×

zbMATH — the first resource for mathematics

The method of fundamental solutions for the inverse conductivity problem. (English) Zbl 1194.65139
Summary: We propose a simple method for detecting an inclusion \(\Omega _{2}\) embedded in a host electrostatic medium \(\Omega _{1}\) from a single Cauchy pair of voltage and current flux measurements on the exterior boundary of \(\Omega _{1}\). A nonlinear constrained minimization regularized method of fundamental solutions is developed for the numerical solution of this inverse problem. The stability of the numerical scheme is investigated by inverting measurements contaminated by random noise.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F05 Direct numerical methods for linear systems and matrix inversion
Software:
HYBRJ; minpack
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1088/0266-5611/18/6/201 · Zbl 1031.35147 · doi:10.1088/0266-5611/18/6/201
[2] Isakov V, Inverse Problems for Partial Differential Equations,, 2. ed. (2006) · Zbl 1092.35001
[3] Kang H, Inverse Problems and Related Topics pp 69– (2000)
[4] Ammari, H. 2008.An Introduction to Mathematics of Emerging Biomedical Imaging, Mathématiques & Applications, Vol. 62, Berlin: Springer-Verlag. · Zbl 1181.92052
[5] Ammari H, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics 1846 (2004) · Zbl 1113.35148 · doi:10.1007/b98245
[6] Ammari H, Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences 162 (2007)
[7] DOI: 10.1137/S003613990343375X · Zbl 1075.35102 · doi:10.1137/S003613990343375X
[8] DOI: 10.1002/cpa.3009 · Zbl 1032.78005 · doi:10.1002/cpa.3009
[9] DOI: 10.1080/17415970600573502 · Zbl 1122.78020 · doi:10.1080/17415970600573502
[10] DOI: 10.4134/JKMS.2008.45.1.097 · Zbl 1138.78012 · doi:10.4134/JKMS.2008.45.1.097
[11] DOI: 10.1002/nme.1620230105 · Zbl 0588.65088 · doi:10.1002/nme.1620230105
[12] DOI: 10.1109/10.844230 · doi:10.1109/10.844230
[13] Borman D, J. Integral Eqns Appl. 21 pp 381– (2009)
[14] DOI: 10.1080/17415970802580263 · Zbl 1175.65130 · doi:10.1080/17415970802580263
[15] Ladyzenskaya OA, The Boundary Value Problems of Mathematical Physics (1985) · doi:10.1007/978-1-4757-4317-3
[16] DOI: 10.1023/A:1018981221740 · Zbl 0922.65074 · doi:10.1023/A:1018981221740
[17] Golberg MA, Boundary Integral Methods: Numerical and Mathematical Aspects, Vol. 1 of Computational Engineering pp 103– (1999)
[18] DOI: 10.1016/0021-9991(92)90178-2 · Zbl 0745.65075 · doi:10.1016/0021-9991(92)90178-2
[19] DOI: 10.1002/(SICI)1097-0207(19990820)45:11<1681::AID-NME649>3.0.CO;2-T · Zbl 0972.80014 · doi:10.1002/(SICI)1097-0207(19990820)45:11<1681::AID-NME649>3.0.CO;2-T
[20] DOI: 10.1137/0714043 · Zbl 0368.65058 · doi:10.1137/0714043
[21] DOI: 10.1016/j.enganabound.2007.05.004 · Zbl 1272.74623 · doi:10.1016/j.enganabound.2007.05.004
[22] DOI: 10.1016/j.enganabound.2006.01.001 · Zbl 1187.65136 · doi:10.1016/j.enganabound.2006.01.001
[23] Garbow BS, MINPACK Project (1980)
[24] DOI: 10.1080/174159701088027750 · doi:10.1080/174159701088027750
[25] DOI: 10.1088/0266-5611/23/2/002 · Zbl 1115.35147 · doi:10.1088/0266-5611/23/2/002
[26] DOI: 10.1216/jiea/1181075363 · Zbl 1139.45003 · doi:10.1216/jiea/1181075363
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.