Numerical computation of approximate generalized polarization tensors

In this article we describe a method to compute generalized polarization tensors. These tensors are the coefficients appearing in the multipolar expansion of the steady-state voltage perturbation caused by an inhomogeneity of constant conductivity. As an alternative to the integral equation approach, we propose an approximate semi-algebraic method which is easy to implement. This method has been integrated in a Myriapole, a Matlab routine with a graphical interface which makes such computations available to non-numerical analysts.

[1]  Michael Vogelius,et al.  Identification of conductivity imperfections of small diameter by boundary measurements. Continuous , 1998 .

[2]  Avner Friedman,et al.  Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence , 1989 .

[3]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[4]  Michael Vogelius,et al.  A direct impedance tomography algorithm for locating small inhomogeneities , 2003, Numerische Mathematik.

[5]  J. Nédélec.  Acoustic and Electromagnetic Equations : Integral Representations for Harmonic Problems , 2001 .

[6]  Habib Ammari,et al.  Boundary layer techniques for deriving the effective properties of composite materials , 2005 .

[7]  Hyeonbae Kang,et al.  Properties of the Generalized Polarization Tensors , 2003, Multiscale Model. Simul..

[8]  Giovanni Alessandrini.  Examples of instability in inverse boundary-value problems , 1997 .

[9]  Habib Ammari,et al.  Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion , 2002 .

[10]  H. Ammari,et al.  Reconstruction of Small Inhomogeneities from Boundary Measurements , 2005 .

[11]  Yves Capdeboscq,et al.  A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction , 2003 .

[12]  Habib Ammari,et al.  The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion , 2012, Math. Comput..

[13]  Kari Astala,et al.  Calderon's inverse conductivity problem in the plane , 2006 .

[14]  Graeme W. Milton,et al.  Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics , 2003 .

[15]  Habib Ammari,et al.  High-Order Terms in the Asymptotic Expansions of the Steady-State Voltage Potentials in the Presence of Conductivity Inhomogeneities of Small Diameter , 2003, SIAM J. Math. Anal..

[16]  Habib Ammari,et al.  The generalized polarization tensors for resolved imaging Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements , 2012, Math. Comput..

[17]  Habib Ammari,et al.  Enhancement of Near Cloaking Using Generalized Polarization Tensors Vanishing Structures. Part I: The Conductivity Problem , 2011, Communications in Mathematical Physics.

[18]  G. Milton.  The Theory of Composites , 2002 .

[19]  J. Craggs.  Applied Mathematical Sciences , 1973 .