Circular resistor networks for electrical impedance tomography with partial boundary measurements

Inverse Problems

Volumn 26, Issue 4, 2010, Pages

  Borcea, L ^a   Druskin, V ^b   Mamonov, A V ^a  

^a Rice University   (United States)

^b SCHLUMBERGER DOLL RESEARCH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATION THEORY; COMPUTER SIMULATION; DIAGNOSTIC RADIOGRAPHY; ELECTRIC IMPEDANCE MEASUREMENT; ELECTRIC IMPEDANCE TOMOGRAPHY; INVERSE PROBLEMS; NUMERICAL ANALYSIS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; REPAIR; RESISTORS;

APPROXIMATE INVERSE; BOUNDARY MEASUREMENTS; ELECTRICAL IMPEDANCE TOMOGRAPHY; ELLIPTIC PARTIAL DIFFERENTIAL EQUATION; EXTREMAL; FINITE VOLUME DISCRETIZATIONS; HOUSTON; ITERATIVE SCHEMES; MEASUREMENT POINTS; NUMERICAL SIMULATION; NUMERICAL SOLUTION; PARTIAL BOUNDARY MEASUREMENTS; PHD THESIS; PRECONDITIONERS; RESISTOR NETWORK; RICE UNIVERSITY; TWO-DIMENSION;

ELECTRIC IMPEDANCE;

EID: 77950549009     PISSN: 02665611     EISSN: 13616420     Source Type: Journal    
DOI: 10.1088/0266-5611/26/4/045010     Document Type: Article

References (49)

1. Alessandrini G 1988 Stable determination of conductivity by boundary measurements Appl. Anal. 27 153-72
2. Astala K and Päivärinta L 2006 Calderón's inverse conductivity problem in the plane Ann. Math.-Second Ser. 163 265-300
3. Asvadurov S, Druskin V and Knizhnerman L 2000 Application of the difference Gaussian rules to solution of hyperbolic problems J. Comput. Phys. 158 116-35
4. Barcelo J A, Barcelo T and Ruiz A 2001 Stability of the inverse conductivity problem in the plane for less regular conductivities J. Differ. Equ. 173 231-70 (Pubitemid 33377548)
5. Biesel O D, Ingerman D V, Morrow J A and Shore W T Layered networks, the discrete Laplacian, and a continued fraction identity http://www.math. washington.edu/∼reu/papers/current/william/layered.pdf
6. Borcea L 2002 Electrical impedance tomography Inverse Problems 18 99-136
7. Borcea L and Druskin V 2002 Optimal finite difference grids for direct and inverse Sturm-Liouville problems Inverse Problems 18 979-1002
8. Borcea L, Druskin V and Knizhnerman L 2005 On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids Commun. Pure Appl. Math. 58 1231 (Pubitemid 41366833)
9. Borcea L, Druskin V and Mamonov A 2007 Solving the discrete EIT problem with optimization techniques Schlumberger-Doll Report
10. Borcea L, Druskin V, Mamonov A V and Vasquez F G 2010 Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements Inverse Problems (submitted) http://www.caam.nce.edu/∼mamonov/pyramidal.pdf
11. Borcea L, Druskin V and Vasquez F G 2008 Electrical impedance tomography with resistor networks Inverse Problems 24 035013 (31pp) (Pubitemid 351763271)
12. Bukhgeim A L and Uhlmann G 2002 Recovering a potential from partial Cauchy data Commun. Part. Differ. Equ. 27 653-68
13. Calderón A P 1980 On an inverse boundary value problem Seminar on Numerical Analysis and Its Applications to Continuum Physics (Soc Brasileira de Matematica, Rio de Janeiro) pp 65-73
14. Curtis E, Mooers E and Morrow J A 1994 Finding the conductors in circular networks from boundary measurements RAIRO-Math. Modelling Numer. Anal. 28 781-814
15. Curtis E B, Ingerman D and Morrow J A 1998 Circular planar graphs and resistor networks Linear Algebr. Appl. 23 115-50 (Pubitemid 128155517)
16. Daubechies I, Defrise M and Mol C De 2004 An iterative thresholding algorithm for linear inverse problems with a sparsity constraint Commun. Pure Appl. Math. 57 1413-57 (Pubitemid 39427442)
17. Verdière Y C de 1994 Reseaux electriques planaires I Comm. Math. Helvetici 69 351-74
18. Verdière Y C de, Gitler I and Vertigan D 1996 Reseaux electriques planaires II Comm. Math. Helvetici 71 144-67
19. Driscoll T A 2002 Schwarz-Christoffel Toolbox User's Guide: Version 2.3 Department of Mathematical Sciences, University of Delaware, Newark, DE
20. Driscoll T A and Trefethen L N 2002 Schwarz-Christoffel Mapping (Cambridge: Cambridge University Press)
21. Druskin V 1982 The unique solution of the inverse problem of electrical surveying and electrical well-logging for piecewise-continuous conductivity Izv. Earth Phys. 18 51-3
22. Druskin V 1985 On uniqueness of the determination of the three-dimensional underground structures from surface measurements with variously positioned steady-state or monochromatic field sources Sov. Phys.-Solid Earth 21 210-4
23. Druskin V and Knizhnerman L 2000 Gaussian spectral rules for second order finite-difference schemes Numer. Algorithms 25 139-59 (Pubitemid 33264757)
24. Druskin V and Knizhnerman L 2000 Gaussian spectral rules for the three-point second differences: I. A two-point positive definite problem in a semi-infinite domain SIAM J. Numer. Anal. 37 403-22
25. Engl H W, Hanke M and Neubauer A 1996 Regularization of inverse problems (Dordretch: Kluwer)
26. Heck H and Wang J 2006 Stability estimates for the inverse boundary value problem by partial Cauchy data Inverse Problems 22 1787 (Pubitemid 44444677)
27. Imanuvilov O Y, Uhlmann G and Yamamoto M 2008 Global uniqueness from partial Cauchy data in two dimensions arXiv:0810.2286
28. Ingerman D 2000 Discrete and continuous Dirichlet-to-Neumann maps in the layered case SIAM J. Math. Anal. 31 1214-34
29. Ingerman D, Druskin V and Knizhnerman L 2000 Optimal finite difference grids and rational approximations of the square root: I. Elliptic problems Commun. Pure Appl. Math. 53 1039-66
30. Ingerman D and Morrow J A 1998 On a characterization of the kernel of the Dirichlet-to-Neumann map for a planar region SIAM J. Appl. Math. 29 106-15 (Pubitemid 128027269)
31. Isaacson D 1986 Distinguishability of conductivities by electric current computed tomography IEEE Trans. Med. Imaging 5 91-5 (Pubitemid 16613325)
32. Knudsen K, Lassas M, Mueller J and Siltanen S 2009 Regularized D-bar method for the inverse conductivity problem Inverse Problems Imaging 3 599-624
33. Kohn R and Vogelius M 1984 Determining conductivity by boundary measurements Commun. Pure Appl. Math. 37 289-98
34. Kohn R and Vogelius M 1985 Determining conductivity by boundary measurements: II. Interior results Commun. Pure Appl. Math. 38 643-67
35. Kolehmainen V, Lassas M and Ola P 2004 Inverse conductivity problem with an imperfectly known boundary arXiv:math.AP/0408237
36. Lavrentiev M A and Shabat B V 1987 Methods of the Complex Variable Function Theory (Moscow: Nauka) (in Russian)
37. MacMillan H R, Manteuffel T A and McCormick S F 2010 First-order system least squares and electrical impedance tomography: Part II (http://amath. colorado.edu/pub/fosis/eit2.ps)
38. MacMillan H R, Manteuffel T A and McCormick S F 2004 First-order system least squares and electrical impedance tomography SIAM J. Numer. Anal. 42 461-83 (Pubitemid 40645201)
39. Mamonov A V 2009 Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements Master's Thesis Rice University, Houston, TX, USA
40. Mandache N 2001 Exponential instability in an inverse problem for the Schrödinger equation Inverse Problems 17 1435-44 (Pubitemid 32992997)
41. Mueller J L, Siltanen S and Isaacson D 2002 A direct reconstruction algorithm for electrical impedance tomography IEEE Trans. Med. Imaging 21 555-9 (Pubitemid 34950072)
42. Nachman A I 1996 Global uniqueness for a two-dimensional inverse boundary value problem Ann. Math. 71-96 (Pubitemid 126342339)
43. Reich E 1976 Quasiconformal mappings of the disk with given boundary values Advances in Complex Function Theory (Lecture Notes in Mathematics vol 505) (Berlin: Springer) pp 101-37
44. Reich E and Strebel K 1973 Extremal plane quasiconformal mappings with given boundary values Am. Math. Soc. 79 488-90
45. Strebel K 1976 On the existence of extremal Teichmüller mappings J. Anal. Math. 30 464-80
46. Strebel K 2002 Extremal quasiconformal polygon mappings for arbitrary subdomains of compact Riemann surfaces Ann. Acad. Sci. Fenn. 27 pp 237-48
47. Sylvester J 1990 An anisotropic inverse boundary value problem Commun. Pure Appl. Math. 43 201-32
48. Sylvester J and Uhlmann G 1987 A global uniqueness theorem for an inverse boundary value problem Ann. Math. 125 153-69
49. Vasquez F G 2006 On the parametrization of ill-posed inverse problems arising from elliptic partial differential equations PhD Thesis Rice University, Houston, TX, USA