Numerical approximation of an SQP-type method for parameter identification

SIAM Journal on Numerical Analysis

Volumn 40, Issue 5, 2002, Pages 1775-1797

  Burger, Martin ^a   Mühlhuber, Wolfram ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

Galerkin methods; Indefinite systems; Iterative regularization; Parameter identification; Sequential quadratic programming

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; FINITE ELEMENT METHOD; GALERKIN METHODS; ITERATIVE METHODS; MATHEMATICAL MODELS; PROBLEM SOLVING; QUADRATIC PROGRAMMING;

DATA NOISE; INDEFINITE SYSTEMS;

IDENTIFICATION (CONTROL SYSTEMS);

EID: 0010048713     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142901389980     Document Type: Article

References (40)

1. W. ALT, The Lagrange-Newton method for infinite-dimensional optimization problems, Numer. Funct. Anal. Optim., 11 (1990), pp. 201-224.
2. H. T. BANKS AND K. KUNISCH, Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Basel, Boston, Berlin, 1989.
3. 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, W. Desch, F. Kappel, and K. Kunisch, eds., Birkhäuser, Basel, Boston, Berlin, 1998, pp. 15-32.
4. R. BECKER, H. KAPP, AND R. RANNACHER, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39 (2000), pp. 113-132.
5. G. BIROS AND O. GHATTAS, Parallel Lagrange-Newton-Krylov-Schur methods for PDE-Constrained Optimization Problems. Part 1: The Krylov-Schur Solver, preprint, Carnegie Mellon University, Pittsburgh, PA, 2000.
6. G. BIROS AND O. GHATTAS, Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization Problems. Part 2: The Lagrange-Newton Solver and Its Application to Optimal Control of Steady Viscous Flow, preprint, Carnegie Mellon University, Pittsburgh, PA, 2001.
7. J. H. BRAMBLE, J. E. PASCIAK, AND A. T. VASSILEV, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072-1092.
8. F. BREZZI, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), pp. 129-151.
9. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer Ser. Comput. Math. 15, Springer, New York, 1991.
10. M. BURGER, Iterative regularization of a parameter identification problem occurring in polymer crystallization, SIAM J. Numer. Anal., 39 (2001), pp. 1029-1055.
11. M. BURGER AND W. MÜHLHUBER, Iterative regularization of parameter identification problems by SQP-methods, Inverse Problems, 18 (2002), pp. 943-969.
12. M. BURGER AND W. MÜHLHUBER, Iterative solution of saddle-point problems arising in the identification and control of distributed parameters, in preparation.
13. D. COLTON, R. EWING, AND W. RUNDELL, EDS., Inverse Problems in Partial Differential Equations, SIAM, Philadelphia, 1990.
14. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT, Trust Region Methods, MPS/SIAM Ser. Optim., SIAM, Philadelphia, 2000.
15. J. E. DENNIS, M. HEINKENSCHLOSS, AND L. N. VICENTE, Trust-region interior-point SQP algorithms for a class of nonlinear programming problems, SIAM J. Control. Optim., 36 (1998), pp. 1750-1794.
16. H. C. ELMAN AND G. H. GOLUB, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 1645-1661.
17. H. W. ENGL AND W. RUNDELL, EDS., Inverse Problems in Diffusion Processes. Proceedings of the GAMM-SIAM Symposium, SIAM, Philadelphia, 1995.
18. H. W. ENGL AND O. SCHERZER, Convergence rate results for iterative methods for solving nonlinear ill-posed problems, in Solution Methods for Inverse Problems D. Colton, H. W. Engl, J. McLaughlin, A. Louis, and W. Rundell, eds., Springer, Vienna, New York, 2000, pp. 7-34.
19. R. W. FREUND AND N. M. NACHTIGAL, QMR: A quasi-minimal residual method for non-Hermitian linear systems, Numer. Math., 60 (1991), pp. 315-339.
20. P. E. GILL, W. MURRAY, AND M. H. WRIGHT, Practical Optimization, Academic Press, London, San Diego, New York, 1981.
21. E. HABER AND U. ASCHER, Preconditioned all-at-once methods for large, sparse parameter estimation problems, Inverse Problems, 17 (2001), pp. 1847-1864.
22. M. HEINKENSCHLOSS, The numerical solution of a control problem governed by a phase field model, Optim. Methods Softw., 7 (1997), pp. 211-263.
23. B. HEISE, Nonlinear field calculations with multigrid Newton methods, Impact Comput. Sci. Engrg., 5 (1993), pp. 75-110.
24. B. KALTENBACHER, On Broyden's method for the regularization of nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 19 (1998), pp. 807-833.
25. B. KALTENBACHER, A projection-regularized Newton method for nonlinear ill-posed problems and its application to parameter identification problems with finite element discretization, SIAM J. Numer. Anal., 37 (2000), pp. 1885-1908.
26. B. KALTENBACHER, M. KALTENBACHER, AND S. REITZINGER, Identification of nonlinear B-H curves based on magnetic field computations and multigrid methods for ill-posed problems, European J. Appl. Math., to appear.
27. B. KALTENBACHER AND J. SCHÖBERL, A saddle point variational formulation for projection-regularized parameter identification, Numer. Math., 91 (2002), pp. 675-697.
28. M. KUHN, U. LANGER, AND J. SCHÖBERL, Scientific computing tools for 3D magnetic field problems, in The Mathematics of Finite Elements and Applications X, J. R. Whiteman, ed., Elsevier, Amsterdam, 2000, pp. 239-258.
29. J. L. LIONS AND O. PIRONNEAU, Sur le controle parallele des systemes distribues, C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 993-998.
30. D. LUKÁŠ, Shape Optimization of Homogeneous Electromagnets, SFB-Report 00-30, University of Linz, Linz, Ausria, 2000.
31. J. NOCEDAL AND S. WRIGHT, Numerical Optimization, Springer Ser. Oper. Res., Springer, New York, Berlin, Heidelberg, 1999.
32. C. C. PAIGE AND M. A. SAUNDERS, Solution of sparse indefinite linear systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.
33. A. QUARTERONI AND A. VALLI, Numerical Approximation of Partial Differential Equations, Springer Series in Comput. Math. 23, Springer, Berlin, Heidelberg, New York, 1994.
34. D. E. REEVE AND M. SPIVACK, Determination of a source term in the linear diffusion equation, Inverse Problems, 10 (1994), pp. 1335-1344
35. Y. SAAD AND M. H. SCHULTZ, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869.
36. E. W. SACHS, Control applications of reduced SQP methods, in Computational Optimal Control, R. Bulirsch and D. Kraft, eds., Birkhäuser, Basel, 1994, pp. 89-104.
37. V. SCHULZ AND H. G. BOCK, Partially reduced SQP methods for large-scale nonlinear optimization problems, Nonlinear Anal., 30 (1997), pp. 4723-4734.
38. V. SCHULZ, Solving discretized optimization problems by partially reduced SQP methods, Comp. and Vis. in Science, 1 (1998), pp. 83-96.
39. 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.
40. W. ZULEHNER, Analysis of iterative methods for saddle point problems: a unified approach, Math. Comput., 71 (2002), pp. 479-505.