Locating an obstacle in a 3D finite depth ocean using the convex scattering support

Bourgeois, Laurent; Chambeyron, Colin; Kusiak, Steven

doi:10.1016/j.cam.2006.01.045

Cited by 10 publications

(12 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that the level set method that we use does not rely on a traditional eikonal equation (see for example [11]) but on a simple Poisson equation, which enables us to base our computations of the regularized solution and of the level set function on a single finite element mesh. Let us remark that, in the present article as in [1], and contrary to [8,9,15,27,29], the data are supported by the boundary of the waveguide, which is realistic in the framework of Non Destructive Testing.…”

Section: Introductioncontrasting

confidence: 51%

“…When only a single pair of Cauchy data is given, the measurement matrix degenerates into a single column and both the LSM and the FM are not applicable any longer. In that context, sampling methods of similar nature like the Convex Scattering support [9] or the Direct Sampling Method [27] can however be applied to the acoustic waveguide. In this paper we propose to apply an alternative iterative method called the "exterior approach".…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems

Bourgeois

Recoquillay

2018

ESAIM: M2AN

View full text Add to dashboard Cite

This paper is dedicated to a new way of presenting the Tikhonov regularization in the form of a mixed formulation. Such formulation is well adapted to the regularization of linear ill-posed partial differential equations because when it comes to discretization, the mixed formulation enables us to use some standard finite elements. As an application of our theory, we consider an inverse obstacle problem in an acoustic waveguide. In order to solve it we use the so-called “exterior approach”, which couples the mixed formulation of Tikhonov regularization and a level set method. Some 2d numerical experiments show the feasibility of our approach.

show abstract

Section: Introductioncontrasting

confidence: 51%

Section: Introductionmentioning

confidence: 99%

A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems

Bourgeois

Recoquillay

2018

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…We assume that (H) is satisfied for an obstacle D with Lipschitz continuous boundary. Let F be the near field operator defined by (5) with u s (., y) being the solution of problem (3)…”

Section: Theoremmentioning

confidence: 99%

The linear sampling method in a waveguide: a modal formulation

2008

View full text Add to dashboard Cite

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.

show abstract

“…The convex scattering support has many useful properties: It is, in essence, the smallest convex set that carries a source that is compatible with the measured data, it is nonempty for nontrivial measurements and, remarkably, it can be defined in a constructive manner that enables a numerical implementation. Generalizations of the convex scattering support formalism have since been studied in several articles: for inverse scattering and the Helmholtz equation in [4,11,23,25,27,28] and for electrostatics and electrical impedance tomography (EIT) in [12,[15][16][17][18], where the term convex source support (CSS) is used instead of convex scattering support.…”

Section: Introductionmentioning

confidence: 99%

Convex source support in three dimensions

et al. 2011

View full text Add to dashboard Cite

This work extends the algorithm for computing the convex source support in the framework of the Poisson equation to a bounded three-dimensional domain. The convex source support is, in essence, the smallest (nonempty) convex set that supports a source that produces the measured (nontrivial) data on the boundary of the object. In particular, it belongs to the convex hull of the support of any source that is compatible with the measurements. The original algorithm for reconstructing the convex source support is inherently two-dimensional as it utilizes Möbius transformations. However, replacing the Möbius transformations by inversions with respect to suitable spheres and introducing the corresponding Kelvin transforms, the basic ideas of the algorithm carry over to three spatial dimensions. The performance of the resulting numerical algorithm is analyzed both for the inverse source problem and for Communicated by Lothar Reichel.

show abstract

Locating an obstacle in a 3D finite depth ocean using the convex scattering support

Cited by 10 publications

References 10 publications

A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems

A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems

The linear sampling method in a waveguide: a modal formulation

Convex source support in three dimensions

Contact Info

Product

Resources

About