Analysis of regularization methods for the solution of ILL-posed problems involving discontinuous operators

SIAM Journal on Numerical Analysis

Volumn 43, Issue 2, 2005, Pages 767-786

  Fruhauf F ^a   Scherzer, O ^a   Leitao A ^b,c  

^a UNIVERSITY OF INNSBRUCK   (Austria)

^b JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^c Federal University of St Catarina   (Brazil)

Author keywords

Ill posed problems; Level set methods; Regularization methods

Indexed keywords

ALGORITHMS; ASYMPTOTIC STABILITY; MATHEMATICAL OPERATORS; SET THEORY; TOPOLOGY;

ILL-POSED PROBLEMS; LEVEL SET METHODS; REGULARIZATION METHODS;

PROBLEM SOLVING;

EID: 27744433287     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903430906     Document Type: Article

References (26)

1. R. ACAR AND C. R. VOGEL, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), pp. 1217-1229.
2. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
3. L. AMBROSIO, Geometric evolution problems, distance function and viscosity solutions, in Calculus of Variations and Partial Differential Equations, L. Ambrosio and N. Dancer, eds., Springer, New York, 1999, pp. 5-94.
4. H. BREZIS, Operateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hubert, North-Holland, Amsterdam, 1973.
5. M. BURGER, A level set method for inverse problems, Inverse Problems, 17 (2001), pp. 1327-1355.
6. H. S. CABAYAN AND G. G. BELFORD, On computing a stable least squares solution to the inverse problem for a planar Newtonian potential, SIAM J. Appl. Math., 20 (1971), pp. 51-61.
7. T. CHAN, J. SHEN, AND L. VESE, Variational PDE models in image processing, Notices Amer. Math. Soc., 50 (2003), pp. 14-26.
8. S. CHEN, B. MERRIMAN, S. OSHER, AND P. SMEREKA, A simple level set method for solving Stefan problems, J. Comput. Phys., 135 (1997), pp. 8-29.
9. M. DELFOUR AND J.-P. ZOLESIO, Shape analysis via oriented distance functions, J. Funct. Anal., 123 (1994), pp. 129-201.
10. H. W. ENGL, M. HANKE, AND A. NEUBAUER, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
11. H. W. ENGL, K. KUNISCH, AND A. NEUBAUER, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), pp. 523-540.
12. F. HETTLICH AND W. RUNDELL, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems, 12 (1996), pp. 251-266.
13. V. ISAKOV, Inverse Source Problems, AMS, Providence, RI, 1990.
14. R. KOHN AND G. STRANG, Optimal design and relaxation of variational problems. I, Comm. Pure Appl. Math., 39 (1986), pp. 113-137.
15. R. KOHN AND G. STRANG, Optimal design and relaxation of variational problems. II, Comm. Pure Appl. Math., 39 (1986), pp. 139-182.
16. R. KOHN AND G., STRANG, Optimal design and relaxation of variational problems. III, Comm. Pure Appl. Math., 39 (1986), pp. 353-377.
17. A. LEITÃO AND O. SCHERZER, On the relation between constraint regularization, level sets, and shape optimization, Inverse Problems, 19 (2003), pp. L1-L11.
18. B. MERRIMAN, J. BENCE, AND S. OSHER, Motion of multiple functions: A level set approach, J. Comput. Phys., 112 (1994), pp. 334-363.
19. V. A. MOROZOV, Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, Berlin, Heidelberg, 1984.
20. V. A. MOROZOV, Regularization Methods for Ill-Posed Problems, CRC Press, Boca Raton, FL, 1993.
21. M. Z. NASHED AND O. SCHERZER, Least squares and bounded variation regularization with nondifferentiable functional, Numer. Funct. Anal. Optim., 19 (1998), pp. 873-901.
22. S. OSHER AND J. A. SETHIAN, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
23. W. RING, Identification of a core from boundary data, SIAM J. Appl. Math., 55 (1995), pp. 677-706.
24. F. SANTOSA, A level-set approach for inverse problems involving obstacles, ESAIM Control Optim. Calc. Var., 1 (1995/96), pp. 17-33.
25. T. I. SEIDMAN AND C. R. VOGEL, Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems, Inverse Problems, 5 (1989), pp. 227-238.
26. H.-K. ZHAO, T. CHAN, B. MERRIMAN, AND S. OSHER, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), pp. 179-195.