A study of iterative methods on forward and inverse scattering problems
Lin, Jiun-Hwa
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Permalink
https://hdl.handle.net/2142/23515
Description
Title
A study of iterative methods on forward and inverse scattering problems
Author(s)
Lin, Jiun-Hwa
Issue Date
1995
Doctoral Committee Chair(s)
Chew, Weng Cho
Department of Study
Engineering, Electronics and Electrical
Discipline
Engineering, Electronics and Electrical
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Electronics and Electrical
Language
eng
Abstract
Iterative methods are suitable for solving large-size problems in the electromagnetic and acoustic wave scattering. Both the conjugate gradient and bi-conjugate gradient methods combined with the fast Fourier transform (CGFFT and BiCGFFT) are employed as efficient solvers in forward and inverse scattering problems for penetrable bodies.
In microwave imaging, material permittivity is the parameter to retrieve. The distorted Born iterative method (DBIM), a nonlinear inverse scattering algorithm that accounts for multiple scattering, can retrieve the permittivity of high contrast. The computational cost for each iteration is $O(N\sp{1.5}\ \log\ N)$ as the number of transmitters is $O(N\sp{0.5}),$ where N is the number of cells. Real experimental data has been processed by the algorithm under a full-view system to obtain images in real and imaginary parts of permittivity. With the aid of the frequency-hopping scheme, large-size objects can be reconstructed with higher fidelity.
The nested equivalence principle algorithm (NEPAL) has been developed to implement the matrix-vector multiply in an O(N log N) fashion. NEPAL can also be applied in the cases of nonuniform grids. With the fast multipole method (FMM) incorporated in NEPAL, an O(N) algorithm can be achieved to perform the matrix-vector multiply.
The T-matrix method is used to formulate the three-dimensional electromagnetic scattering problems. Exploiting the Toeplitz structure of the translation matrix, BiCGFFT is invoked as the solver, which requires only O(N log N) operations at each iteration and O(N) memory storage. Efficiency in computation and storage enables the algorithm to solve large problems in real practice.
Acoustic wave equations possess the same features as the electromagnetic wave equations for $H\sb{z}$ polarization in two dimensions. The local shape function method (LSF) developed for inverse electromagnetic scattering is adapted to reconstructing both the density and compressibility of soft tissues in ultrasonic imaging. CGFFT is utilized as the forward solver as required to implement inverse operators. The capability of the algorithm has been demonstrated in the reconstructions from the experimental data as well as the synthetic data, and its complexity can be $O(N\sp{1.5}\ {\rm log}\ N)$ at each iteration. A multiple-frequency scheme, such as the frequency-hopping method, provides better reconstruction than a single-frequency scheme.
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