71 research outputs found
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
We consider an inverse boundary value problem for the equation
in a given bounded domain
at a fixed . and denote the conductivity
and permittivity of the material forming , respectively. We give some
formulae for extracting information about the location of the discontinuity
surface of 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 with
in the first integral on line 10 dow
Inverse obstacle scattering problems with a single incident wave and the logarithmic differential of the indicator function in the Enclosure Method
This paper gives a remark on the Enclosure Method by considering inverse
obstacle scattering problems with a single incident wave whose governing
equation is given by the Helmholtz equation in two dimensions. It is concerned
with the indicator function in the Enclosure Method. The previous indicator
function is essentially real-valued since only its absolute value is used. In
this paper, another method for the use of the indicator function is introduced.
The method employs the logarithmic differential with respect to the independent
variable of the indicator function and yields directly the coordinates of the
vertices of the convex hull of unknown polygonal sound-hard obstacles or thin
ones. The convergence rate of the formulae is better than that of the previous
indicator function. Some other applications of this method are also given.Comment: submitted to Inverse Problems on April 4, 2011. 26 page
- β¦