Masaru Ikehata

Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data

We consider a reconstruction problem of the shape of an unknown open set D in a two-dimensional bounded domain from the Cauchy data on of a nonconstant solution u of the equation u = 0 in D. We assume that the Neumann derivative of u vanishes on D and that D is a convex open polygon. We give a formula for the calculation of the support function of D from such data.

Reconstruction of the support function for inclusion from boundary measurements

Abstract - First we give a formula (procedure) for the reconstruction of the support function for unknown inclusion by means of the Dirichlet to Neumann map. In the procedure we never make use of the unique continuation property or the Runge approximation property of the governing equation. Second we apply the method to a similar problem for the Helmholtz equation.

Reconstruction of obstacle from boundary measurements

Abstract We give a reconstruction formula for the three-dimensional sound-soft/sound-hard obstacle by employing the surface data of the scattering solution generated by a point source.

On reconstruction in the inverse conductivity problem with one measurement

We consider an inverse problem for electrically conductive material occupying a domain ? in 2. Let ? be the conductivity of ?, and D a subdomain of ?. We assume that ? is a positive constant k on D, k?1 and is 1 on ?D; both D and k are unknown. The problem is to find a reconstruction formula of D from the Cauchy data on ?? of a non-constant solution u of the equation ????u = 0 in ?. We prove that if D is known to be a convex polygon such that diamD<dist(D,??), there are two formulae for calculating the support function of D from the Cauchy data.

How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms

We report two new mathematical inversion algorithms for the electric impedance tomography. An application to the reconstruction problem of the unknown boundary on which the Neumann derivative of the solution of the Helmholtz equation vanishes is included.

Reconstruction of inclusion from boundary measurements

Abstract - We consider the problem of reconstructing of the boundary of an unknown inclusion together with its conducticity from the localized Dirichlet-to- Neumann map. We give an exact reconstruction proceduer and apply the method to an inverse boundary value problem for the system of the equations in the theory of elasticity.

On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval

The enclosure method was originally introduced for inverse problems of concerning non destructive evaluation governed by elliptic equations. It was developed as one of useful approach in inverse problems and applied for various equations. In this article, an application of the enclosure method to an inverse initial boundary value problem for a parabolic equation with a discontinuous coefficients is given. A simple method to extract the depth of unknown inclusions in a heat conductive body from a single set of the temperature and heat flux on the boundary observed over a finite time interval is introduced. Other related results with infinitely many data are also reported. One of them gives the minimum radius of the open ball centered at a given point that contains the inclusions. The formula for the minimum radius is newly discovered.

Reconstruction of a source domain from the Cauchy data

We consider an inverse source problem for the Helmholtz equation in a bounded domain. The problem is to reconstruct the shape of the support of a source term from the Cauchy data on the boundary of the solution of the governing equation. We prove that if the shape is a polygon, one can calculate its support function from such data. An application to the inverse boundary value problem is also included.

Inverse conductivity problem in the infinite slab

We consider the inverse conductivity problem in the infinite slab which is important from a practical point of view. We give formulae for extracting information about the location of an inclusion in the infinite slab from infinitely many pairs of the voltage potentials on the whole boundary and the corresponding electric current densities on a bounded part of the boundary. In order to establish the formulae we make use of a special version of Yarmukhamedovs Green function which is a generalization of Faddeevs Green function. Using the function, we give an explicit sequence of harmonic functions with finite energy that approximates the exponentially growing solution of the Laplace equation in a bounded part of the infinite slab and zero in an unbounded part of the infinite slab. This gives a new role for Yarmukhamedovs Green function.

The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval

A simple method for some classes of inverse obstacle scattering problems is introduced. The observation data are given by a wave field measured on a known surface, surrounding unknown obstacles over a finite time interval. The wave is generated by the initial data with compact support outside the surface. The method yields the distance from a given point outside the surface to obstacles and thus more than the convex hull.