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Section: New Results
Methods for inverse problems
Identifying defects in an unknown background using differential measurements
L. Audibert and H. Haddar
In the framework of the PhD thesis of Lorenzo Audibert we studied non destructive testing of concrete using ultrasonic waves, and more generallly imaging in complex heterogeneous media. We assume that measurements are multistatic, which means that we record the scattered field on different points by using several sources. For this type of data we wish to build methods that are able to image the obstacle that created the scattered field. We use qualitative methods in this work, which only provide the support of the object independently from its physical property. The first part of this thesis consists of a theoretical analysis of the Linear Sampling Method. Such analysis is done in the framework of regularization theory, and our main contribution is to provide and analyze a regularization term that ensures good theoretical properties. Among those properties we were able to demonstrate that when the regularization parameter goes to zero, we actually construct a sequence of functions that strongly converges to the solution of the interior transmission problem. This behavior gives a central place to the interior transmission problem as it allows describing the asymptotic solution of our regularized problem. Using this characterization of our solution, we are able to give the optimal reconstruction we can get from our method. More importantly this description of the solution allows us to compare the solution coming from two different datasets. Based on the result of this comparison, we manage to produce an image of the connected component that contains the defect which appears between two measurement campaigns and this regardless of the medium. This method is well suited for the characteristics of the microstructure of concrete as shown on several numerical examples with realistic concrete-like microstructure. Finally, we extend our theoretical results to the case of limited aperture, anisotropic medium and elastic waves, which correspond to the real physics of the ultrasounds
Invisibility in scattering theory for small obstacles
L. Chesnel, X. Claeys and S.A. Nazarov
We are interested in a time harmonic acoustic problem in a waveguide containing flies. The flies are modelled by small sound soft obstacles. We explain how they should arrange to become invisible to an observer sending waves from and measuring the resulting scattered field at the same position. We assume that the flies can control their position and/or their size. On the other hand, we show that any sound soft obstacle (non necessarily small) embedded in the waveguide always produces some non exponentially decaying scattered field at . As a consequence, the flies cannot be made completely invisible to an observer equipped with a measurement device located at .
New notion of regularization for Poisson data with an application to nanoparticle volume determination
F. Benvenuto, H. Haddar and B. Lantz
The aim of this work is to develop a fully automatic method for the reconstruction of the volume distribution of diluted polydisperse non-interacting 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 the positivity constraint 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.
A conformal mapping algorithm for the Bernoulli free boundary value problem
H. Haddar and R. Kress
We propose a new numerical method for the solution of Bernoulli's free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary is determined by prescribed Cauchy data on in addition to a Dirichlet condition on the known boundary . Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar and Kress for the solution of a related inverse boundary value problem. For this we interpret the free boundary as the unknown boundary in the inverse problem to construct from the Dirichlet condition on and Cauchy data on the known boundary . Our method for the Bernoulli problem iterates on the missing normal derivative on by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach.
Identification of small objects with near-field data in quasi-backscattering configurations
H. Haddar and M. Lakhal
We present a new sampling method for detecting targets (small inclusions or defects) immersed in a homogeneous medium in three-dimensional space, from measurements of acoustic scattered fields created by point source incident waves. We consider the harmonic regime and a data setting that corresponds with quasi-backscattering configuration: the data is collected by a set a receivers that are distributed on a segment centered at the source position and the device is swept along a path orthogonal to the receiver line. We assume that the aperture of the receivers is small compared with the distance to the targets. Considering the asymptotic form of the scattered field as the size of the targets goes to zero and the small aperture approximation, one is able to derive a special expression for the scattered field. In this expression a separation of the dependence of scattered field on the source location and the distance source-target is performed. This allows us to propose a sampling procedure that characterizes the targets location in terms of the range of a near-field operator constructed from available data. Our procedure is similar to the one proposed by Haddar-Rezac for far-field configurations. The reconstruction algorithm is based on the MUSIC (Multiple SIgnal Classification) algorithm.