arXiv subjects
Publications and source records attributed to Masaru Ikehata.
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.
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.
The aim of this chapter is to make a review of the recent results using the Enclosure Method on inverse obstacle problems governed by the wave equation and the Maxwell system in time domain. We also describe some of unsolved problems related to further possibility of the Enclosure Method itself.
This paper is concerned with the numerical implementation of a formula in the enclosure method as applied to a prototype inverse initial boundary value problem for thermal imaging in a one-space dimension. A precise error estimate of the formula is given and the effect on the discretization of the used integral of the measured data in the formula is studied. The formula requires a large frequency to converge; however, the number of time interval divisions grows exponetially as the frequency increases. Therefore, for a given number of divisions, we fixed the trusted frequency region of convergence with some given error bound. The trusted frequency region is computed theoretically using theorems provided in this paper and is numerically implemented for various cases.
This paper studies a prototype of inverse initial boundary value problems whose governing equation is the heat equation in three dimensions. An unknown discontinuity embedded in a three-dimensional heat conductive body is considered. A {\it single} set of the temperature and heat flux on the lateral boundary for a fixed observation time is given as an observation datum. It is shown that this datum yields the minimum length of broken paths that start at a given point outside the body, go to a point on the boundary of the unknown discontinuity and return to a point on the boundary of the body under some conditions on the input heat flux, the unknown discontinuity and the body. This is new information obtained by using enclosure method.
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.
An inverse obstacle scattering problem for the wave governed by the Maxwell system in the time domain, in particular, over a finite time interval is considered. It is assumed that the electric field $\mbox{\boldmath $E$}$ and magnetic field $\mbox{\boldmath $H$}$ which are solutions of the Maxwell system are generated only by a current density at the initial time located not far a way from an unknown obstacle. The obstacle is embedded in a medium like air which has constant electric permittivity $ε$ and magnetic permeability $μ$. It is assumed that the fields on the surface of the obstacle satisfy the impedance-or the Leontovich boundary condition $\mbox{\boldmath $ν$}\times\mbox{\boldmath $H$} -λ\,\mbox{\boldmath $ν$}\times(\mbox{\boldmath $E$}\times\mbox{\boldmath $ν$})=\mbox{\boldmath $0$}$ with $λ$ an unknown positive function and $\mbox{\boldmath $ν$}$ the unit outward normal. The observation data are given by the electric field observed at the same place as the support of the current density over a finite time interval. It is shown that an indicator function computed from the electric fields corresponding two current densities enables us to know: the distance of the center of the common spherical support of the current densities to the obstacle; whether the value of the impedance $λ$ is greater or less than the special value $\sqrt{ε/μ}$.
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.
This paper considers an inverse problem for the classical wave equation in an exterior domain. It is a mathematical interpretation of an inverse obstacle problem which employs the dynamical scattering data of acoustic wave over a finite time interval. It is assumed that the wave satisfies a Robin type boundary condition with an unknown variable coefficient. The wave is generated by the initial data localized outside the obstacle and observed over a finite time interval at the same place as the support of the initial data. It is already known that, using the enclosure method, one can extract the maximum sphere whose exterior encloses the obstacle, from the data. In this paper, it is shown that the enclosure method enables us to extract also: (i) a quantity which indicates the deviation of the geometry between the maximum sphere and the boundary of the obstacle at the first reflection points of the wave; (ii) the value of the coefficient of the boundary condition at an arbitrary first reflection point of the wave provided, for example, the surface of the obstacle is known in a neighbourhood of the point. Another new obtained knowledge is that: the enclosure method can cover the case when the data are taken over a sphere whose centre coincides with that of the support of an initial data and yields corresponding results to (i) and (ii).
This paper is concerned with an inverse obstacle problem which employs the dynamical scattering data of acoustic wave over a finite time interval. The unknown obstacle is assumed to be sound-soft one. The governing equation of the wave is given by the classical wave equation. The wave is generated by the initial data localized outside the obstacle and observed over a finite time interval at a place which is not necessary the same as the support of the initial data. The observed data are the so-called bistatic data. In this paper, an enclosure method which employs the bistatic data and is based on two main analytical formulae, is developed. The first one enables us to extract the maximum spheroid with focal points at the center of the support of the initial data and that of the observation points whose exterior encloses the unknown obstacle of general shape. The second one, under some technical assumption for the obstacle including convexity as an example, indicates the deviation of the geometry of the boundary of the obstacle and the maximum spheroid at the contact points. Several implications of those two formulae are also given. In particular, a constructive proof of a uniqueness of a spherical obstacle using the bistatic data is given.
In this paper a wave is generated by an initial data whose support is localized at the outside of unknown obstacles and observed in a limited time on a known closed surface or the same position as the support of the initial data. The observed data in the latter process are nothing but the back-scattering data. Two types of obstacles are considered. One is obstacles with a dissipative boundary condition which is a generalization of the sound-hard obstacles; another is obstacles with a finite refractive index, so-called, transparent obstacles. For each type of obstacles two formulae which yield explicitly the distance from the support of the initial data to unknown obstacles are given.
This paper gives a note on an application of the enclosure method to an inverse obstacle scattering problem governed by the Helmholtz equation in two dimensions. It is shown that one can uniquely determine the convex hull of an unknown sound-hard polygonal obstacle from the trace of the total wave that was exerted by a single point source onto a known circle surrounding the obstacle provided the source is sufficiently far from the obstacle. The result contains a formula that extracts the value of the support function of the obstacle at a generic direction. Some other applications to thin obstacles, obstacles in a layered medium and the far-field equation in the linear sampling method are also included.
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.
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.
More than ten years ago Ikehata discovered two mathematical methods for the purpose of extracting information about the location and shape of unknown discontinuity embedded in a known background medium from observation data. The methods are called the probe and enclosure methods. This paper presents their past and recent applications to inverse obstacle scattering problems of acoustic wave.
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.
This paper studies a prototype of inverse obstacle scattering problems whose governing equation is the Helmholtz equation in two dimensions. An explicit method to extract information about the location and shape of unknown obstacles from the far field operator with a fixed wave number is given. The method is based on: an explicit construction of a modification of Mittag-Leffler's function via the Vekua transform and the study of the asymptotic behaviour; an explicit density in the Herglotz wave function that approximates the modification of Mittag-Leffler's function in the bounded domain surrounding unknown obstacles; a system of inequalities derived from Kirsch's factorization formula of the far field operator. Then an indicator function which can be calculated from the far field operator acting on the density is introduced. It is shown that the asymptotic behaviour of the indicator function yields information about the visible part of the exterior of the obstacles.
This paper considerers the problem of computing the value of a solution of the heat equation at a given point inside a bounded domain after the initial time. It is assumed that the initial value of the solution inside the domain (possibly in a part of the domain) is known; the boundary value and the normal derivative on a part of the boundary of the domain over a finite time interval are known. Two analytical formulae for the problem are given. Both formulae make use of a special fundamental solution having a large parameter of the backward heat equation.