19 research outputs found
Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity
We consider the inverse problem of the reconstruction of the spatially
distributed dielectric constant $\varepsilon_{r}\left(\mathbf{x}\right), \
\mathbf{x}\in \mathbb{R}^{3}n\left(\mathbf{x}\right) =\sqrt{\varepsilon_{r}\left(\mathbf{x}\right)}.\varepsilon_{r}\left(\mathbf{x}\right) $ is reconstructed using a
two-stage reconstruction procedure. In the first stage an approximately
globally convergent method proposed is applied to get a good first
approximation of the exact solution. In the second stage a locally convergent
adaptive finite element method is applied, taking the solution of the first
stage as the starting point of the minimization of the Tikhonov functional.
This functional is minimized on a sequence of locally refined meshes. It is
shown here that all three components of interest of targets can be
simultaneously accurately imaged: refractive indices, shapes and locations
A posteriori error estimate in the lagrangian setting for an inverse problem based on a new formulation of Maxwellâs system
In this paper we consider an inverse problem of determination of a dielectric permittivity function from a backscattered electromagnetic wave. The inverse problem is formulated as an optimal control problem for a certain partial differential equation derived from Maxwellâs system. We study a solution method based on finite element approximation and provide a posteriori error estimate for the use in an adaptive algorithm
A Two-stage Numerical Procedure for an Inverse Scattering Problem
In this thesis we study a numerical procedure for the solution of the inverse problem of reconstructing location, shape and material properties (in particular refractive indices) of scatterers located in a known background medium. The data consist of time-resolved backscattered radar signals from a single source position. This relatively small amount of data and the ill-posed nature of the inversion are the main challenges of the problem. Mathematically, the problem is formulated as a coefficient inverse problem for a system of partial differential equations derived from Maxwell\u27s equations.The numerical procedure is divided into two stages. In the first stage, a good initial approximation for the unknown coefficient is computed by an approximately globally convergent algorithm. This initial approximation is refined in the second stage, where an adaptive finite element method is employed to minimize a Tikhonov functional. An important tool for the second stage is a posteriori error estimates -- estimates in terms of known (computed) quantities -- for the difference between the computed coefficient and the true minimizing coefficient. This thesis includes four papers. In the first two, the a posteriori error analysis required for the adaptive finite element method in the second stage is extended from the previously existing indirect error estimators to direct ones. The last two papers concern verification of the two-stage numerical procedure on experimental data. We find that location and material properties of scatterers are obtained already in the first stage, while shapes are significantly improved in the second stage
A two-stage numerical procedure for an inverse scattering problem
In this thesis we study a numerical procedure for the solution of the
inverse problem of reconstructing location, shape and material properties
(in particular refractive indices) of scatterers located in a known background
medium. The data consist of time-resolved backscattered radar
signals from a single source position. This relatively small amount of
data and the ill-posed nature of the inversion are the main challenges of
the problem. Mathematically, the problem is formulated as a coefficient
inverse problem for a system of partial differential equations derived from
Maxwellâs equations.
The numerical procedure is divided into two stages. In the first stage,
a good initial approximation for the unknown coefficient is computed by
an approximately globally convergent algorithm. This initial approximation
is refined in the second stage, where an adaptive finite element
method is employed to minimize a Tikhonov functional. An important
tool for the second stage is a posteriori error estimates â estimates in
terms of known (computed) quantities â for the difference between the
computed coefficient and the true minimizing coefficient.
This thesis includes four papers. In the first two, the a posteriori error
analysis required for the adaptive finite element method in the second
stage is extended from the previously existing indirect error estimators
to direct ones. The last two papers concern verification of the two-stage
numerical procedure on experimental data. We find that location and
material properties of scatterers are obtained already in the first stage,
while shapes are significantly improved in the second stage
Approximate globally convergent algorithm with applications in electrical prospecting
In this paper we present at the first time an approximate globally convergent method for the reconstruction of an unknown conductivity function from backscattered electric field measured at the boundary of geological medium under assumptions that dielectric permittivity and magnetic permeability functions are known. This is the typical case of an coefficient inverse problem in electrical prospecting. We consider a simplified mathematical model assuming that geological medium is isotropic and non-dispersive
An Adaptive Finite Element Method in Quantitative Reconstruction of Small Inclusions from Limited Observations
We consider a coefïŹcient inverse problem to determine the dielectric permittivity in Maxwellâs equations, with data consisting of boundary measurements. The true dielectric permittivity is assumed to belong to an ideal space of very ïŹne ïŹnite elements. The problem is treated using a Lagrangian approach to the minimization of a Tikhonov functional, where an adaptive ïŹnite element method forms the basis of the computations. A new a posteriori error estimate for the norm of the error in the reconstructed permittivity is derived. The adaptive algorithm is formulated and tested successfully in numerical experiments for the reconstruction of two, three, and four small inclusions with low contrast, as well as the reconstruction of a superposition of two Gaussian functions
Iterative regularization and adaptivity for an electromagnetic coefficient inverse problem
We study how the choice of the regularization parameter affects the quality of the reconstruction of the dielectric permittivity for an inhomogeneous medium, with data consisting of boundary observations of the electric field. Our method is based on the minimization of a Tikhonov functional and uses a finite element method for computations of the electric field. We conclude that the choice of the regularization parameter does not affect the quality of the reconstruction significantly in the studied cases, and can even be removed with results not significantly different from those with regularization
An adaptive finite element method in quantitative reconstruction of small inclusions from limited observations
We consider a coefficient inverse problem to determine the dielectric permittivity in Maxwell\u27s equations, with data consisting of boundary measurements. The true dielectric permittivity is assumed to belong to an ideal space of very fine finite elements. The problem is treated using a Lagrangian approach to the minimization of a Tikhonov functional, where an adaptive finite element method forms the basis of the computations. A new a posteriori error estimate for the norm of the error in the reconstructed permittivity is derived. The adaptive algorithm is formulated and tested successfully in numerical experiments for the reconstruction of two, three, and four small inclusions with low contrast, as well as the reconstruction of a superposition of two Gaussian functions