11,534 research outputs found
A computational geometry method for the inverse scattering problem
In this paper we demonstrate a computational method to solve the inverse
scattering problem for a star-shaped, smooth, penetrable obstacle in 2D. Our
method is based on classical ideas from computational geometry. First, we
approximate the support of a scatterer by a point cloud. Secondly, we use the
Bayesian paradigm to model the joint conditional probability distribution of
the non-convex hull of the point cloud and the constant refractive index of the
scatterer given near field data. Of note, we use the non-convex hull of the
point cloud as spline control points to evaluate, on a finer mesh, the volume
potential arising in the integral equation formulation of the direct problem.
Finally, in order to sample the arising posterior distribution, we propose a
probability transition kernel that commutes with affine transformations of
space. Our findings indicate that our method is reliable to retrieve the
support and constant refractive index of the scatterer simultaneously. Indeed,
our sampling method is robust to estimate a quantity of interest such as the
area of the scatterer. We conclude pointing out a series of generalizations of
our method.Comment: 20 pages, figures
A Direct Sampling Method for Inverse Scattering Using Far-Field Data
This work is concerned with a direct sampling method (DSM) for inverse
acoustic scattering problems using far-field data. The method characterizes
some unknown obstacles, inhomogeneous media or cracks, directly through an
indicator function computed from the measured data. Using one or very few
incident waves, the DSM provides quite reasonable profiles of scatterers in
time-harmonic inverse acoustic scattering without a priori knowledge of either
the physical properties or the number of disconnected components of the
scatterer. We shall first derive the DSM using far-field data, then carry out a
systematic evaluation of the performances and distinctions of the DSM using
both near-field and far-field data. The numerical simulations are shown to
demonstrate interesting and promising potentials of the DSM: a) ability to
identify not only medium scatterers, but also obstacles, and even cracks, using
measurement data from one or few incident directions, b) robustness with
respect to large noise, and c) computational efficiency with only inner
products involved
Near field imaging of small isotropic and extended anisotropic scatterers
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 field
operator. This provides a rigorous characterization of the support of the
anisotropic obstacle in terms of a range test derived from the measured data.
Under appropriate assumptions on the material parameters, the derived
factorization can be used to determine the support of the medium without
a-priori knowledge of the material properties
Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data
This paper is concerned with inverse scattering of plane waves by a locally
perturbed infinite plane (which is called a locally rough surface) with the
modulus of the total-field data (also called the phaseless near-field data) at
a fixed frequency in two dimensions. We consider the case where a Dirichlet
boundary condition is imposed on the locally rough surface. This problem models
inverse scattering of plane acoustic waves by a one-dimensional sound-soft,
locally rough surface; it also models inverse scattering of plane
electromagnetic waves by a locally perturbed, perfectly reflecting, infinite
plane in the TE polarization case. We prove that the locally rough surface is
uniquely determined by the phaseless near-field data generated by a countably
infinite number of plane waves and measured on an open domain above the locally
rough surface. Further, a direct imaging method is proposed to reconstruct the
locally rough surface from the phaseless near-field data generated by plane
waves and measured on the upper part of the circle with a sufficiently large
radius. Theoretical analysis of the imaging algorithm is derived by making use
of properties of the scattering solution and results from the theory of
oscillatory integrals (especially the method of stationary phase). Moreover, as
a by-product of the theoretical analysis, a similar direct imaging method with
full far-field data is also proposed to reconstruct the locally rough surface.
Finally, numerical experiments are carried out to demonstrate that the imaging
algorithm with phaseless near-field data and full far-field data are fast,
accurate and very robust with respect to noise in the data
On a novel inverse scattering scheme using resonant modes with enhanced imaging resolution
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 obstacle, we
propose an inverse scattering scheme of using those resonant modes for the
reconstruction. To that end, we first develop a numerical procedure in
determining the interior eigenvalues associated with an unknown obstacle from
its far-field data based on the validity of the factorization method. Then we
propose two efficient optimization methods in further determining the
corresponding eigenfunctions. Using the afore-determined interior resonant
modes, we show that the shape of the underlying obstacle can be effectively
recovered. Moreover, the reconstruction yields enhanced imaging resolution,
especially for the concave part of the obstacle. We provide rigorous
theoretical justifications for the proposed method. Numerical examples in 2D
and 3D verify the theoretically predicted effectiveness and efficiency of the
method.Comment: 21 pages, 7 figures, 1 tabl
A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data
In this paper, we consider the inverse problem of determining the location
and the shape of a sound-soft obstacle from the modulus of the far-field data
for a single incident plane wave. By adding a reference ball artificially to
the inverse scattering system, we propose a system of nonlinear integral
equations based iterative scheme to reconstruct both the location and the shape
of the obstacle. The reference ball technique causes few extra computational
costs, but breaks the translation invariance and brings information about the
location of the obstacle. Several validating numerical examples are provided to
illustrate the effectiveness and robustness of the proposed inversion
algorithm.Comment: 20 pages, 13 figure
Numerical Solution of Obstacle Scattering Problems
Some novel numerical approaches to solving direct and inverse obstacle
scattering problems (IOSP) are presented. Scattering by finite obstacles and by
periodic structures is considered. The emphasis for solving direct scattering
problem is on the Modified Rayleigh Conjecture (MRC) method, recently
introduced and tested by the authors. This method is used numerically in
scattering by finite obstacles and by periodic structures. Numerical results it
produces are very encouraging. The support function method (SFM) for solving
the IOSP is described and tested in some examples. Analysis of the various
versions of linear sampling methods for solving IOSP is given and the
limitations of these methods are described.Comment: 25 page
Inverse Obstacle Scattering for Elastic Waves in Three Dimensions
Consider an exterior problem of the three-dimensional elastic wave equation,
which models the scattering of a time-harmonic plane wave by a rigid obstacle.
The scattering problem is reformulated into a boundary value problem by
introducing a transparent boundary condition. Given the incident field, the
direct problem is to determine the displacement of the wave field from the
known obstacle; the inverse problem is to determine the obstacle's surface from
the measurement of the displacement on an artificial boundary enclosing the
obstacle. In this paper, we consider both the direct and inverse problems. The
direct problem is shown to have a unique weak solution by examining its
variational formulation. The domain derivative is studied and a frequency
continuation method is developed for the inverse problem. Numerical experiments
are presented to demonstrate the effectiveness of the proposed method
Limited Aperture Inverse Scattering Problems using Bayesian Approach and Extended Sampling Method
Inverse scattering problems have many important applications. In this paper,
given limited aperture data, we propose a Bayesian method for the inverse
acoustic scattering to reconstruct the shape of an obstacle. The inverse
problem is formulated as a statistical model using the Baye's formula. The
well-posedness is proved in the sense of the Hellinger metric. The extended
sampling method is modified to provide the initial guess of the target
location, which is critical to the fast convergence of the MCMC algorithm. An
extensive numerical study is presented to illustrate the performance of the
proposed method
A Newton method for simultaneous reconstruction of an interface and a buried obstacle from far-field data
This paper is concerned with the inverse problem of scattering of
time-harmonic acoustic waves from a penetrable and buried obstacles. By
introducing a related transmission scattering problem, a Newton iteration
method is proposed to simultaneously reconstruct both the penetrable interface
and the buried obstacle inside from far-field data. A main feature of our
method is that we do not need to know the type of boundary conditions on the
buried obstacle. In particular, the boundary condition on the buried obstacle
can also be determined simultaneously by the method. Finally, numerical
examples using multi-frequency data are carried out to illustrate the
effectiveness of our method.Comment: 25 pages, 8 figure
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