Levenberg-Marquardt level set methods for inverse obstacle problems

Inverse Problems

Volumn 20, Issue 1, 2004, Pages 259-282

  Burger, Martin ^a,b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

ILL-POSED SOURCE RECONSTRUCTION; INVERSE OBSTACLE PROBLEMS; LEVEL SET METHODS; NUMERICAL DISCERTIZATION;

CONSTRAINT THEORY; GRADIENT METHODS; ITERATIVE METHODS; LEAST SQUARES APPROXIMATIONS; LINEAR SYSTEMS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; SET THEORY; TOPOLOGY;

INVERSE PROBLEMS;

EID: 1342329760     PISSN: 02665611     EISSN: None     Source Type: Journal    
DOI: 10.1088/0266-5611/20/1/016     Document Type: Article

References (46)

1. Adams R A 1975 Sobolev Spaces (New York: Academic)
2. Allaire G 2001 Shape Optimization by the Homogenization Method (New York: Springer)
3. Allaire G, Jouve F and Toader A M 2002 A level-set method for shape optimization C. R. Acad. Sci. I 334 1125-30
4. Alessandrini G 2001 Luca Rondi, Optimal stability for the inverse problem of multiple cavities J. Diff. Eqns 176 356-86
5. Almgren F and Taylor J E 1995 Optimal geometry in equilibrium and growth Fractals 3 713-23
6. Ascher U and Haber E 2003 A multigrid method for distributed parameter estimation problems El. Trans. Numer. Anal. 18 1-18
7. Barles G, Soner H M and Souganidis P E 1993 Front propagation and phase field theory SIAM J. Control Optim. 31 439-69
8. Battermann A and Sachs E W 2000 An indefinite preconditioner for KKT systems arising in optimal control problems Technical Report Universität Trier
9. Battermann A and Sachs E W 2001 Block preconditioners for KKT systems in PDE-governed optimal control problems Workshop on Fast Solutions of Discretized Optimization Problems ed R H W Hoppe, K H Hoffmann and V Schulz (Basle: Birkhäuser) pp 1-18
10. Benameur H, Burger M and Hackl B 2003 Level set methods for geometric inverse problems in linear elasticity SFB-Report 29003-12 SFB F 013, Universität Linz, submitted
11. Bramble J H, Pasciak J E and Vassilevski P S 2000 Computational scales of Sobolev norms with application to preconditioning Math. Comput. 69 463-80
12. Brezzi F and Fortin M 1991 Mixed and Hybrid Finite Element Methods (New York: Springer)
13. Burger M 2001 A level set method for inverse problems Inverse Problems 17 1327-56
14. Burger M 2002 Growth fronts of first-order Hamilton-Jacobi equations SFB Report 02-8 University Linz
15. Burger M 2003 A framework for the construction of level set methods for shape optimization and reconstruction Interfaces Free Boundaries 5 301-329
16. Burger M and Mühlhuber W 2002 Iterative regularization of parameter identification problems by SQP-methods Inverse Problems 18 943-70
17. Burger M and Mühlhuber W 2002 Numerical approximation of an SQP-type method for parameter identification problems SIAM J. Numer. Anal. 40 1775-97
18. Crandall M G and Lions P L 1983 Viscosity solutions of Hamilton-Jacobi equations Trans. Am. Math. Soc. 277 1-42
19. Delfour M C and Zolésio J P 2001 Shapes and Geometries. Analysis, Differential Calculus, and Optimization (Philadelphia: SIAM)
20. DiCristo M and Rondi L 2003 Examples of exponential instability for elliptic inverse problems Preprint
21. Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Dordrecht: Kluwer)
22. Evans L C and Gariepy R F 1992 Measure Theory and Fine Properties of Functions (Boca Raton, FL: Chemical Rubber Company Press)
23. Haber E and Ascher U 2001 Preconditioned all-at-once methods for large, sparse parameter estimation problems Inverse Problems 17 1847-64
24. Haber E, Ascher U and Oldenburg D 2000 On optimization techniques for solving nonlinear inverse problems Inverse Problems 16 1263-80
25. Hanke M 1997 A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems Inverse Problems 13 79-95
26. Hanke M and Vogel C R 1999 Two-level preconditioners for regularized inverse problems I: theory Numer. Math. 83 385-402
27. Hettlich F and Rundell B 1996 Iterative methods for the reconstruction of an inverse potential problem Inverse Problems 12 251-66
28. Hintermüller M and Ring W 2003 A second order shape optimization approach for image segmentation SIAM J. Appl. Math. at press
29. Hintermüller M and Ring W 2003 An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional J. Math. Imaging Vis. at press
30. Ito K, Kunisch K and Li Z 2001 Level-set function approach to an inverse interface problem Inverse Problems 17 1225-42
31. Jakobsen E R and Karlsen K H 2002 Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations J. Diff. Eqns 183 497-525
32. Jiang G S and Peng D 2000 Weighted ENO-schemes for Hamilton-Jacobi equations SIAM J. Sci. Comput. 21 2126-43
33. Lions J L and Magenes E 1972 Non-Homogenous Boundary Value Problems and Applications (Berlin: Springer)
34. Litman A, Lesselier D and Santosa F 1998 Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set Inverse Problems 14 685-706
35. Mohamadi B and Pironneau O 2001 Applied Shape Optimization for Fluids (Oxford: Clarendon)
36. Ramananjaona C, Lambert M and Lesselier D 2001 Shape inversion from TM and TE real data by controlled evolution of level sets Inverse Problems 17 1585-95
37. Ramananjaona C, Lambert M, Lesselier D and Zolesio J P 2001 Shape reconstruction of buried obstacles by controlled evolution of a level set: from a min-max formulation to numerical experimentation Inverse Problems 17 1087-112
38. Osher S and Fedkiw R P 2002 Level Set Methods and Dynamic Implicit Surfaces (New York: Springer)
39. Osher S and Santosa F 2001 Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum J. Comput. Phys. 171 272-88
40. Osher S and Sethian J A 1988 Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations J. Comput. Phys. 79 12-49
41. Osher S and Sethian J A 1988 Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations J. Comput. Phys. 79 12-49
42. Saad Y 1996 Iterative Methods for Sparse Linear Systems (Amsterdam: PWS)
43. Santosa F 1996 A level-set approach for inverse problems involving obstacles ESAIM: Control, Optimisation Calculus Variations 1 17-33
44. Sokolowski J and Zolesio J P 1992 Introduction to Shape Optimization (Berlin: Springer)
45. Trottenberg U, Oosterlee C W and Schöller A 2001 Multigrid (Lond: Academic)
46. Vogel C R and Hanke M 1999 Two-level preconditioners for regularized inverse problems II: implementation and numerical results Preprint