154 research outputs found
A coefficient inverse problem with a single measurement of phaseless scattering data
This paper is concerned with a numerical method for a 3D coefficient inverse
problem with phaseless scattering data. These are multi-frequency data
generated by a single direction of the incident plane wave. Our numerical
procedure consists of two stages. The first stage aims to reconstruct the
(approximate) scattered field at the plane of measurements from its intensity.
We present an algorithm for the reconstruction process and prove a uniqueness
result of this reconstruction. After obtaining the approximate scattered field,
we exploit a newly developed globally convergent numerical method to solve the
coefficient inverse problem with the phased scattering data. The latter is the
second stage of our algorithm. Numerical examples are presented to demonstrate
the performance of our method. Finally, we present a numerical study which aims
to show that, under a certain assumption, the solution of the scattering
problem for the 3D scalar Helmholtz equation can be used to approximate the
component of the electric field which was originally incident upon the medium
A numerical method to solve a phaseless coefficient inverse problem from a single measurement of experimental data
We propose in this paper a globally numerical method to solve a phaseless
coefficient inverse problem: how to reconstruct the spatially distributed
refractive index of scatterers from the intensity (modulus square) of the full
complex valued wave field at an array of light detectors located on a
measurement board. The propagation of the wave field is governed by the 3D
Helmholtz equation. Our method consists of two stages. On the first stage, we
use asymptotic analysis to obtain an upper estimate for the modulus of the
scattered wave field. This estimate allows us to approximately reconstruct the
wave field at the measurement board using an inversion formula. This reduces
the phaseless inverse scattering problem to the phased one. At the second
stage, we apply a recently developed globally convergent numerical method to
reconstruct the desired refractive index from the total wave obtained at the
first stage. Unlike the optimization approach, the two-stage method described
above is global in the sense that it does not require a good initial guess of
the true solution. We test our numerical method on both computationally
simulated and experimental data. Although experimental data are noisy, our
method produces quite accurate numerical results
Uniqueness of a 3-D coefficient inverse scattering problem without the phase information
We use a new method to prove uniqueness theorem for a coefficient inverse
scattering problem without the phase information for the 3-D Helmholtz
equation. We consider the case when only the modulus of the scattered wave
field is measured and the phase is not measured. The spatially distributed
refractive index is the subject of the interest in this problem. Applications
of this problem are in imaging of nanostructures and biological cells
Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients
In this work we shall review the (phased) inverse scattering problem and then
pursue the phaseless reconstruction from far-field data with the help of the
concept of scattering coefficients. We perform sensitivity, resolution and
stability analysis of both phased and phaseless problems and compare the degree
of ill-posedness of the phased and phaseless reconstructions. The phaseless
reconstruction is highly nonlinear and much more severely ill-posed. Algorithms
are provided to solve both the phased and phaseless reconstructions in the
linearized case. Stability is studied by estimating the condition number of the
inversion process for both the phased and phaseless cases. An optimal strategy
is suggested to attain the infimum of the condition numbers of the phaseless
reconstruction, which may provide an important guidance for efficient phaseless
measurements in practical applications. To the best of our knowledge, the
stability analysis in terms of condition numbers are new for the phased and
phaseless inverse scattering problems, and are very important to help us
understand the degree of ill-posedness of these inverse problems. Numerical
experiments are provided to illustrate the theoretical asymptotic behavior, as
well as the effectiveness and robustness of the phaseless reconstruction
algorithm
Uniqueness in inverse scattering problems with phaseless far-field data at a fixed frequency
This paper is concerned with uniqueness in inverse acoustic scattering with
phaseless far-field data at a fixed frequency. The main difficulty of this
problem is the so-called translation invariance property of the modulus of the
far-field pattern generated by one plane wave as the incident field. Based on
our previous work (J. Comput. Phys. 345 (2017), 58-73), the translation
invariance property of the phaseless far-field pattern can be broken by using
infinitely many sets of superpositions of two plane waves as the incident
fields at a fixed frequency. In this paper, we prove that the obstacle and the
index of refraction of an inhomogeneous medium can be uniquely determined by
the phaseless far-field patterns generated by infinitely many sets of
superpositions of two plane waves with different directions at a fixed
frequency under the condition that the obstacle is a priori known to be a
sound-soft or non-absorbing impedance obstacle and the index of refraction
of the inhomogeneous medium is real-valued and satisfies that either or in the support of for some positive constant .
To the best of our knowledge, this is the first uniqueness result in inverse
scattering with phaseless far-field data. Our proofs are based essentially on
the limit of the normalized eigenvalues of the far-field operators which is
also established in this paper by using a factorization of the far-field
operators
Uniqueness of a phaseless inverse scattering problem for the generalized 3-D Helmholtz equation
An inverse scattering problems for the 3-D generalized Helmholtz equation is
considered. Only the modulus of the complex valued scattered wave field is
assumed to be measured and the phase is not measured. Uniqueness theorem is
proved.Comment: 18 page
Target reconstruction with a reference point scatterer using phaseless far field patterns
An important property of the phaseless far field patterns with incident plane
waves is the translation invariance. Thus it is impossible to reconstruct the
location of the underlying scatterers. By adding a reference point scatterer
into the model, we design a novel direct sampling method using the phaseless
data directly. The reference point technique not only overcomes the translation
invariance, but also brings a practical phase retrieval algorithm. Based on
this, we propose a hybrid method combining the novel phase retrieval algorithm
and the classical direct sampling methods. Numerical examples in two dimensions
are presented to demonstrate their effectiveness and robustness
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
Inverse elastic scattering problems with phaseless far field data
This paper is concerned with uniqueness, phase retrieval and shape
reconstruction methods for inverse elastic scattering problems with phaseless
far field data. Systematically, we study two basic models, i.e., inverse
scattering of plane waves by rigid bodies and inverse scattering of sources
with compact support. For both models, we show that the location of the objects
can not be uniquely recovered by the data. To solve this problem, we consider
simultaneously the incident point sources with one fixed source point and at
most three scattering strengths. We then establish some uniqueness results for
source scattering problem with multi-frequency phaseless far field data.
Furthermore, a fast and stable phase retrieval approach is proposed based on a
simple geometric result which provides a stable reconstruction of a point in
the plane from three distances to given points. Difficulties arise for inverse
scattering by rigid bodies due to the additional unknown far field pattern of
the point sources. To overcome this difficulty, we introduce an artificial
rigid body into the system and show that the underlying rigid bodies can be
uniquely determined by the corresponding phaseless far field data at a fixed
frequency. Noting that the far field pattern of the scattered field
corresponding to point sources is very small if the source point is far away
from the scatterers, we propose an appropriate phase retrieval method for
obstacle scattering problems, without using the artificial rigid body. Finally,
we propose several sampling methods for shape reconstruction with phaseless far
field data. Extended numerical examples in two dimensions are conducted with
noisy data, and the results further verify the effectiveness and robustness of
the proposed phase retrieval techniques and sampling methods.Comment: 37 page
Uniqueness of two phaseless inverse acoustics problems in 3-d
Uniqueness is proven for two 3-d inverse problems of the determination of the
spatially distributed sound speed in the frequency dependent acoustic PDE. The
main new point is the assumption that only the modulus of the scattered complex
valued wave field is measured on a certain set
- β¦