45,342 research outputs found
Direct and Inverse Computational Methods for Electromagnetic Scattering in Biological Diagnostics
Scattering theory has had a major roll in twentieth century mathematical
physics. Mathematical modeling and algorithms of direct,- and inverse
electromagnetic scattering formulation due to biological tissues are
investigated. The algorithms are used for a model based illustration technique
within the microwave range. A number of methods is given to solve the inverse
electromagnetic scattering problem in which the nonlinear and ill-posed nature
of the problem are acknowledged.Comment: 61 pages, 5 figure
Application of the inhomogeneous Lippmann-Schwinger equation to inverse scattering problems
In this paper we present a hybrid approach to numerically solve
two-dimensional electromagnetic inverse scattering problems, whereby the
unknown scatterer is hosted by a possibly inhomogeneous background. The
approach is `hybrid' in that it merges a qualitative and a quantitative method
to optimize the way of exploiting the a priori information on the background
within the inversion procedure, thus improving the quality of the
reconstruction and reducing the data amount necessary for a satisfactory
result. In the qualitative step, this a priori knowledge is utilized to
implement the linear sampling method in its near-field formulation for an
inhomogeneous background, in order to identify the region where the scatterer
is located. On the other hand, the same a priori information is also encoded in
the quantitative step by extending and applying the contrast source inversion
method to what we call the `inhomogeneous Lippmann-Schwinger equation': the
latter is a generalization of the classical Lippmann-Schwinger equation to the
case of an inhomogeneous background, and in our paper is deduced from the
differential formulation of the direct scattering problem to provide the
reconstruction algorithm with an appropriate theoretical basis. Then, the point
values of the refractive index are computed only in the region identified by
the linear sampling method at the previous step. The effectiveness of this
hybrid approach is supported by numerical simulations presented at the end of
the paper.Comment: accepted in SIAM Journal on Applied Mathematic
Initial-Boundary Value Problems for Linear and Soliton PDEs
Evolution PDEs for dispersive waves are considered in both linear and
nonlinear integrable cases, and initial-boundary value problems associated with
them are formulated in spectral space. A method of solution is presented, which
is based on the elimination of the unknown boundary values by proper
restrictions of the functional space and of the spectral variable complex
domain. Illustrative examples include the linear Schroedinger equation on
compact and semicompact n-dimensional domains and the nonlinear Schroedinger
equation on the semiline.Comment: 18 pages, LATEX, submitted to the proccedings of NEEDS 2001 - Special
Issue, to be published in the Journal of Theoretical and Mathematical Physic
A new phase space method for recovering index of refraction from travel times
We develop a new phase space method for reconstructing the index of refraction of a medium from travel time measurements. The method is based on the so-called Stefanov–Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The new algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for the index of refraction allows us to compute the solution in physical space. Numerical examples including isotropic metrics and the Marmousi synthetic model are shown to validate the new method
On the Derivation of Vector Radiative Transfer Equation for Polarized Radiative Transport in Graded Index Media
Light transport in graded index media follows a curved trajectory determined
by the Fermat's principle. Besides the effect of variation of the refractive
index on the transport of radiative intensity, the curved ray trajectory will
induce geometrical effects on the transport of polarization ellipse. This paper
presents a complete derivation of vector radiative transfer equation for
polarized radiation transport in absorption, emission and scattering graded
index media. The derivation is based on the analysis of the conserved
quantities for polarized light transport along curved trajectory and a novel
approach. The obtained transfer equation can be considered as a generalization
of the classic vector radiative transfer equation that is only valid for
uniform refractive index media. Several variant forms of the transport equation
are also presented, which include the form for Stokes parameters defined with a
fixed reference and the Eulerian forms in the ray coordinate and in several
common orthogonal coordinate systems.Comment: This paper has been submitted to JQSR
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