A El Badia

An inverse source problem in potential analysis

This paper discusses some aspects of an inverse source problem for elliptic equations, with observations on the boundary of the domain. The main application aimed at is the problem of identifying electrostatic dipoles in the human head where the boundary data are collected via electrodes placed on a part of the head. An uniqueness result is established for dipolar sources. Through solving a finite number of Cauchy problems, one arrives at an inverse problem in the homogeneous case. Assuming the number of dipoles bounded by a known integer M, we have established an algorithm which allows us to identify the number, the locations and moments of the dipoles by algebraic considerations. Other types of sources are also considered.

Identification of a point source in a linear advection–dispersion–reaction equation: application to a pollution source problem

We consider the problem of identification of a pollution source in a river. The mathematical model is a one-dimensional linear advection?dispersion?reaction equation with the right-hand side spatially supported at a point (the source) and a time-dependent intensity, both unknown. Assuming that the source becomes inactive after the time T*, we prove that it can be identified by recording the evolution of the concentration at two points, one of which is strategic.

Some remarks on the problem of source identification from boundary measurements

We consider the problem of determining a source term from boundary measurements, in an elliptic problem. In general, this source is unattainable except for its harmonic component. We then turn ourselves to the problem in some special cases when a priori information is available. In particular, when separation of variables is possible.

Determination of point wave sources by boundary measurements

This paper is concerned with an inverse point wave sources problem in a bounded domain Ω⊂3 from boundary observations. Assuming that all point sources vanish after a certain time T1, we prove first an identifiability result provided that some condition is satisfied between the time T1, the observation interval (0,T) and the observation domain on the boundary. This condition is reduced to the inequality T>T1 + diam(Ω) when the observation domain is the whole boundary of Ω. In this case, we propose a method to identify completely the sources: their number, locations and intensities.

Identification of moving pointwise sources in an advection-dispersion-reaction equation

We consider the inverse problem of identifying a moving source in a linear advection?dispersion?reaction equation. The main application, but not the only one, is the identification of an environmental pollution source in a river. An identifiability result is established and an identification method proposed using measurement records at two locations, one upstream and the other downstream from the source. Finally, numerical simulations are performed to assess the identification process.

A one-phase inverse Stefan problem

The moving solid/liquid interface of a melting solid in the one-dimensional case is identified from temperature and flux measurements performed solely on the solid part. An algorithm is used, based on the least-square approach using a constrained optimization method and sensitivity equation. A comparison with the numerical results obtained by A Afshari (1990 Thesis Universitede Paris-Sud) is given.

Identification of multiple moving pollution sources in surface waters or atmospheric media with boundary observations

We consider the inverse problem of identifying multiple moving pollution sources in a linear advection–dispersion–reaction equation. Although we consider the specific application of pollution source identification in surface waters or atmospheric media, this problem has many other important applications in ecological and diffusive systems. We establish an identifiability result using observations on a non-empty subset of the domain boundary and develop an identification method by reformulating the inverse problem into a minimization problem. Finally, we provide numerical results to support the theoretical results.

Identification of Dislocations in Materials from Boundary Measurements

In this paper, we are interested in the study of an inverse source problem that occurs in a microscopic dislocation model where dislocations are some defects in the material. Three questions will be addressed: the identifiability, the identification, and the stability of the inverse problem. Our objective is to identify the number and the locations of dislocations in the material. Physically, this leads us to determine the initial data in a microscopic model of evolution of discrete dislocation. This model is important for the understanding of the elasto-visco-plasticity behavior of materials.

On some inverse EEG problems

Publisher Summary The main purpose of this chapter is the inverse source problem when the source terms are dipolar. The problem comes from usual models in Electroencephalographie. After resolving the question of identifiability, an algebraic method to identify the number of dipoles, their locations, and moments are also presented. One of the first studies in inverse source problem for elliptic equation is the work of Novikov, where F is the characteristic function of a domain, and the available observation consists in the Newtonian potential created by this function source in a vicinity of infinity.

Coefficient identification in some partial differential equations from partial boundary measurements

We consider the problem of finding a finite number of controls such that the overposed data uniquely determine the coefficient q(x) in the parabolic equation subject to homogeneous Dirichlet boundary conditions. Where the sources form a complete set in and is an arbitrary small region of the boundary assumed to be controlled by a positive function .