Maya de Buhan

Global Carleman Estimates for Waves and Applications

In this article, we extensively develop Carleman estimates for the wave equation and give some applications. We focus on the case of an observation of the flux on a part of the boundary satisfying the Gamma conditions of Lions. We will then consider two applications. The first one deals with the exact controllability problem for the wave equation with potential. Following the duality method proposed by Fursikov and Imanuvilov in the context of parabolic equations, we propose a constructive method to derive controls that weakly depend on the potentials. The second application concerns an inverse problem for the waves that consists in recovering an unknown time-independent potential from a single measurement of the flux. In that context, our approach does not yield any new stability result, but proposes a constructive algorithm to rebuild the potential. In both cases, the main idea is to introduce weighted functionals that contain the Carleman weights and then to take advantage of the freedom on the Carleman parameters to limit the influences of the potentials.

Exponential convergence of an observer based on partial field measurements for the wave equation

We analyze an observer strategy based on partial—that is, in a subdomain—measurements of the solution of a wave equation, in order to compensate for uncertain initial conditions. We prove the exponential convergence of this observer under a nonstandard observability condition, whereas using measurements of the time derivative of the solution would lead to a standard observability condition arising in stabilization and exact controlabillity. Nevertheless, we directly relate our specific observability condition to the classical geometric control condition. Finally, we provide some numerical illustrations of the effectiveness of the approach.

A new approach to solve the inverse scattering problem for waves: combining the TRAC and the adaptive inversion methods

The aim of this paper is to propose a new method to solve the inverse scattering problem. This method works directly in the time-dependent domain, using the wave equation and proceeds in two steps. The first step is the time-reversed absorbing condition (TRAC) method to reconstruct and regularize the signal and to reduce the computational domain. The second step is the adaptive inversion method to solve the inverse problem from the TRAC data, by using basis and mesh adaptation. This strategy allows us to recover the position, the shape and the properties of the scatterer in a precise and robust manner.

Convergent Algorithm Based on Carleman Estimates for the Recovery of a Potential in the Wave Equation

This article develops the numerical and theoretical study of the reconstruction algorithm of a potential in a wave equation from boundary measurements, using a cost functional built on weighted energy terms coming from a Carleman estimate. More precisely, this inverse problem for the wave equation consists in the determination of an unknown time-independent potential from a single measurement of the Neumann derivative of the solution on a part of the boundary. While its uniqueness and stability properties are already well known and studied, a constructive and globally convergent algorithm based on Carleman estimates for the wave operator was recently proposed in [BdBE13]. However, the numerical implementation of this strategy still presents several challenges, that we propose to address here.

Whole-microwave system modeling for brain imaging

In this paper, we present the results of a whole-system modeling of a microwave measurement prototype for brain imaging, consisting of 160 ceramic-loaded antennas working around 1 GHz. The modelization has been performed using open source FreeFem++ solver. Quantitative comparisons were performed using commercial software Ansys-HFSS and measurements. Coupling effects between antennas are studied with the empty system (without phantom) and simulations have been carried out with a fine numerical brain phantom model issued from scanner and MRI data for determining the sensitivity of the system in realistic configurations.

Numerical Modeling and High-Speed Parallel Computing: New Perspectives on Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring.

This article deals with microwave tomography for brain stroke imaging using state-of-the-art numerical modeling and massively parallel computing. Iterative microwave tomographic imaging requires the solution of an inverse problem based on a minimization algorithm (e.g., gradient based) with successive solutions of a direct problem such as the accurate modeling of a whole-microwave measurement system. Moreover, a sufficiently high number of unknowns is required to accurately represent the solution. As the system will be used for detecting a brain stroke (ischemic or hemorrhagic) as well as for monitoring during the treatment, the running times for the reconstructions should be reasonable. The method used is based on high-order finite elements, parallel preconditioners from the domain decomposition method and domain-specific language with the opensource FreeFEM++ solver.

A facial reconstruction method based on new mesh deformation techniques

ABSTRACT This article presents a new numerical method for facial reconstruction. The problem is the following: given a dry skull, reconstruct a virtual face that would help in the identification of the subject. The approach combines classical features as the use of a skulls/faces database and more original aspects: (1) an original shape matching method is used to link the unknown skull to the database templates; (2) the final face is seen as an elastic 3D mask that is deformed and adapted onto the unknown skull. In this method, the skull is considered as a whole surface and not restricted to some anatomical landmarks, allowing a dense description of the skull/face relationship. Also, the approach is fully automated. Various results are presented to show its efficiency.