833 research outputs found
Convexification of a 3-D coefficient inverse scattering problem
A version of the so-called "convexification" numerical method for a
coefficient inverse scattering problem for the 3D Hemholtz equation is
developed analytically and tested numerically. Backscattering data are used,
which result from a single direction of the propagation of the incident plane
wave on an interval of frequencies. The method converges globally. The idea is
to construct a weighted Tikhonov-like functional. The key element of this
functional is the presence of the so-called Carleman Weight Function (CWF).
This is the function which is involved in the Carleman estimate for the Laplace
operator. This functional is strictly convex on any appropriate ball in a
Hilbert space for an appropriate choice of the parameters of the CWF. Thus,
both the absence of local minima and convergence of minimizers to the exact
solution are guaranteed. Numerical tests demonstrate a good performance of the
resulting algorithm. Unlikeprevious the so-called tail functions globally
convergent method, we neither do not impose the smallness assumption of the
interval of wavenumbers, nor we do not iterate with respect to the so-called
tail functions.Comment: 27 pages, 6 figure
A globally convergent method for a 3-D inverse medium problem for the generalized Helmholtz equation
A 3-D inverse medium problem in the frequency domain is considered. Another
name for this problem is Coefficient Inverse Problem. The goal is to
reconstruct spatially distributed dielectric constants from scattering data.
Potential applications are in detection and identification of explosive-like
targets. A single incident plane wave and multiple frequencies are used. A new
numerical method is proposed. A theorem is proved, which claims that a small
neigborhood of the exact solution of that problem is reached by this method
without any advanced knowledge of that neighborhood. We call this property of
that numerical method "global convergence". Results of numerical experiments
for the case of the backscattering data are presented
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
Convergence of a series associated with the convexification method for coefficient inverse problems
This paper is concerned with the convergence of a series associated with a
certain version of the convexification method. That version has been recently
developed by the research group of the first author for solving coefficient
inverse problems. The convexification method aims to construct a globally
convex Tikhonov-like functional with a Carleman Weight Function in it. In the
previous works the construction of the strictly convex weighted Tikhonov-like
functional assumes a truncated Fourier series (i.e. a finite series instead of
an infinite one) for a function generated by the total wave field. In this
paper we prove a convergence property for this truncated Fourier series
approximation. More precisely, we show that the residual of the approximate PDE
obtained by using the truncated Fourier series tends to zero in as the
truncation index in the truncated Fourier series tends to infinity. The proof
relies on a convergence result in the -norm for a sequence of
-orthogonal projections on finite-dimensional subspaces spanned by
elements of a special Fourier basis. However, due to the ill-posed nature of
coefficient inverse problems, we cannot prove that the solution of that
approximate PDE, which results from the minimization of that Tikhonov-like
functional, converges to the correct solution.Comment: 15 page
Convexification and experimental data for a 3D inverse scattering problem with the moving point source
Inverse scattering problems of the reconstructions of physical properties of
a medium from boundary measurements are substantially challenging ones. This
work aims to verify the performance on experimental data of a newly developed
convexification method for a 3D coefficient inverse problem for the case of
objects buried in a sandbox a fixed frequency and the point source moving along
an interval of a straight line. Using a special Fourier basis, the method of
this work strongly relies on a new derivation of a boundary value problem for a
system of coupled quasilinear elliptic equations. This problem, in turn, is
solved via the minimization of a Tikhonov-like functional weighted by a
Carleman Weight Function. The global convergence of the numerical procedure is
established analytically. The numerical verification is performed using
experimental data, which are raw backscatter data of the electric field. These
data were collected using a microwave scattering facility at The University of
North Carolina at Charlotte.Comment: 35 pages, 22 figures, 2 table
An inverse problem of a simultaneous reconstruction of the dielectric constant and conductivity from experimental backscattering data
This report extends our recent progress in tackling a challenging 3D inverse
scattering problem governed by the Helmholtz equation. Our target application
is to reconstruct dielectric constants, electric conductivities and shapes of
front surfaces of objects buried very closely under the ground. These objects
mimic explosives, like, e.g., antipersonnel land mines and improvised explosive
devices. We solve a coefficient inverse problem with the backscattering data
generated by a moving source at a fixed frequency. This scenario has been
studied so far by our newly developed convexification method that consists in a
new derivation of a boundary value problem for a coupled quasilinear elliptic
system. However, in our previous work only the unknown dielectric constants of
objects and shapes of their front surfaces were calculated. Unlike this, in the
current work performance of our numerical convexification algorithm is verified
for the case when the dielectric constants, the electric conductivities and
those shapes of objects are unknown. By running several tests with
experimentally collected backscattering data, we find that we can accurately
image both the dielectric constants and shapes of targets of interests
including a challenging case of targets with voids. The computed electrical
conductivity serves for reliably distinguishing conductive and non-conductive
objects. The global convergence of our numerical procedure is shortly
revisited.Comment: Coefficient inverse problem, multiple point sources, experimental
data, Carleman weight, global convergence, Fourier series. arXiv admin note:
text overlap with arXiv:2003.1151
The gradient descent method for the convexification to solve boundary value problems of quasi-linear PDEs and a coefficient inverse problem
We study the global convergence of the gradient descent method of the
minimization of strictly convex functionals on an open and bounded set of a
Hilbert space. Such results are unknown for this type of sets, unlike the case
of the entire Hilbert space. The proof of this convergence is based on the
classical contraction principle. Then, we use our result to establish a general
framework to numerically solve boundary value problems for quasi-linear partial
differential equations (PDEs) with noisy Cauchy data. The procedure involves
the use of Carleman weight functions to convexify a cost functional arising
from the given boundary value problem and thus to ensure the convergence of the
gradient descent method above. We prove the global convergence of the method as
the noise tends to 0. The convergence rate is Lipschitz. Next, we apply this
method to solve a highly nonlinear and severely ill-posed coefficient inverse
problem, which is the so-called back scattering inverse problem. This problem
has many real-world applications. Numerical examples are presented
Convexification for a 3D inverse scattering problem with the moving point source
For the first time, we develop in this paper the globally convergent
convexification numerical method for a Coefficient Inverse Problem for the 3D
Helmholtz equation for the case when the backscattering data are generated by a
point source running along an interval of a straight line and the wavenumber is
fixed. Thus, by varying the wavenumber, one can reconstruct the dielectric
constant depending not only on spatial variables but the wavenumber (i.e.
frequency) as well. Our approach relies on a new derivation of a boundary value
problem for a system of coupled quasilinear elliptic partial differential
equations. This is done via an application of a special truncated Fourier-like
method. First, we prove the Lipschitz stability estimate for this problem via a
Carleman estimate. Next, using the Carleman Weight Function generated by that
estimate, we construct a globally strictly convex cost functional and prove the
global convergence to the exact solution of the gradient projection method.
Finally, our theoretical finding is verified via several numerical tests with
computationally simulated data. These tests demonstrate that we can accurately
recover all three important components of targets of interest: locations,
shapes and dielectric constants. In particular, large target/background
contrasts in dielectric constants (up to 10:1) can be accurately calculated.Comment: 32 pages; 11 figures; 2 table
Convexification for an Inverse Problem for a 1D Wave Equation with Experimental Data
The forward problem here is the Cauchy problem for a 1D hyperbolic PDE with a
variable coefficient in the principal part of the operator. That coefficient is
the spatially distributed dielectric constant. The inverse problem consists of
the recovery of that dielectric constant from backscattering boundary
measurements. The data depend on one variable, which is time. To address this
problem, a new version of the convexification method is analytically developed.
The theory guarantees the global convergence of this method. Numerical testing
is conducted for both computationally simulated and experimental data.
Experimental data, which are collected in the field, mimic the problem of the
recovery of the spatially distributed dielectric constants of antipersonnel
land mines and improvised explosive devices.Comment: 28 pages, 7 figures, to be published in SIAM Journal on Imaging
Sciences (SIIMS
Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data
This paper is concerned with the inverse scattering problem which aims to
determine the spatially distributed dielectric constant coefficient of the 2D
Helmholtz equation from multifrequency backscatter data associated with a
single direction of the incident plane wave. We propose a globally convergent
convexification numerical algorithm to solve this nonlinear and ill-posed
inverse problem. The key advantage of our method over conventional optimization
approaches is that it does not require a good first guess about the solution.
First, we eliminate the coefficient from the Helmholtz equation using a change
of variables. Next, using a truncated expansion with respect to a special
Fourier basis, we approximately reformulate the inverse problem as a system of
quasilinear elliptic PDEs, which can be numerically solved by a weighted
quasi-reversibility approach. The cost functional for the weighted
quasi-reversibility method is constructed as a Tikhonov-like functional that
involves a Carleman Weight Function. Our numerical study shows that, using a
version of the gradient descent method, one can find the minimizer of this
Tikhonov-like functional without any advanced \emph{a priori} knowledge about
it.Comment: 25 page
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