIK, L. C. MCINNES, AND B. F. SMITH, PETSc home page, http://www.mcs.anl.gov/petsc (2001).
5. S. BALAY, W. D. GROPP, L. C. MCINNES, AND B. F. SMITH, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser, Boston, 1997, pp. 163-202.
6. S. BALAY, W. D. GROPP, L. C. McINNES, AND B. F. SMITH, PETSc Users Manual, Technical report ANL-95/11, Revision 2.1.1, Argonne National Laboratory, Argonne, IL, 2001.
7. A. BATTERMANN AND M. HEINKENSCHLOSS, Preconditioned for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems, in Optimal Control of Partial Differential Equations. Internat. Ser. Numer. Math. 126, W. Desch, F. Kappel, and K. Kunisch, eds., Birkhäuser-Verlag, 1998, pp. 15-32.
8. L. T. BIEGLER, J. NOCEDAL, AND C. SCHMID, A reduced Hessian method for large-scale constrained optimization, SIAM J. Optim., 5 (1995), pp. 314-347.
9. G. BIROS AND O. GHATTAS, Parallel Newton-Krylov algorithms for PDE-constrained optimization, in Proceedings of SC99, The SCxy Conference Series, Portland, OR, 1999, ACM/IEEE.
10. G. BIROS AND O. GHATTAS, Parallel preconditioners for KKT systems arising in optimal control of viscous incompressible flows, in Parallel Computational Fluid Dynamics 1999, D. E. Keyes, A. Ecer, J. Periaux, and N. Satofuka, eds., North-Holland, Amsterdam, 1999.
11. G. BIROS AND O. GHATTAS, Parallel Langrange-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.
12. R. BYRD AND J. NOCEDAL, An analysis of reduced Hessian methods for constrained optimization, Math. Program., 49 (1991), pp. 285-323.
13. X.-C. CAI AND O. B. WIDLUND, Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 243-258.
14. E. CHOW AND Y. SAAD, Approximate inverse techniques for block-partitioned matrices, SIAM J. Sci. Comput., 18 (1997), pp. 1657-1675.
15. J. E. DENNIS AND R. M. LEWIS, A Comparison of Nonlinear Programming Approaches to an Elliptic Inverse Problem and a New Domain Decomposition Approach, Technical report TR-9433, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1994.
16. C. FARHAT, R. TEZAUR, AND R. DJELLOULI, On the Solution of Three Dimensional Inverse Obstacle Scattering Problems by a Regularized Newton Method, Technical report CU-CAS-01-11, Center for Aerospace Structures, University of Colorado, Boulder, CO, 2001.
17. R. FLETCHER, Practical Methods of Optimization, 2nd ed. John Wiley and Sons, Chichester, UK, 1987.
18. R. W. FREUND AND N. M. NACHTIGAL, An implementation of the QMR method based on coupled two-term recurrences, SIAM J. Sci. Comput., 15 (1994), pp. 313-337.
19. O. GHATTAS AND J.-H. BARK, Optimal control of two- and three-dimensional incompressible Navier-Stokes flows, J. Comput. Phys., 136 (1997), pp. 231-244.
20. O. GHATTAS AND C. E. OROZCO, A parallel reduced Hessian SQP method for shape optimization, in Multidisciplinary Design Optimization: State of the Art, N. M. Alexandrov and M. Y. Hussaini, eds., SIAM, Philadelphia, 1997, pp. 133-152.
21. G. H. GOLUB AND C. H. V. LOAN, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
22. A. GREENBAUM, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, 1997.
23. M. D. GUNZBURGER, Finite Element Methods for Viscous Incompressible Flows, Academic Press, Boston, 1989.
24. E. HABER AND U. ASCHER, Preconditioned all-at-once methods for large sparse parameter estimation problems, Inverse Problems, 17 (2000), pp. 1847-1864.
25. D. E. KEYES, How scalable is domain decomposition in practice?, in Proceedings of the 11th International Conference on Domain Decomposition Methods, C.-H. Lai, M. Cross, and O. Widlund, eds., 1998.
26. D. E. KEYES AND W. D. GROPP, A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation, SIAM J. Sci. Statist. Comput., 8 (1987), pp. S166-S202.
27. K. KUNISCH AND E. W. SACHS, Reduced SQP methods for parameter identification problems, SIAM J. Numer. Anal., 29 (1992), pp. 1793-1820.
28. F.-S. KUPFER AND E. W. SACHS, Numerical solution of a nonlinear parabolic control problem by a reduced SQP method, Comput. Optim. Appl., 1 (1992), pp. 113-135.
29. I. MALČEVIĆ, Large-Scale Unstructured Mesh Shape Optimization on Parallel Computers, Master's thesis, Carnegie Mellon University, Pittsburgh, PA, 1997.
30. J. L. MORALES AND J. NOCEDAL, Automatic preconditioning by limited memory quasi-Newton updating, SIAM J. Optim., 10 (2000), pp. 1079-1096.
31. S. G. NASH AND A. SOFER, Preconditioning reduced matrices, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 47-68.
32. J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, Springer, New York, 1999.
33. C. E. OROZCO AND O. GHATTAS, Massively parallel aerodynamic shape optimization, Comput. Systems Engrg., 1-4 (1992), pp. 311-320.
34. C. E. OROZCO AND O. GHATTAS, A reduced SAND method for optimal design of nonlinear structures, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 2759-2774.
35. J. REUTHER, J. ALONSO, M. J. RIMLINGER, AND A. J. JAMESON, Aerodynamic shape optimization of supersonic aircraft configurations via an adjoint formulation on parallel computers, Comput. & Fluids, 29 (1999), pp. 675-700.
36. U. RINGERTZ, Optimal Design of Nonlinear Shell Structures, Technical report FFA TN 91-18, The Aeronautical Research Institute of Sweden, 1991.
37. U. RINGERTZ, An algorithm for optimization of nonlinear shell structures, Internat. J. Numer. Methods Engrg., 38 (1995), pp. 299-314.
38. Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS, Boston, 1996.
39. A. SALINGER, R. PAWLOWSKI, J. SHADID, B. VAN BLOEMEN WAANDERS, R. BARLETT, G. ITLE, AND L. BIEGLER, RSQP optimization of large-scale reacting flow applications with MP-Salsa, in Proceedings of First Sandia Workshop on PDE-Constrained Optimization, Santa Fe, NM, 2001.
40. V. SCHULZ, Solving discretized optimization problems by partially reduced SQP methods, Comput. Vis. Sci., 1 (1997), pp. 83-96.
41. V. H. SCHULZ AND H. G. BOCK, Partially reduced SQP methods for large-scale nonlinear optimization problems, in Proceedings of the Second World Congress of Nonlinear Analysis, Elsevier, 1997.