This paper concerns the Linear Sampling Method to retrieve obstacles in a 2D or 3D acoustic waveguide. The classical mathematical results concerning the identifiability of the obstacle and the justification of the inverse method are established for this particular geometry. Our main concern is to derive a modal formulation of the Linear Sampling Method that is well adapted to the waveguide configuration. In particular, thank's to such formulation, we highlight the fact that finding some obstacles from remote scattering data is more delicate in a waveguide than in free space. Indeed, the presence of evanescent modes increases the ill-posedness of the inverse problem. However, we show that numerical reconstruction of obstacles by using the far field is feasible, even by using a few incident waves.

This work concerns the use of the method of quasi-reversibility to solve the Cauchy problem for Laplace's equation. We describe a mixed formulation of that method and its relationship with a classical formulation. A discretized formulation using finite elements of class C 0 is derived from the mixed formulation, and convergence of the solution of that discretized problem with noisy data to the exact solution is analyzed. Finally, a simple numerical example is implemented in order to show the feasibility of the method.

We consider the quasi-reversibility method to solve the Cauchy problem for Laplace's equation in a smooth bounded domain. We assume that the Cauchy data are contaminated by some noise of amplitude σ, so that we make a regular choice of ε as a function of σ, where ε is the small parameter of the quasi-reversibility method. Specifically, we present two different results concerning the convergence rate of the solution of quasi-reversibility to the exact solution when σ tends to 0. The first result is a convergence rate of type 1/(log 1 σ ) β in a truncated domain, the second one holds when a source condition is assumed and is a convergence rate of type σ 1 2 in the whole domain. The Cauchy problem for Laplace's equationLet Ω be an open set of R N , N ≥ 2, which satisfies the following assumptions, referred by (A) from now on. Assumptions (A): Ω is bounded, connected and of class C 2 [3]. Furthermore, Γ 0 and Γ 1 are two open subsets of ∂Ω with mes(Γ 0 ) > 0 and mes(Γ 1 ) > 0, Γ 0 ∩ Γ 1 = ∅ and Γ 0 ∪ Γ 1 = ∂Ω.

International audienceWe introduce a new approach based on the coupling of the method of quasi-reversibility and a simple level set method in order to solve the inverse obstacle problem with Dirichlet boundary condition. We provide a theoretical justification of our approach and illustrate its feasibility with the help of numerical experiments in 2D. © 2010 American Institute of Mathematical Sciences