Fréchet derivatives for some bilinear inverse problems

SIAM Journal on Applied Mathematics

Volumn 62, Issue 6, 2002, Pages 2092-2113

  Dierkes, Thomas ^a   Dorn, Oliver ^b   Natterer, Frank ^c   Palamodov, Victor ^d   Sielschott, Helmut ^c  

^a UNIVERSITY COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^c UNIVERSITY OF MÜNSTER   (Germany)

^d NONE

Author keywords

Adjoint field methods; Mathematical imaging; Numerical inverse scattering; Tomography

Indexed keywords

COMPUTER SIMULATION; HEURISTIC METHODS; PARTIAL DIFFERENTIAL EQUATIONS; TOMOGRAPHY;

MATHEMATICAL TOMOGRAPHY;

INVERSE PROBLEMS;

EID: 0036665081     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139901386375     Document Type: Article

References (31)

1. H. AMMARI AND G. BAO, Analysis of the scattering map of a linearized inverse medium problem for electromagnetic waves, Inverse Problems, 17 (2001), pp. 219-234.
2. S. R. ARRIDGE, Optical tomography in medical imaging, Inverse Problems, 15 (1999), pp. R41-R93.
3. A. BAMBERGER, G. CHAVENT, AND P. LAILLY, About the stability of the inverse problem in 1-d wave equations-application to the interpretation of seismic profiles, Appl. Math. Optim., 5 (1979), pp. 1-47.
4. G. BAO AND W. W. SYMES, On the sensitivity of hyperbolic equations to the coefficient, Comm. Partial Differential Equations, 21 (1996), pp. 395-422.
5. G. BAO AND W. W. SYMES, Regularity of an inverse problem in wave propagation, in Inverse Problems of Wave Propagation and Diffraction, G. Chavent and P. C. Sabatier, eds., Springer, New York, 1997.
6. H. D. BUI, Inverse Problems in the Mechanics of Materials, CRC Press, Boca Raton, FL, 1994.
7. K. M. CASE AND P. F. ZWEIFEL, Linear Transport Theory, Plenum Press, New York, 1967.
8. T. J. CONNOLLY AND J. N. WALL, On Fréchet differentiability of some nonlinear operators occurring in inverse problems: An implicit function theorem approach, Inverse Problems, 6 (1990), pp. 949-966.
9. R. DAUTRAY AND J. L. LIONS, Evolution Problems II, Mathematical Analysis and Numerical Methods for Science and Technology 6, Springer, Berlin, 1993.
10. T. DIERKES, Rekonstruktionsverfahren zur optischen Tomographie, Ph.D. thesis, Preprints Angewandte Mathematik und Informatik, 16/00-N, Münster, Germany, 2000; also available online at www.math.uni-muenster.de/num/Preprints/2000/dierkes.
11. D. C. DOBSON, Convergence of a reconstruction method for the inverse conductivity problem, SIAM J. Appl. Math., 52 (1992), pp. 442-458.
12. O. DORN, Das inverse Transportproblem in der Lasertomographie, Ph.D. thesis, Preprints Angewandte Mathematik und Informatik, 7/97-N, Münster, Germany, 1997; also available online at www.math.uni-muenster.de/num/Preprints/1997/dorn.
13. O. DORN, A transport-backtransport method for optical tomography, Inverse Problems, 14 (1998), pp. 1107-1130.
14. L. C. EVANS, Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI, 1998.
15. F. HETTLICH, Frechet derivatives in inverse obstacle scattering, Inverse Problems, 11 (1995), pp. 371-382.
16. J. P. KALTENBACH AND M. KASCHKE, Frequency and time-domain modelling of light transport in random media, in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller et al., eds., SPIE Proceedings IS11, SPIE, Bellingham, WA, 1993, pp. 65-86.
17. A. KIRSCH, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), pp. 81-96.
18. R. KRESS AND L. PÄIVÄRINTA, On the far field in obstacle scattering, SIAM J. Appl. Math., 59 (1999), pp. 1413-1426.
19. 1, Comm. Pure Appl. Math., 28 (1975), pp. 729-746.
20. R. M. LEWIS AND W. W. SYMES, On the relation between the velocity coefficient and boundary value for solutions of the one-dimensional wave equation, Inverse Problems, 7 (1991), pp. 597-631.
21. J. L. LIONS AND E. MAGENES, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer, New York, 1972.
22. P. M. MORSE AND K. U. INGARD, Theoretical Acoustics, Princeton University Press, Princeton, NJ, 1986.
23. W. MUNK AND P. WORCHESTER, Ocean Acoustic Tomography, Cambridge University Press, Cambridge, UK, 1995.
24. F. NATTERER, Numerical Solution of Bilinear Inverse Problems, Preprints Angewandte Mathematik und Informatik, 19/96-N, Münster, Germany, 1996; also available online at www.math.uni-muenster.de/num/Preprints/1997/natterer_3.
25. F. NATTERER AND F. WÜBBELING, A propagation-backpropagation method for ultrasound tomography, Inverse Problems, 11 (1995), pp. 1225-1232.
26. R. POTTHAST, Fréchet differentiability of boundary integral operators, Inverse Problems, 10 (1994), pp. 431-447.
27. R. POTTHAST, Electromagnetic scattering from an orthotropic medium, J. Integral Equations Appl., 11 (1999), pp. 197-215.
28. R. POTTHAST, Integral equation methods in electromagnetic scattering from anisotropic media, Math. Methods Appl. Sci., 23 (2000), pp. 1145-1159.
29. H. SIELSCHOTT, Measurement of horizontal flow in a large scale furnace using acoustic vector tomography, Flow Meas. Instrum., 8 (1997), pp. 191-197.
30. H. SIELSCHOTT, Rückpropagationsverfahren für die Wellengleichung in bewegtem Medium, Ph.D. thesis, Preprints Angewandte Mathematik und Informatik, 15/00-N, Münster, Germany, 2000; also available online at www.math.uni-muenster.de/num/Preprints/2000/sielsch.
31. J. SYLVESTER AND G. UHLMANN, Inverse boundary value problems at the boundary Continuous dependence, Comm. Pure Appl. Math., 41 (1988), pp. 197-219.