The Manakov system appears in the physics of optical fibers, as well as in quantum mechanics, as multi-component versions of the Nonlinear Schr\"odinger and the Gross-Pitaevskii equations.Although the Manakov system is completely integrable its solutions are far from being explicit in most cases. However, the Inverse Scattering Transform (IST) can be exploited to obtain asymptotic information about solutions.This thesis will describe the IST of the Manakov system, and its asymptotic behavior at short times. I will compare the focusing and defocusing behavior, numerically and analytically, for squared barrier initial potentials. Finally, I will show that the continuous spectrum gives the dominant contribution at short-times.

Wang, Yiran. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 70-73). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Overview of the subject --- p.6 / Chapter 1.2 --- Motivation --- p.8 / Chapter 2 --- Inverse Medium Scattering Problem --- p.11 / Chapter 2.1 --- Mathematical Formulation --- p.11 / Chapter 2.1.1 --- Absorbing Boundary Conditions --- p.12 / Chapter 2.1.2 --- Applications --- p.14 / Chapter 2.2 --- Preliminary Results --- p.17 / Chapter 2.2.1 --- Weak Formulation --- p.17 / Chapter 2.2.2 --- About the Unique Determination --- p.21 / Chapter 3 --- Unconstrained Optimization: Steepest Decent Method --- p.25 / Chapter 3.1 --- Recursive Linearization Method Revisited --- p.25 / Chapter 3.1.1 --- Frechet differentiability --- p.26 / Chapter 3.1.2 --- Initial guess --- p.28 / Chapter 3.1.3 --- Landweber iteration --- p.30 / Chapter 3.1.4 --- Numerical Results --- p.32 / Chapter 3.2 --- Steepest Decent Analysis --- p.35 / Chapter 3.2.1 --- Single Wave Case --- p.36 / Chapter 3.2.2 --- Multiple Wave Case --- p.39 / Chapter 3.3 --- Numerical Experiments and Discussions --- p.43 / Chapter 4 --- Constrained Optimization: Augmented Lagrangian Method --- p.51 / Chapter 4.1 --- Method Review --- p.51 / Chapter 4.2 --- Problem Formulation --- p.54 / Chapter 4.3 --- First Order Optimality Condition --- p.56 / Chapter 4.4 --- Second Order Optimality Condition --- p.60 / Chapter 4.5 --- Modified Algorithm --- p.62 / Chapter 5 --- Conclusions and Future Work --- p.68 / Bibliography --- p.70

The goal of this research is to achieve accurate characterization of multi-layered soft tissues in three dimensions using focused ultrasound. The characterization of the acoustic parameters of each tissue layer is formulated as recursive processes of forward- and inverse- scattering. Forward scattering deals with the modeling of focused ultrasound wave propagation in multi-layered tissues, and the computation of the focused wave amplitudes in the tissues based on the acoustic parameters of the tissue as generated by inverse scattering. The model for mapping the tissue acoustic parameters to focused waves is highly nonlinear and stochastic. In addition, solving (or inverting) the model to obtain tissue acoustic parameters is an ill-posed problem. Therefore, a nonlinear stochastic inverse scattering method is proposed such that no linearization and mathematical inversion of the model are required. Inverse scattering aims to estimate the tissue acoustic parameters based on the forward scattering model and ultrasound measurements of the tissues. A multi-scale stochastic filter (MSF) is proposed to perform inverse scattering. MSF generates a set of tissue acoustic parameters, which are then mapped into focused wave amplitudes in the multi-layered tissues by forward scattering. The tissue acoustic parameters are weighted by comparing their focused wave amplitudes to the actual ultrasound measurements. The weighted parameters are used to estimate a weighted Gaussian mixture as the posterior probability density function (PDF) of the parameters. This PDF is optimized to achieve minimum estimation error variance in the sense of the posterior Cramer-Rao bound. The optimized posterior PDF is used to produce minimum mean-square-error estimates of the tissue acoustic parameters. As a result, both the estimation error and uncertainty of the parameters are minimized. PDF optimization is formulated based on a novel multi-scale PDF analysis framework. This framework is founded based on exploiting the analogy between PDFs and analog (or digital) signals. PDFs and signals are similar in the sense that they represent characteristics of variables in their respective domains, except that there are constraints imposed on PDFs. Therefore, it is reasonable to consider a PDF as a signal that is subject to amplitude constraints, and as such apply signal processing techniques to analyze the PDF. The multi-scale PDF analysis framework is proposed to recursively decompose an arbitrary PDF from its fine to coarse scales. The recursive decompositions are designed so as to ensure that requirements such as PDF constraints, zero-phase shift and non-creation of artifacts are satisfied. The relationship between the PDFs at consecutive scales is derived in order for the PDF optimization process to recursively reconstruct the posterior PDF from its coarse to fine scales. At each scale, PDF reconstruction aims to reduce the variances of the posterior PDF Gaussian components, and as a result the confidence in the estimate is increased. The overall posterior PDF variance reduction is guided by the posterior Cramer-Rao bound. A series of experiments is conducted to investigate the performance of the proposed method on ultrasound multi-layered soft tissue characterization. Multi-layered tissue phantoms that emulate ocular components of the eye are fabricated as test subjects. Experimental results confirm that the proposed MSF inverse scattering approach is well suited for three-dimensional ultrasound tissue characterization. In addition, performance comparisons between MSF and a state-of-the-art nonlinear stochastic filter are conducted. Results show that MSF is more accurate and less computational intensive than the state-of-the-art filter.

The goal of this research is to achieve accurate characterization of multi-layered soft tissues in three dimensions using focused ultrasound. The characterization of the acoustic parameters of each tissue layer is formulated as recursive processes of forward- and inverse- scattering. Forward scattering deals with the modeling of focused ultrasound wave propagation in multi-layered tissues, and the computation of the focused wave amplitudes in the tissues based on the acoustic parameters of the tissue as generated by inverse scattering. The model for mapping the tissue acoustic parameters to focused waves is highly nonlinear and stochastic. In addition, solving (or inverting) the model to obtain tissue acoustic parameters is an ill-posed problem. Therefore, a nonlinear stochastic inverse scattering method is proposed such that no linearization and mathematical inversion of the model are required. Inverse scattering aims to estimate the tissue acoustic parameters based on the forward scattering model and ultrasound measurements of the tissues. A multi-scale stochastic filter (MSF) is proposed to perform inverse scattering. MSF generates a set of tissue acoustic parameters, which are then mapped into focused wave amplitudes in the multi-layered tissues by forward scattering. The tissue acoustic parameters are weighted by comparing their focused wave amplitudes to the actual ultrasound measurements. The weighted parameters are used to estimate a weighted Gaussian mixture as the posterior probability density function (PDF) of the parameters. This PDF is optimized to achieve minimum estimation error variance in the sense of the posterior Cramer-Rao bound. The optimized posterior PDF is used to produce minimum mean-square-error estimates of the tissue acoustic parameters. As a result, both the estimation error and uncertainty of the parameters are minimized. PDF optimization is formulated based on a novel multi-scale PDF analysis framework. This framework is founded based on exploiting the analogy between PDFs and analog (or digital) signals. PDFs and signals are similar in the sense that they represent characteristics of variables in their respective domains, except that there are constraints imposed on PDFs. Therefore, it is reasonable to consider a PDF as a signal that is subject to amplitude constraints, and as such apply signal processing techniques to analyze the PDF. The multi-scale PDF analysis framework is proposed to recursively decompose an arbitrary PDF from its fine to coarse scales. The recursive decompositions are designed so as to ensure that requirements such as PDF constraints, zero-phase shift and non-creation of artifacts are satisfied. The relationship between the PDFs at consecutive scales is derived in order for the PDF optimization process to recursively reconstruct the posterior PDF from its coarse to fine scales. At each scale, PDF reconstruction aims to reduce the variances of the posterior PDF Gaussian components, and as a result the confidence in the estimate is increased. The overall posterior PDF variance reduction is guided by the posterior Cramer-Rao bound. A series of experiments is conducted to investigate the performance of the proposed method on ultrasound multi-layered soft tissue characterization. Multi-layered tissue phantoms that emulate ocular components of the eye are fabricated as test subjects. Experimental results confirm that the proposed MSF inverse scattering approach is well suited for three-dimensional ultrasound tissue characterization. In addition, performance comparisons between MSF and a state-of-the-art nonlinear stochastic filter are conducted. Results show that MSF is more accurate and less computational intensive than the state-of-the-art filter.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 134-137).

Three related electromagnetic scattering problems, namely, high frequency electromagnetic (EM) ray tracing, scattering feature extraction, and inverse scattering are studied in this dissertation. New approaches are presented to advance the state of the art in each of the areas. The presented study in electromagnetic ray tracing leads to an alternative ray tracing algorithm which can outperform the traditional algorithms for complex targets. The performance of the proposed techniques demonstrates their potential application to the study of high-frequency EM scattering prediction. Second, a genetic algorithm (GA)-based algorithm with an adaptive-feeding technique is developed to simultaneously extract both scattering centers and resonances. Scattering feature extraction algorithms are then developed with the consideration of the visibility of scattering centers. Inverse scattering problems with strong multiple scattering effects are also studied. A GA-based method is presented to invert the shapes with multiple scattering effects. An approach combining hybrid GA with the tabu list idea are then developed to further improve the performance of the GA-based inversion algorithms. / text

published_or_final_version / abstract / toc / Physics / Master / Master of Philosophy

No description available.

A non-linear edge-preserving solution to the least-squares migration problem with sparseness & illumination constraints is proposed. The applied formalism explores Curvelets as basis functions. By virtue of their sparseness and locality, Curvelets not only reduce the dimensionality of the imaging problem but they also naturally lead to a dense preconditioning that almost diagonalizes the normal/Hessian operator. This almost diagonalization allows us to recast the imaging problem into a ’simple’ denoising problem. As such, we are in the position to use non-linear estimators based on thresholding. These estimators exploit the sparseness and locality of Curvelets and allow us to compute a first estimate for the reflectivity, which approximates the least-squares solution of the seismic inverse scattering problem. Given this estimate, we impose sparseness and additional amplitude corrections by solving a constrained optimization problem. This optimization problem is initialized and constrained by the thresholded image and is designed to remove remaining imaging artifacts and imperfections in the estimation and reconstruction.

Thesis (M. Phil.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 102-120).