zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Second-order topological expansion for electrical impedance tomography. (English) Zbl 1243.49049
Summary: Second-order topological expansions in electrical impedance tomography problems with piecewise constant conductivities are considered. First-order expansions usually consist of local terms typically involving the state and the adjoint solutions and their gradients estimated at the point where the topological perturbation is performed. In the case of second-order topological expansions, non-local terms which have a higher computational cost appear. Interactions between several simultaneous perturbations are also considered. The study is aimed at determining the relevance of these non-local and interaction terms from a numerical point of view. A level set based shape algorithm is proposed and initialized by using topological sensitivity analysis.
MSC:
49Q10Optimization of shapes other than minimal surfaces
49K21Optimal control problems involving relations other than differential equations
49N45Inverse problems in calculus of variations
References:
[1]Ammari, H., Kang, H.: High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter. SIAM J. Math. Anal. 34(5), 1152–1166 (electronic) (2003) · Zbl 1036.35050 · doi:10.1137/S0036141001399234
[2]Ammari, H., Moskow, S., Vogelius, M.S.: Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM Control Optim. Calc. Var. 9, 49–66 (2003) · Zbl 1075.78010 · doi:10.1051/cocv:2002071
[3]Bonnaillie-Noël, V., Dambrine, M., Tordeux, S., Vial, G.: On moderately close inclusions for the Laplace equation. C. R. Math. Acad. Sci. Paris 345(11), 609–614 (2007) · Zbl 1129.35024 · doi:10.1016/j.crma.2007.10.037
[4]Bonnet, M.: Higher-order topological sensitivity for 2-D potential problems. Application to fast identification of inclusions. Int. J. Solids Struct. 46(11–12), 2275–2292 (2009) · Zbl 1217.74095 · doi:10.1016/j.ijsolstr.2009.01.021
[5]Borcea, L.: Electrical impedance tomography. Inverse Probl. 18(6), R99–R136 (2002)
[6]Brühl, M., Hanke, M., Vogelius, M.S.: A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93(4), 635–654 (2003) · Zbl 1016.65079 · doi:10.1007/s002110200409
[7]Cedio-Fengya, D.J., Moskow, S., Vogelius, M.S.: Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Probl. 14(3), 553–595 (1998) · Zbl 0916.35132 · doi:10.1088/0266-5611/14/3/011
[8]Cheney, M., Isaacson, D., Newell, J.C.: Electrical impedance tomography. SIAM Rev. 41(1), 85–101 (electronic) (1999) · Zbl 0927.35130 · doi:10.1137/S0036144598333613
[9]Chung, E.T., Chan, T.F., Tai, X.-C.: Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys. 205(1), 357–372 (2005) · Zbl 1072.65143 · doi:10.1016/j.jcp.2004.11.022
[10]de Faria, J.R., Novotny, A.A., Feijóo, R.A., Taroco, E., Padra, C.: Second order topological sensitivity analysis. Int. J. Solids Struct. 44(14–15), 4958–4977 (2007) · Zbl 1166.74406 · doi:10.1016/j.ijsolstr.2006.12.013
[11]Delfour, M.C., Zolésio, J.-P.: Shapes and geometries. In: Advances in Design and Control, vol. 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)
[12]Eschenauer, H., Kobelev, V., Schumacher, A.: Bubble method for topology and shape optimization of structures. J. Struct. Optim. 8, 42–51 (1994) · doi:10.1007/BF01742933
[13]Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39(6), 1756–1778 (electronic) (2001) · Zbl 0990.49028 · doi:10.1137/S0363012900369538
[14]Henrot, A.: Extremum problems for eigenvalues of elliptic operators. In: Frontiers in Mathematics. Birkhäuser, Basel (2006)
[15]Hintermüller, M., Laurain, A.: Electrical impedance tomography: from topology to shape. Control Cybern. 37(4), 913–933 (2008)
[16]Il’in, A.M.: Matching of asymptotic expansions of solutions of boundary value problems. In: Transl. Math. Monog., vol. 102. American Mathematical Society, Providence (1992)
[17]Mazja, W.G., Nasarow, S.A., Plamenewski, B.A.: Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. I. In: Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien, vol. 82. Akademie, Berlin (1991)
[18]Nazarov, S.A., Sokolowski, J.: Spectral problems in shape optimization. Singular boundary perturbations. Asymptot. Anal. 56(3–4), 159–196 (2008)
[19]Nazarov, S.A., Sokolowski, J.: Shape sensitivity analysis of eigenvalues revisited. Control Cybern. 37(4), 999–1012 (2008)
[20]Osher, S., Fedkiw, R.: Level set methods and dynamic implicit surfaces. In: Applied Mathematical Sciences, vol. 153. Springer, New York (2003)
[21]Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988) · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[22]Sokołowski, J., Żochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (electronic) (1999) · Zbl 0940.49026 · doi:10.1137/S0363012997323230
[23]Sokołowski, , Zolésio, J.-P.: Introduction to shape optimization. In: Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992). Shape sensitivity analysis