Ting Zhou
We present a reconstruction algorithm for recovering both "magnetic-hard" and "magnetic-soft" obstacles in a background domain with known isotropic medium from the boundary impedance map. We use in our algorithm complex geometric optics solutions constructed for Maxwell's equation.
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An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the a…
The wave equation is time-reversal invariant. The enclosure method using a Neumann data generated by this invariance is introduced. The method yields the minimum ball that is centered at a given arbitrary point and encloses an unknown obstacle embedded in a known bounded domain from a single point on the graph of the so-called response operator on the boundary of the domain over a finite time interval. The occurrence of the lacuna in the solution of the free space wave equation is positively used.
A mathematical method for through-wall imaging via wave phenomena in the time domain is introduced. The method makes use of a single reflected wave over a finite time interval and gives us a criterion whether a penetrable obstacle exists or not in a general rough background medium. Moreover, if the obstacle exists, the lower and upper estimates of the distance between the obstacle and the center point of the support of the initial data are given. As an evidence of the potential of the method two applications are also given.
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 obst…
This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts. (i) The first part considers the foundation of the probe and enclosure methods for an impenetrable obstacle embedded in a medium governed by the stationary Schrödinger equation. Under a general framework, some natural estimates for a quantity computed from a pair of the Dirichlet and Neumann data on the outer surface of the body occupied by the medium are given. The estimates enables us to derive almost immediately the necessary asymptotic behavio…
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 d…
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…
An inverse obstacle problem for the wave governed by the wave equation in a two layered medium is considered under the framework of the time domain enclosure method. The wave is generated by an initial data supported on a closed ball in the upper half-space, and observed on the same ball over a finite time interval. The unknown obstacle is penetrable and embedded in the lower half-space. It is assumed that the propagation speed of the wave in the upper half-space is greater than that of the wave in the lower half-space, which is excluded in the previous study: Ikehata and Kawashita (2018) to a…
An inverse obstacle scattering problem for the electromagnetic wave governed by the Maxwell system over a finite time interval is considered. It is assumed that the wave satisfies the Leontovich boundary condition on the surface of an unknown obstacle. The condition is described by using an unknown positive function on the surface of the obstacle which is called the surface admittance. The wave is generated at the initial time by a volumetric current source supported on a very small ball placed outside the obstacle and only the electric component of the wave is observed on the same ball over a…
This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space dependent variable-order of fractional time derivative of the governing equation deviates from a known homogeneous background one. The observation data is given by the Neumann data of the solution of the governing equation for a specially designed Dirichlet data. Under a suitable jump condition on the deviation, it is shown that the most recent version of the time domain enclosure method enables o…
A simple idea of finding a domain that encloses an unknown discontinuity embedded in a body is introduced by considering an inverse boundary value problem for the heat equation. The idea gives a design of a special heat flux on the surface of the body such that from the corresponding temperature field on the surface one can extract the smallest radius of the sphere centered at an arbitrary given point in the whole space and enclosing unknown inclusions. Unlike before, the designed flux is free from a large parameter. An application of the idea to a coupled system of the elastic wave and heat e…
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.
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.
The enclosure method using a single point on the graph of the response operator for the Stokes system
本文针对时域Stokes系统的反障碍物问题，利用最新时域包络法，仅基于边界上有限时间内的速度与Cauchy应力数据，提出了两类提取未知障碍物几何信息的公式。
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.
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 obsta…
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 gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal sound-hard obstacles and piecewise linear cracks from the far field pattern of the scattered field at a fixed wave number and at most two incident directions are given. The main new points of this paper are: a combination of the enclosure method and the Herglotz wave function; explicit construction of the density in the Herglotz wave function by using the idea of the Vekua…
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 …
On finding a penetrable obstacle using a single electromagnetic wave in the time domain
本文针对时域 Maxwell 方程组的可穿透障碍反演问题，建立基于单组有限时间解的时域包围法理论，并在均匀各向同性背景介质中加以实现。
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 co…
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 orie…
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 th…
Reconstructing unknown inclusions for the biharmonic equation
本文研究利用包围法和Dirichlet-to-Neumann映射检测双调和方程中未知可穿透障碍物的反问题，通过具有对数相位的复几何光学解重构障碍物的非凸部分。
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 sat…
We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigourous by using systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann (DNO) and Neumann-to-Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or ND…
An inverse obstacle problem for the magnetic Schrödinger equation
本文研究磁薛定谔方程的反障碍物问题，利用边界上的两次测量重构障碍物边界上的未知函数，并证明了该反问题在时间局部具有Lipschitz稳定性、在时间全局具有对数稳定性。
We deal with the problem of reconstructing interfaces using complex geometrical optics solutions for the Maxwell system. The contributions are twofold. First, we justify the enclosure method for the impenetrable obstacle case avoiding any assumption on the directions of the phases of the CGO's (or the curvature of obstacle's surface). In addition, we need only a Lipschitz regularity of this surface. The analysis is based on some fine properties of the corresponding layer potentials in appropriate Sobolev spaces. Second, we justify this method also for the penetrable case, where the interface i…
This paper is dedicated to design a direct sampling method of inverse electromagnetic scattering problems, which uses multi-frequency sparse backscattering far field data for reconstructing the boundary of perfectly conducting obstacles. We show that a smallest strip containing the unknown object can be approximately determined by the multi-frequency backscattering far field data at two opposite observation directions. The proof is based on the Kirchhoff approximation and Fourier transform. Such a strip is then reconstructed by an indicator, which is the absolute value of an integral of the pr…
We develop a novel wave imaging scheme for reconstructing the shape of an inhomogeneous scatterer and we consider the inverse acoustic obstacle scattering problem as a prototype model for our study. There exists a wealth of reconstruction methods for the inverse obstacle scattering problem and many of them intentionally avoid the interior resonant modes. Indeed, the occurrence of the interior resonance may cause the failure of the corresponding reconstruction. However, based on the observation that the interior resonant modes actually carry the geometrical information of the underlying obstacl…
In this paper, we consider two time-harmonic inverse scattering problems of reconstructing penetrable inhomogeneous obstacles from near field measurements. First we appeal to the Born approximation for reconstructing small isotropic scatterers via the MUSIC algorithm. Some numerical reconstructions using the MUSIC algorithm are provided for reconstructing the scatterer and piecewise constant refractive index using a Bayesian method. We then consider the reconstruction of an anisotropic extended scatterer by {a modified linear sampling method and the factorization method applied }to the near fi…
We consider the electromagnetic scattering from a convex polyhedral PEC or PMC obstacle due to a time-harmonic incident plane wave. It is shown that the modulus of the far-field pattern in the backscattering aperture possesses a certain local maximum behavior. Using the local maximum indicating phenomena, one can determine the exterior unit normal directions, as well as the face areas, of the front faces of the obstacle. Then we propose a recovery scheme of reconstructing the obstacle by phaseless backscattering measurements. This work significantly extends our recent study in [12] from two di…
Inverse Obstacle Scattering from Multi-Frequency Near-Field Backscattering Data
本文研究多频近场背散射数据下同时恢复障碍几何形状和边界条件的问题，通过高频率渐近展开和伪微分算子分析，证明了凸性假设下的全局唯一性定理，并提出三阶段数值重建框架。
The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insulating. The main obstacle has arguably been establishing suitable monotonicity principles for the corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this shortcoming by first giving a convergence result in the operator norm for …
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