The enclosure method for inverse obstacle scattering over a finite time interval: IV. Extraction from a single point on the graph of the response operator

Now a final and maybe simplest formulation of the enclosure method applied to inverse obstacle problems governed by partial differential equations in a {\it spacial domain with an outer boundary} over a finite time interval is fixed. The method employs only a single pair of a quite natural Neumann data prescribed on the outer boundary and the corresponding Dirichlet data on the same boundary of the solution of the governing equation over a finite time interval, that is a single point on the graph of the so-called {\it response operator}. It is shown that the methods enables us to extract the distance of a given point outside the domain to an embedded unknown obstacle, that is the maximum sphere centered at the point whose exterior encloses the unknown obstacle. To make the explanation of the idea clear only an inverse obstacle problem governed by the wave equation is considered.Comment: typo corrected on p.1

The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain

In this paper, a time domain enclosure method for an inverse obstacle scattering problem of electromagnetic wave is introduced. The wave as a solution of Maxwell's equations is generated by an applied volumetric current having an {\it orientation} and supported outside an unknown obstacle and observed on the same support over a finite time interval. It is assumed that the obstacle is a perfect conductor. Two types of analytical formulae which employ a {\it single} observed wave and explicitly contain information about the geometry of the obstacle are given. In particular, an effect of the orientation of the current is catched in one of two formulae. Two corollaries concerning with the detection of the points on the surface of the obstacle nearest to the centre of the current support and curvatures at the points are also given.Comment: Corrected proof of Lemma 2.1; improved proof of Lemma 2.

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.Comment: 10 page

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

A simple method for some class 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 an 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.Comment: 24pages, revise

Detecting a hidden obstacle via the time domain enclosure method. A scalar wave case

The characterization problem of the existence of an unknown obstacle behind a known obstacle is considered by using a singe observed wave at a place where the wave is generated. The unknown obstacle is invisible from the place by using visible ray. A mathematical formulation of the problem using the classical wave equation is given. The main result consists of two parts: (i) one can make a decision whether the unknown obstacle exists or not behind a known impenetrable obstacle by using a single wave over a finite time interval under some a-priori information on the position of the unknown obstacle; (ii) one can obtain a lower bound of the Euclidean distance of the unknown obstacle to the center point of the support of the initial data of the wave. The proof is based on the idea of the time domain enclosure method and employs some previous results on the Gaussian lower/upper estimates for the heat kernels and domination of semigroups.Comment: 26 pages, typo correcte

On reconstruction from a partial knowledge of the Neumann-to-Dirichlet operator

We give formulae that yield an information about the location of an unknown polygonal inclusion having unknown constant conductivity inside a known conductive material having known constant conductivity from a partial knowledge of the Neumann -to-Dirichlet operator.Comment: 7 page

A remark on finding the coefficient of the dissipative boundary condition via the enclosure method in the time domain

An inverse problem for the wave equation outside an obstacle with a {\it dissipative boundary condition} is considered. The observed data are given by a single solution of the wave equation generated by an initial data supported on an open ball. An explicit analytical formula for the computation of the coefficient at a point on the surface of the obstacle which is nearest to the center of the support of the initial data is given.Comment: added Corollary 1.

The framework of the enclosure method with dynamical data and its applications

The aim of this paper is to establish the framework of the enclosure method for some class of inverse problems whose governing equations are given by parabolic equations with discontinuous coefficients. The framework is given by considering a concrete inverse initial boundary value problem for a parabolic equation with discontinuous coefficients. The problem is to extract information about the location and shape of unknown inclusions embedded in a known isotropic heat conductive body from a set of the input heat flux across the boundary of the body and output temperature on the same boundary. In the framework the original inverse problem is reduced to an inverse problem whose governing equation has a large parameter. A list of requirements which enables one to apply the enclosure method to the reduced inverse problem is given. Two new results which can be considered as the application of the framework are given. In the first result the background conductive body is assumed to be homogeneous and a family of explicit complex exponential solutions are employed. Second an application of the framework to inclusions in an isotropic inhomogeneous heat conductive body is given. The main problem is the construction of the special solution of the governing equation with a large parameter for the background inhomogeneous body required by the framework. It is shown that, introducing another parameter which is called the virtual slowness and making it sufficiently large, one can construct the required solution which yields an extraction formula of the convex hull of unknown inclusions in a known isotropic inhomogeneous conductive body.Comment: This paper has been submitted to Inverse Problems on 10 November 2010: 19pages, shorten

Extraction formulae for an inverse boundary value problem for the equation ∇⋅(σ−iωϵ)∇u=0\nabla\cdot(\sigma-i\omega\epsilon)\nabla u=0

We consider an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u=0$ in a given bounded domain $\Omega$ at a fixed $\omega>0$. $\sigma$ and $\epsilon$ denote the conductivity and permittivity of the material forming $\Omega$, respectively. We give some formulae for extracting information about the location of the discontinuity surface of $(\sigma,\epsilon)$ from the Dirichlet-to-Neumann map. In order to obtain results we make use of two methods. The first is the enclosure method which is based on a new role of the exponentially growing solutions of the equation for the background material. The second is a generalization of the enclosure method based on a new role of Mittag-Leffler's function.Comment: (2.3) on page 7, fixed the same as the final version: put $+$ in front of the first integral on line 8 down; replaced $u_1-u_2$ with $u_2-u_1$ in the first integral on line 10 dow

Travel Time and Heat Equation. One space dimensional case

The extraction problem of information about the location and shape of the cavity from a single set of the temperature and heat flux on the boundary of the conductor and finite time interval is a typical and important inverse problem. Its one space dimensional version is considered. It is shown that the enclosure method developed by the author for elliptic equations yields the extraction formula of a quantity which can be interpreted as the {\it travel time} of a {\it virtual} signal with an arbitrary fixed propagation speed that starts at the known boundary and the initial time, reflects at another unknown boundary and returns to the original boundary.Comment: 19page