Lorenzo Audibert

We introduce a new family of artificial backgrounds corresponding to averaged impedance boundary conditions formulated in an abstract framework. These backgrounds are used to define a finite number of averaged Steklov eigenvalues, which are associated with inverse scattering problems from inhomogeneous media. We prove that these special eigenvalues can be determined from full-aperture, fixed-frequency far fields using the inside-outside duality method. We then numerically demonstrate how this method can be used to reconstruct averaged values of the refractive index.

In this work, we introduce a novel low-rank structure tailored for solving the inverse scattering problem. The particular low-rank structure is given by the generalized prolate spheroidal wave functions, computed stably and accurately via a Sturm–Liouville problem. We first process the far-field data to obtain a postprocessed data set within a disk domain. Subsequently, the postprocessed data are projected onto a low-rank space given by the low-rank structure. The unknown is approximately solved in this low-rank space by dropping higher-order terms. The low-rank structure leads to an explicit stability estimate for unknown functions belonging to standard Sobolev spaces and a Lipschitz stability estimate for unknowns belonging to a finite-dimensional low-rank space. Various numerical experiments are conducted to validate its performance, encompassing assessments of resolution capability, robustness against randomly added noise and modeling errors, and demonstration of increasing stability.

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.

The application motivating this work is related to the identification of deposits inside nuclear power plant steam generators using eddy-current probes. We consider a realistic experimental process that relies on the scan of a domain by sweeping along a tube axis a probe made out of coils, playing the role of the sources/receivers. Solving the inverse shape problem associated with these measurements using a least squares method requires solutions to the eddy-current and the adjoint problems for a large number of right-hand sides at each gradient-descent iteration. Additional cost in the forward solver comes from the use of a potential formulation of the problem that has the advantage of being independent from the topology of the conductive media (that may vary during iterations). We use a level-set approach to avoid remeshing and handle unknown topologies. The crucial ingredient in our algorithm is an optimized way of handling high numbers of right-hand sides for iterative solvers of large-scale problems. We first benchmark various block Krylov methods, block GMRES, and block BGCRODR to test their effectiveness compared to their standard counterpart, i.e., GMRES and GCRODR. Then we propose for BGCRODR a new implementation for recycling information from previously generated Krylov bases that scales better than traditional approaches. This part is independent from the practical inverse problem at hand. The efficiency of the overall inversion procedure is finally demonstrated on realistic synthetic 3D examples.

We propose a rapid and robust iterative algorithm to solve inverse acoustic scattering problems formulated as a PDE constrained shape optimization problem. We use a level-set method to represent th...

We consider the problem of imaging a crack network embedded in some homogeneous background from measured multistatic far field data generated by acoustic plane waves. We propose two novel approache...

We use the inside–outside duality approach proposed by Kirsch–Lechleiter to identify transmission eigenvalues (TEs) associated with artificial backgrounds. We prove that for well chosen artificial backgrounds, in particular for the ones with zero index of refraction at the inclusion location, one obtains a necessary and sufficient condition characterizing TEs via the spectrum of the modified far field operator. We also complement the existing literature with a convergence result for the invisible generalized incident field associated with the TEs. This work is based on several of the pioneering works of our dearest colleague and friend Armin Lechleiter and is dedicated to his memory.

We use the inside-outside duality approach proposed by Kirsch-Lechleiter to identify transmission eigenvalues associated with artificial backgrounds. We prove that for well chosen artificial backgrounds, in particular for the ones with zero index of refraction at the inclusion location, one obtains a necessary and sufficient condition characterizing transmission eigenvalues via the spectrum of the modified far field operator. We also complement the existing literature with a convergence result for the invisible generalized incident field associated with the transmission eigenvalues. This work is based on several of the pioneering works of our dearest colleague and friend Armin Lechleiter and is dedicated to his memory.