A newton method for reconstructing non star-shaped domains in electrical impedance tomography

Inverse Problems and Imaging

Volumn 3, Issue 2, 2009, Pages 353-371

  Harbrecht, Helmut ^a   Hohage, Thorsten ^b  

^a UNIVERSITY OF BONN   (Germany)

^b UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

Electrical impedance to mography; Inverse obstacle problems; Nonlinear inverse problems; Regularization

Indexed keywords

EID: 84855387447     PISSN: 19308337     EISSN: 19308345     Source Type: Journal    
DOI: 10.3934/ipi.2009.3.353     Document Type: Article

References (32)

1. A. B. Bakushinskii, The problem of the convergence of the iteratively regularized Gauss-Newton method, Comput. Maths. Math. Phys., 32 (1992), 1353–1359.
2. F. Bauer, T. Hohage and A. Munk, Regularized Newton methods for nonlinear inverse problems with random noise, Technical Report 2008-06, NAM, Univ. Göttingen, 2008, ftp://ftp.num.math.uni-goettingen.de/pub/preprints/hohage/main.pdf.
3. A. B. Bakushinsky and M. Y. Kokurin, “Iterative Methods for Approximate Solution of Inverse Problems,” Springer, Dordrecht, 2004.
4. B. Blaschke, A. Neubauer and O. Scherzer, On convergence rates for the iteratively regularized Gauß-Newton method, IMA Journal of Numerical Analysis, 17 (1997), 421–436.
5. M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327–1341.
6. M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Problems, 16 (2000), 1029–1042.
7. R. Chapko and R. Kress, A hybrid method for inverse boundary value problems in potential theory, J. Inverse Ill-Posed Probl., 13 (2005), 27–40.
8. K. Colton and R. Kress, “Integral Equation Methods in Scattering Theory,” Pure and Applied Mathematics. Wiley, Chichester, 1983.
9. W. Dahmen, H. Harbrecht and R. Schneider, Compression techniques for boundary integral equations – optimal complexity estimates, SIAM J. Numer. Anal., 43 (2006), 2251–2271.
10. W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math., 63 (1992), 315–344.
11. W. Dahmen, A. Kunoth and K. Urban, Biorthogonal spline-wavelets on the interval – stability and moment conditions, Appl. Comp. Harm. Anal., 6 (1999), 259–302.
12. K. Eppler, Optimal shape design for elliptic equations via BIE-methods, J. of Applied Mathematics and Computer Science, 10 (2000), 487–516.
13. H. Harbrecht and R. Schneider, Wavelet Galerkin schemes for boundary integral equations – Implementation and quadrature, SIAM J. Sci. Comput., 27 (2006), 1347–1370.
14. H. Harbrecht and R. Schneider, Biorthogonal wavelet bases for the boundary element method, Math. Nachr., 269–270 (2004), 167–188.
15. Helmut Harbrecht, “Wavelet Galerkin Schemes for the Boundary Element Method in Three Dimensions,” PhD thesis, Technical University of Chemnitz, Str. d. Nationen 62, D-09111 Chemnitz, 2001.
16. H. Harbrecht and T. Hohage, Fast methods for three-dimensional inverse obstacle scattering problems, J. Int. Eq. Appl., 19 (2007), 237–260.
17. F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14 (1998), 67–82.
18. T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauß-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279–1299.
19. T. Hohage, On the numerical solution of a three–dimensional inverse medium scattering problem, Inverse Problems, 17 (2001), 1743–1763.
20. D. S. Holder, “Electrical Impedance Tomography: Methods, History and Applications,” In-stitute of Physics, 2004.
21. B. Kaltenbacher, A. Neubauer and O. Scherzer, “Iterative Regularization Methods for Non-linear ill-posed Problems,” Radon Series on Computational and Applied Mathematics. de Gruyter, Berlin, 2008.
22. H. Kersten, Grenz-und Sprungrelationen für Potentiale mit quadrat–summierbarer Dichte, Resultate d. Math., 3 (1980), 17–24.
23. A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems, 9 (1993), 81–96.
24. R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comp. Simulation, 66 (2004), 255–265.
25. R. Kress, “Linear Integral Equations,” 2nd edition, Springer, Berlin-Heidelberg, 1999.
26. R. Potthast, On the convergence of a new Newton-type method in inverse scattering, Inverse Problems, 17 (2001), 1419–1434.
27. S. A. Sauter and C. Schwab, “Randelementmethoden. Analyse, Numerik und Implementierung schneller Algorithmen,” B. G. Teubner, Stuttgart, 2004.
28. R. Schneider, “Multiskalen-und Wavelet-Matrixkompression: Analysisbasierte Methoden zur Lösung großer Vollbesetzter Gleichungssysteme,” B. G. Teubner, Stuttgart, 1998.
29. E. M. Stein, “Singular Integrals and Differentiability Properties of Functions,” Princeton University Press, 1970.
30. M. Taylor, “Partial Differential Equations: Basic Theory,” volume 1, Springer, New York, 1996.
31. E. Zeidler, “Nonlinear Functional Analysis and Applications,” volume 3, Springer, New York, 1985.
32. E. Zeidler, “Nonlinear Functional Analysis and Applications,” volume 4, Springer, New York, 1988.