Deng, Xiaomao. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 98-104). / Chapter 1 --- Introduction to Inverse Scattering Problems --- p.6 / Chapter 1.1 --- Direct Problems --- p.6 / Chapter 1.1.1 --- Far-field Patterns --- p.10 / Chapter 1.2 --- Inverse Problems --- p.16 / Chapter 1.2.1 --- Introduction --- p.16 / Chapter 2 --- Numerical Methods in Inverse Obstacle Scattering --- p.19 / Chapter 2.1 --- Linear Sampling Method --- p.19 / Chapter 2.1.1 --- History Review --- p.19 / Chapter 2.1.2 --- Numerical Scheme of LSM --- p.21 / Chapter 2.1.3 --- Theoretic Justification --- p.25 / Chapter 2.1.4 --- Summarize --- p.38 / Chapter 2.2 --- Point Source Method --- p.38 / Chapter 2.2.1 --- Historical Review --- p.38 / Chapter 2.2.2 --- Superposition of Plane Waves --- p.40 / Chapter 2.2.3 --- Approximation of Domains --- p.42 / Chapter 2.2.4 --- Algorithm --- p.44 / Chapter 2.2.5 --- Summarize --- p.49 / Chapter 2.3 --- Singular Source Method --- p.49 / Chapter 2.3.1 --- Historical Review --- p.49 / Chapter 2.3.2 --- Algorithm --- p.51 / Chapter 2.3.3 --- Far-field Data --- p.54 / Chapter 2.3.4 --- Summarize --- p.55 / Chapter 2.4 --- Probe Method --- p.57 / Chapter 2.4.1 --- Historical Review --- p.57 / Chapter 2.4.2 --- Needle --- p.58 / Chapter 2.4.3 --- Algorithm --- p.59 / Chapter 3 --- Numerical Experiments --- p.61 / Chapter 3.1 --- Discussions on Linear Sampling Method --- p.61 / Chapter 3.1.1 --- Regularization Strategy --- p.61 / Chapter 3.1.2 --- Cut off Value --- p.70 / Chapter 3.1.3 --- Far-field data --- p.76 / Chapter 3.2 --- Numerical Verification of PSM and SSM --- p.80 / Chapter 3.3 --- Inverse Medium Scattering --- p.83 / Bibliography --- p.98

Cheung, Man Wah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 89-91). / Abstracts in English and Chinese. / Chapter 1 --- Introduction to Inverse Problems --- p.1 / Chapter 1.1 --- Major properties --- p.1 / Chapter 1.2 --- Typical examples --- p.3 / Chapter 1.3 --- Thesis outline --- p.5 / Chapter 2 --- Some Operator Theory --- p.6 / Chapter 2.1 --- Fredholm integral equation of the first kind --- p.6 / Chapter 2.2 --- Compact operator theory --- p.8 / Chapter 2.3 --- Singular system --- p.12 / Chapter 2.4 --- Moore-Penrose generalized inverse --- p.14 / Chapter 3 --- Regularization Theory for First Kind Equations --- p.19 / Chapter 3.1 --- General regularization theory --- p.19 / Chapter 3.2 --- Tikhonov regularization --- p.24 / Chapter 3.3 --- Landweber iteration --- p.26 / Chapter 3.4 --- TSVD --- p.28 / Chapter 4 --- Multilevel Algorithms for Ill-posed Problems --- p.30 / Chapter 4.1 --- Basic assumptions and definitions --- p.31 / Chapter 4.2 --- Multilevel analysis --- p.33 / Chapter 4.3 --- Applications --- p.37 / Chapter 4.3.1 --- Preconditioned iterative methods with nonzero regularization parameter --- p.38 / Chapter 4.3.2 --- Preconditioned iterative methods with zero regularization parameter --- p.38 / Chapter 4.3.3 --- Full multilevel algorithm --- p.40 / Chapter 5 --- Numerical Experiments --- p.41 / Chapter 5.1 --- Integral equations --- p.41 / Chapter 5.1.1 --- Discretization --- p.42 / Chapter 5.1.2 --- Test problems --- p.43 / Chapter 5.1.3 --- "Singular values, singular vectors and condition numbers" --- p.45 / Chapter 5.1.4 --- Effect of condition numbers on numerical accuracies --- p.49 / Chapter 5.2 --- Differential equations --- p.50 / Chapter 5.2.1 --- Discretization --- p.51 / Chapter 5.2.2 --- "Singular values, singular vectors and condition numbers" --- p.53 / Chapter 5.3 --- Numerical experiments by classical methods --- p.55 / Chapter 5.3.1 --- Tikhonov regularization --- p.55 / Chapter 5.3.2 --- TSVD --- p.56 / Chapter 5.3.3 --- Landweber iteration --- p.63 / Chapter 5.4 --- Numerical experiments by multilevel methods --- p.63 / Chapter 5.4.1 --- General convergence --- p.63 / Chapter 5.4.2 --- Numerical results --- p.65 / Chapter 5.4.3 --- Effect of multilevel parameters on convergence --- p.76 / Bibliography --- p.89

Neste trabalho são apresentados alguns resultados matemáticos relevantes para a aplicação do método de espalhamento inverso à resolução de uma classe de equações de evolução não-lineares. É demonstrada a propriedade isoespectral para certas famílias de operadores lineares não auto-adjuntos. Esta propriedade tem um papel central na aplicação do método acima a equações de evolução não-lineares de interesse físico, tais como a equação de sine-Gordon e a equação de Schrödinger não-linear. É feito também, uma teoria de espalhamento inverso rigorosa para sistemas do tipo Zakharov-Shabat, o que inclui uma análise qualitativa do espectro de operadores deste tipo. / This Thesis presents some mathematical results relevant in applications of the inverse scattering transform to the solution of a class of non-linear evolution equations. First, it is shown that certain families of non-selfadjoint linear operators have the isospectral property, which is fundamental for the above applications. These families include various operators related to no-linear equations of great physical interest, such as the sine-Gordon and the non-linear Schrödinger equations. In the sequel, a rigorous inverse scattering theory, including a qualitative spectral analysis, is developed for systems of Zakharov-Shabat type.

We give a brief overview of Bass-Serre theory and introduce a method of adding a limit point to graphs of groups. We explore a basic example of this method, and find that while the fundamental theorem of Bass-Serre theory no longer applies in this case we still recover a group action on a covering space of sorts with a subgroup isomorphic to the fundamental group of our new base space with added limit point. We also quantify how much larger the fundamental group of a graph of groups becomes after this construction, and discuss the effects of adding and identifying together such limit points in more general graphs of groups. We conclude with a theorem stating that the cokernel of the map on fundamental groups induced by collapsing an arc between two limit points contains a certain fundamental group of a double cone of graphs of groups, and we conjecture that this cokernel is isomorphic to this double cone group.

A common problem in science and engineering applications deals with finding information about a system where only limited information is known. One example of this problem is determining the geometry of an aquifer or oil reservoir based on well tests taken at the site. The Conditional Coding Method attacks this type of problem. This method uses the Simulated Annealing Algorithm in conjunction with a probability model which generates possible solutions based on a uniform random number list. The Annealing Algorithm generates a conditional probability distribution on all possible solutions generated by the probability model, conditioned on the observed data set. The problem is attacked by sampling from this distribution. This method accounts for the noise inherent in the data set as well as the uncertainty due to the limited amount of data available.

ABSTRACT The quest to model wave propagation in various physical systems has produced a large set of diverse nonlinear equations. Nonlinear singular integro-differential equations rank amongst the intricate nonlinear wave equations available to study the classical problem of wave propagation in physical systems. Integro-differential equations are characterized by the simultaneous presence of integration and differentiation in a single equation. Substantial interest exists in nonlinear wave equations that are amenable to the Inverse Scattering Transform (IST). The IST is an adroit mathematical technique that delivers analytical solutions of a certain type of nonlinear equation: soliton equation. Initial value problems of numerous physically significant nonlinear equations have now been solved through elegant and novel implementations of the IST. The prototype nonlinear singular integro-differential equation receptive to the IST is the Intermediate Long Wave (ILW) equation, which models one-dimensional weakly nonlinear internal wave propagation in a density stratified fluid of finite total depth. In the deep water limit the ILW equation bifurcates into a physically significant nonlinear singular integro-differential equation known as the 'Benjamin-Ono' (BO) equation; the shallow water limit of the ILW equation is the famous Korteweg-de Vries (KdV) equation. Both the KdV and BO equations have been solved by dissimilar implementations of the IST. The Modified Korteweg-de Vries (MKdV) equation is a nonlinear partial differential equation, which was significant in the historical development of the IST. Solutions of the MKdV equation are mapped by an explicit nonlinear transformation known as the 'Miura transformation' into solutions of the KdV equation. Historically, the Miura transformation manifested the intimate connection between solutions of the KdV equation and the inverse problem for the one-dimensional time independent Schroedinger equation. In light of the MKdV equation's significance, it is natural to seek 'modified' versions of the ILW and BO equations. Solutions of each modified nonlinear singular integro-differential equation should be mapped by an analogue of the original Miura transformation into solutions of the 'unmodified' equation. In parallel with the limiting cases of the ILW equation, the modified version of the ILW equation should reduce to the MKdV equation in the shallow water limit and to the modified version of the BO equation in the deep water limit. The Modified Intermediate Long Wave (MILW) and Modified Benjamin-Ono (MBO) equations are the two nonlinear singular integro-differential equations that display all the required attributes. Several researchers have shown that the MILW and MBO equations exhibit the signature characteristic of soliton equations. Despite the significance of the MILW and MBO equations to soliton theory, and the possible physical applications of the MILW and MBO equations, the initial value problems for these equations have not been solved. In this thesis we use the IST to solve the initial value problems for the MILW and MBO equations on the real-line. The only restrictions that we place on the initial values for the MILW and MBO equations are that they be real-valued, sufficiently smooth and decay to zero as the absolute value of the spatial variable approaches large values.

Thesis (Sc. D.)--Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 1987. / "January 1987." Bibliography: p. 193-200.

<p>The main contribution of this thesis is on the study of optimization methods in aft hull design. The optimization methods are inverse geometry design methods to find an aft hull with the flow velocities we specify. The analytic foundation for the flow is given by Stratford in [31], and gives a prescribed velocity distribution on the aft body. With the parameter β we have adjusted this flow to have a certain margin to separation along the pressure recovery region.</p><p>This principle and optimization method are successfully applied to design of ships with pram-type aft hull. The 2D optimized profiles corresponds to centerline buttock, and 3D hull sections are extended from this centerline buttock with a bilge radius. </p><p>Stratfords original pressure distribution for pressure recovery region were meant for Reynolds numbers up to 107. We have extended Stratfords formula to yield for ship full scale Reynolds numbers to 109. </p><p>Different optimization methods were programmed and tested. The best routine for our optimization of aft hull with Stratford flow, was when the offset y-value were the optimization parameter to be changed. When we tried to optimize a complete 2D profile with a given pressure distribution, it worked best to use the variables in a B-spline as the optimization parameter.</p><p>Extensive windtunnel tests and towing tank tests are carried out. The tests verified the hydrodynamic properties of the hulls.</p><p>Towing tests indicates that the optimized hull lines have lower total resistance than conventional ships with the same main dimensions. Both the frictional, viscous pressure resistance and wave making resistance are significantly lower. Further we can increase cargo capacity with the same power consumption, and achieve a more favourable distribution of the displacement in the aft hull.</p><p>This study has shown us that the slant angle for the bottom of the aft hull should not excess 15º with horizontal plane due to danger of separation over the bilge, and longitudinal vortices forming. </p>

The main contribution of this thesis is on the study of optimization methods in aft hull design. The optimization methods are inverse geometry design methods to find an aft hull with the flow velocities we specify. The analytic foundation for the flow is given by Stratford in [31], and gives a prescribed velocity distribution on the aft body. With the parameter β we have adjusted this flow to have a certain margin to separation along the pressure recovery region. This principle and optimization method are successfully applied to design of ships with pram-type aft hull. The 2D optimized profiles corresponds to centerline buttock, and 3D hull sections are extended from this centerline buttock with a bilge radius. Stratfords original pressure distribution for pressure recovery region were meant for Reynolds numbers up to 107. We have extended Stratfords formula to yield for ship full scale Reynolds numbers to 109. Different optimization methods were programmed and tested. The best routine for our optimization of aft hull with Stratford flow, was when the offset y-value were the optimization parameter to be changed. When we tried to optimize a complete 2D profile with a given pressure distribution, it worked best to use the variables in a B-spline as the optimization parameter. Extensive windtunnel tests and towing tank tests are carried out. The tests verified the hydrodynamic properties of the hulls. Towing tests indicates that the optimized hull lines have lower total resistance than conventional ships with the same main dimensions. Both the frictional, viscous pressure resistance and wave making resistance are significantly lower. Further we can increase cargo capacity with the same power consumption, and achieve a more favourable distribution of the displacement in the aft hull. This study has shown us that the slant angle for the bottom of the aft hull should not excess 15º with horizontal plane due to danger of separation over the bilge, and longitudinal vortices forming.

Blood oxygen level dependent (BOLD) functional magnetic resonance imaging is a non-invasive technique used to trace changes in neural dynamics in reaction to mental activity caused by perceptual, motor or cognitive tasks. The BOLD response is a complex signal, a consequence of a series of physiological events regulated by increased neural activity. A method to infer from the BOLD signal onto underlying neuronal activity (hemodynamic inverse problem) is proposed in Chapter 2 under the assumption of a previously proposed mathematical model on the transduction of neural activity to the BOLD signal. Also, in this chapter we clarify the meaning of the neural activity function used as the input for an intrinsic dynamic system which can be viewed as an advanced substitute for the impulse response function. Chapter 3 describes an approach for recovering neural timing information (mental chronometry) in an object interaction decision task via solving the hemodynamic inverse problem. In contrast to the hemodynamic level, at the neural level, we were able to determine statistically significant latencies in activation between functional units in the model used. In Chapter 4, two approaches for regularization parameter tuning in a regularized-regression analysis are compared in an attempt to find the optimal amount of smoothing to be imposed on fMRI data in determining an empirical hemodynamic response function. We found that the noise autocorrelation structure can be improved by tuning the regularization parameter but the whitening-based criterion provides too much smoothing when compared to cross-validation. Chapter~5 illustrates that the smoothing techniques proposed in Chapter 4 can be useful in the issue of correlating behavioral and hemodynamic characteristics. Specifically, Chapter 5, based on the smoothing techniques from Chapter 4, seeks to correlate several parameters characterizing the hemodynamic response in Broca's area to behavioral measures in a naming task. In particular, a condition for independence between two routes of converting print to speech in a dual route cognitive model was verified in terms of hemodynamic parameters.