Inverse problem for the Helmholtz equation and singular sources in the divergence form

International audienceWe shall discuss an inverse problem where the underlying model is related to sources generated by currents on an anisotropic layer. This problem is a generalization of another motivated by the recovering of magnetization distribution in a rock sample from outer measurements of the generated static magnetic field. The original problem can be formulated as inverse source problem for the Laplace equation [1,2] with sources being the divergence of the magnetization whereas the generalization comes from taking the Helmholtz equation. Either inverse problem is non uniquely solvable with a kernel of infinite dimension. We shall present a decomposition of the space of sources that will allow us to discuss constraints that may restore uniqueness and propose regularization schemes adapted to these assumptions. We then present some validating experiments and some related open questions

Reconstruction of discontinuous parameters in a second order impedance boundary operator

International audienceWe consider the inverse problem of retrieving the coefficients of a second order boundary operator from Cauchy data associated with the Laplace operator at a measurement curve. We study the identifiability and reconstruction in the case of piecewise continuous parameters. We prove in particular the differentiability of the Khon-Vogelius functional with respect to the discontinuity points and employ the result in a gradient type minimizing algorithm. We provide validating numerical results discussing in particular the case of unknown number of discontinuity points

Determining the shape of defects in non-absorbing inhomogeneous media from far-field measurements

International audienceWe consider non-absorbing inhomogeneous media represented by some refraction index. We have developed a method to reconstruct, from far-field measurements, the shape of the areas where the actual index differs from a reference index. Following the principle of the Factorization Method, we present a fast reconstruction algorithm relying on far field measurements and near field values, easily computed from the reference index. Our reconstruction result is illustrated by several numerical test cases

Numerical analysis of the Factorization Method forElectrical Impedance Tomography ininhomogeneous medium

The retrieval of informations on the coefficient in Electrical Impedance Tomography is a severely ill-posed problem, and often leads to inaccurate solutions. It is well-known that numerical methods provide only low-resolution reconstructions. The aim of this work is to analyze the Factorization Method in the case of inhomogeneous background. We propose a numerical scheme to solve the dipole-like Neumann boundary-value problem, when the background coefficient is inhomogeneous. Several numerical tests show that the method is capable of recovering the location and the shape of the inclusions, in many cases where the diffusion coefficient is nonlinearly space-dependent. In addition, we test the numerical scheme after adding artificial noise

A NEW LINEAR SAMPLING METHOD FOR THE ELECTROMAGNETIC IMAGINING OF BURIED OBJECTS

We present a new linear sampling method for determining the shape of scattering objects imbedded in a known inhomogeneous medium from a knowledge of the scattered electromagnetic field due to a point source incident field at fixed frequency. The method does not require any a prior information on the physical properties of the scattering object and, under some restrictions, avoids the need to compute the Green’s tensor for the background medium. 1

A robust inversion method according to a new notion of regularization for poisson data with an application to nanoparticle volume determination

In this paper we present an efficient method for the reconstruction of the volume distribution of diluted polydisperse noninteracting nanoparticles with identical shapes from small angle X-ray scattering measurements. The described method solves a maximum likelihood problem with a positivity constraint on the solution by means of an expectation maximization iterative scheme coupled with a robust stopping criterion. We prove that this is a regularization method according to an innovative notion of regularization specifically defined for inverse problems with Poisson data. Such a regularization, together with an upper bound to the largest retrievable particle size given by the Shannon theorem, results in high fidelity quantitative reconstructions of particle volume distributions, making the method particularly effective in real applications. We test the performance of the method on synthetic data in the case of uni-and bi-modal particle volume distributions. Moreover, we show the reliability of the method on real data provided by a Xenocs device prototype