168 research outputs found
Application of the Finite Element Method in a Quantitative Imaging technique
We present the Finite Element Method (FEM) for the numerical solution of the
multidimensional coefficient inverse problem (MCIP) in two dimensions. This
method is used for explicit reconstruction of the coefficient in the hyperbolic
equation using data resulted from a single measurement. To solve our MCIP we
use approximate globally convergent method and then apply FEM for the resulted
equation. Our numerical examples show quantitative reconstruction of the sound
speed in small tumor-like inclusions
Numerical analysis of least squares and perceptron learning for classification problems
This work presents study on regularized and non-regularized versions of
perceptron learning and least squares algorithms for classification problems.
Fr'echet derivatives for regularized least squares and perceptron learning
algorithms are derived. Different Tikhonov's regularization techniques for
choosing the regularization parameter are discussed. Decision boundaries
obtained by non-regularized algorithms to classify simulated and experimental
data sets are analyzed
Domain decomposition finite element/finite difference method for the conductivity reconstruction in a hyperbolic equation
We present domain decomposition finite element/finite difference method for
the solution of hyperbolic equation. The domain decomposition is performed such
that finite elements and finite differences are used in different subdomains of
the computational domain: finite difference method is used on the structured
part of the computational domain and finite elements on the unstructured part
of the domain. The main goal of this method is to combine flexibility of finite
element method and efficiency of a finite difference method.
An explicit discretization schemes for both methods are constructed such that
finite element and finite difference schemes coincide on the common structured
overlapping layer between computational subdomains. Then the resulting scheme
can be considered as a pure finite element scheme which allows avoid
instabilities at the interfaces.
We illustrate efficiency of the domain decomposition method on the
reconstruction of the conductivity function in the hyperbolic equation in three
dimensions
Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide
We consider an inverse problem of reconstructing the conductivity function in
a hyperbolic equation using single space-time domain noisy observations of the
solution on the backscattering boundary of the computational domain. We
formulate our inverse problem as an optimization problem and use Lagrangian
approach to minimize the corresponding Tikhonov functional. We present a
theorem of a local strong convexity of our functional and derive error
estimates between computed and regularized as well as exact solutions of this
functional, correspondingly. In numerical simulations we apply domain
decomposition finite element-finite difference method for minimization of the
Lagrangian. Our computational study shows efficiency of the proposed method in
the reconstruction of the conductivity function in three dimensions
Computational design of nanophotonic structures using an adaptive finite element method
We consider the problem of the construction of the nanophotonic structures of
arbitrary geometry with prescribed desired properties. We reformulate this
problem as an optimization problem for the Tikhonov functional which is
minimized on adaptively locally refined meshes. These meshes are refined only
in places where the nanophotonic structure should be designed. Our special
symmetric mesh refinement procedure allows the construction of different
nanophotonic structures. We illustrate efficiency of our adaptive optimization
algorithm on the construction of nanophotonic structure in two dimensions
Uniqueness and stability of time and space-dependent conductivity in a hyperbolic cylindrical domain
This paper is devoted to the reconstruction of the time and space-dependent
coefficient in an infinite cylindrical hyperbolic domain. Using a local
Carleman estimate we prove the uniqueness and a H\"older stability in the
determining of the conductivity by a single measurement on the lateral
boundary. Our numerical examples show good reconstruction of the location and
contrast of the conductivity function in three dimensions.Comment: arXiv admin note: text overlap with arXiv:1501.0138
Lipschitz stability for an inverse hyperbolic problem of determining two coefficients by a finite number of observations
We consider an inverse problem of reconstructing two spatially varying
coefficients in an acoustic equation of hyperbolic type using interior data of
solutions with suitable choices of initial condition. Using a Carleman
estimate, we prove Lipschitz stability estimates which ensures unique
reconstruction of both coefficients. Our theoretical results are justified by
numerical studies on the reconstruction of two unknown coefficients using noisy
backscattered data
Reconstruction of dielectric constants of multi-layered optical fibers using propagation constants measurements
We present new method for the numerical reconstruction of the variable
refractive index of multi-layered circular weakly guiding dielectric waveguides
using the measurements of the propagation constants of their eigenwaves. Our
numerical examples show stable reconstruction of the dielectric permittivity
function for random noise level using these measurements
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