NVERSE CONDUCTIVITY PROBLEM WITH INTERNAL DATA

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data.This problem finds applications in multi-wave imaging,greedy methods to approximate parameter-dependent elliptic problems,and image treatment with partial differential equations.We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation.Assuming that the coefficient is known near the boundary,we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method.We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter.We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient,and using synthetic data.

ERROR ESTIMATES FOR THE RECURSIVE LINEARIZATION OF INVERSE MEDIUM PROBLEMS

This paper is devoted to the mathematical analysis of a general recursive linearization algorithm for solving inverse medium problems with multi-frequency measurements. Under some reasonable assumptions, it is shown that the algorithm is convergent with error estimates. The work is motivated by our effort to analyze recent significant numerical results for solving inverse medium problems. Based on the uncertainty principle, the recursive linearization allows the nonlinear inverse problems to be reduced to a set of linear problems and be solved recursively in a proper order according to the measurements. As an application, the convergence of the recursive linearization algorithm [Chen, Inverse Problems 13（1997）, pp.253-282] is established for solving the acoustic inverse scattering problem.

The Plasmonic Resonances of a Bowtie Antenna

Metallic bowtie-shaped nanostructures are very interesting objects in optics,due to their capability of localizing and enhancing electromagnetic fields in the vicinity of their central neck.In this article,we investigate the electrostatic plasmonic resonances of two-dimensional bowtie-shaped domains by looking at the spectrum of their Poincare variational operator.In particular,we show that the latter only consists of essential spectrum and fills the whole interval[0,1].This behavior is very different from what occurs in the counterpart situation of a bowtie domain with only close-totouching wings,a case where the essential spectrum of the Poincare variational operator is reduced to an interval oess strictly containing in[0,1].We provide an explanation for this difference by showing that the spectrum of the Poincare variational operator of bowtie-shaped domains with close-to-touching wings has eigenvalues which densify and eventually fill the remaining parts of[0,1]\σess as the distance between the two wings tends to zero.

EIGENVALUES OF THE NEUMANN-POINCARE OPERATOR FOR TWO INCLUSIONS WITH CONTACT OF ORDER m： A NUMERICAL STUDY

In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance S between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincare operator converge to ±1/2 as δ→ 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance 5 〉 0 from each other. When δ=0, the contact between the inclusions is of order m 〉 2. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann- Poincare operator, in terms of 5 and rn, and we check that we recover the estimates obtained in [10].