Thesis (Ph. D.)--University of Hong Kong, 1988.

Mantle convection is the principal control on the thermal and geological evolution of the earth, including the motion of the tectonic plates, which in turn influences earthquakes, tsunamis, and volcanic eruptions. This system is governed by the equations for balance of mass, momentum, and energy for a viscous incompressible non-Newtonian fluid. Taking present-day temperatures as given, the time dependence can be neglected, eliminating the energy equation. In this case, the physics of the mantle are modeled by the Stokes equation with nonlinear rheology (the so-called forward problem). This dissertation focuses on solving the mantle convection inverse problem governed by the nonlinear Stokes forward problem with full nonlinear rheology, with an infinite-dimensional adjoint-based inversion method. The need for inverse methods in the study of mantle convection stems from the fact that the constitutive parameters are subject to uncertainty. Inversion for nonlinear rheology parameters presents considerable difficulties, which are explored in this dissertation. A spectral analysis of the Hessian operator is performed to investigate the ill-posedness of the inverse problem. The general form of the numerical eigenvalues is found to agree with that of the theoretically-derived ones (based on a model 1D Stokes problem), both of which collapse rapidly to zero, suggesting a high degree of ill-posedness. This motivates the use in this thesis of regularizations that are of Tikhonov type (favoring smooth viscosity) and total variation type (favoring piecewise-smooth viscosity). In addition, the eigenfunctions of the Hessian indicate that increasingly smaller length scales of viscosity are increasingly less observable, and that resolution decays with depth. The wide range of spatial scales of interest (varying from 1 km scale associated with plate boundaries to 10⁴ km global scales) prompts the use of adaptive mesh refinement in a parallel framework. The results show that both higher levels of nonlinearity and larger orders of magnitude of variation in the viscosity cause the inverse problem to be more ill-conditioned, increasing the difficulty of solving the inverse problem. Despite the severe ill-posedness of the inverse problem, stemming from the small number of observations compared to large number of degrees of freedom of the viscosity parameters, with the correct regularization weight and the right type of regularization, it is possible to reasonably infer information about the viscosity of the mantle, particularly in shallow regions. A number of 2D and 3D inversions are shown to demonstrate these capabilities. / text

This dissertation focuses on inverse problems for partial differential equations with multiscale coefficients in which the goal is to determine the coefficients in the equation using solution data. Such problems pose a huge computational challenge, in particular when the coefficients are of multiscale form. When faced with balancing computational cost with accuracy, most approaches only deal with models of large scale behavior and, for example, account for microscopic processes by using effective or empirical equations of state on the continuum scale to simplify computations. Obtaining these models often results in the loss of the desired fine scale details. In this thesis we introduce ways to overcome this issue using a multiscale approach. The first part of the thesis establishes the close relation between computational grids in multiscale modeling and sampling strategies developed in information theory. The theory developed is based on the mathematical analysis of multiscale functions of the type that are studied in averaging and homogenization theory and in multiscale modeling. Typical examples are two-scale functions f (x, x/[epsilon]), (0 < [epsilon] ≪ 1) that are periodic in the second variable. We prove that under certain band limiting conditions these multiscale functions can be uniquely and stably recovered from nonuniform samples of optimal rate. In the second part, we present a new multiscale approach for inverse homogenization problems. We prove that in certain cases where the specific form of the multiscale coefficients is known a priori, imposing an additional constraint of a microscale parametrization results in a well-posed inverse problem. The mathematical analysis is based on homogenization theory for partial differential equations and classical theory of inverse problems. The numerical analysis involves the design of multiscale methods, such as the heterogeneous multiscale method (HMM). The use of HMM solvers for the forward model has unveiled theoretical and numerical results for microscale parameter recovery, including applications to inverse problems arising in exploration seismology and medical imaging. / text

This dissertation addresses forward and inverse modeling of Rayleigh waves for near surface investigation. Results were obtained by imaging abandoned mine openings using Rayleigh waves in the laterally inhomogeneous medium. The efficient staggered grid stencil method to solve elastic wave equations using 2-D finite difference technique is presented. This numerical scheme is used to conduct a series of parametric studies on the propagation of Rayleigh waves. The first parametric study was conducted on a flat layered model of increasing and decreasing velocity with depth. A Rayleigh waves dispersion curve is found to be sensitive on a layer’s depth up to half of the minimum wavelength of Rayleigh waves. The phase velocity in the dispersion curve of Rayleigh waves is inversely and directly proportional to the frequency, depending on velocity increase or decrease with depth. The parametric study was carried out by introducing dipping layers in the model with increasing dip. The front (near the shot point) and back (at the end of receiver line) shot records are different if the subsurface contains dip. Dispersion is observed in near offset for down dip and in the far offset for up dip, computed from front and back shots respectively. Finally, a parametric study looked at subsurface anomalies with different shapes and sizes as well as their material properties. A Rayleigh wave is sensitive to very high material contrast and very low material contrast of the anomaly from it surrounding medium. The presence of a low material contrast anomaly from the surrounding medium traps the energy which causes reverberation. A Rayleigh wave is sensitive to an anomaly which is placed within the depth between one-third to half of minimum wavelength of Rayleigh wave from the surface. In order to resolve lateral heterogeneity, a new method is developed in this research which allows localization of the multichannel record in different panels. The dispersion curve of Rayleigh waves is computed in each panel using the slant stack technique. On the basis of parametric studies, an innovative inversion algorithm has been developed to minimize the error norm; ”the sum of the squares of the difference of reference and model dispersion curves” in an iterative way using a Very Fast Simulated Re-annealing (VFSR) technique.

A double semigroup is a set equipped with two associative binary operations satisfying the middle-four interchange law. A double inverse semigroup is a double semigroup in which both operations are inverse semigroup operations. It is shown by Kock (2007) that all double inverse semigroups must be commutative. In this thesis, we define the notion of a double inductive groupoid which admits both a construction of double inverse semigroups given any double inductive groupoid, and vice-versa. These constructions are functorial and induce an isomorphism of categories between the category of double inductive groupoids with inductive functors and double inverse semigroups with double semigroup homomorphisms. By a further investigation of double inverse semigroups, we are able to show that the two operations of any double inverse semigroups must coincide and thus double inverse semigroups are commutative inverse semigroups.

Generalized inverses of matrices are of great importance in the resolution of linear systems and have been extensively studied by many researchers. A collection of some results on generalized inverses of matrices over commutative rings has been provided by K. P. S. Bhaskara Rao (2002). In this thesis, we consider constructing algorithms for finding generalized inverses and generalizing the results collected in Rao's book to the non-commutative case. We first construct an algorithm by using the greatest common divisor to find a generalized inverse of a given matrix over a commutative Euclidean domain. We then build an algorithm for finding a generalized inverse of a matrix over a non-commutative Euclidean domain by using the one-sided greatest common divisor and the least common left multiple. Finally, we explore properties of various generalized inverses including the Moore-Penrose inverse, the group inverse and the Drazin inverse in the non-commutative case.

Volumetric modulated arc therapy delivers highly conformal radiotherapy treatments to cancer patients in a continuous arc whilst dynamically varying the multi-leaf collimator (MLC) position, dose rate and rotational angular velocity. The present master’s thesis seeks to develop a better understanding of delivering treatment in this manner, ranging from progressive resolution inverse optimisation, class solutions, clinical application and the ability of dose calculation algorithms to model such a complex modality. A progressive resolution based class solution for inverse planning has been developed, outlining contouring, field set-up and optimisation. This class solution was then applied to 10 prostate patients and subjected to an inter-comparative planning study with static gantry intensity-modulated radiotherapy. The results of this justification study showed the presented class solution produces plans that are generally and directly comparable with previously published data. Following this result, the class solution was applied to a previously uninvestigated clinical site (treatment of prostate bed following radical prostatectomy) in an effort to solve persistent clinical problems involving target volumes and dose escalation. The results of this secondary study provisionally showed the feasibility of treating prostate beds with rotational intensity-modulated techniques whilst maintaining the integrity of the target volumes and escalating the delivered dose. The potential for improving the accuracy of the dose calculation analytic anisotropic algorithm for volumetric modulated plans was also investigated, through configuration of two independent algorithms containing beam data taken with either the linac jaws or MLCs defining the field. The two algorithms were inter-compared in virtual water phantoms and against physical verification measurements. The configuration process has shown to be sensitive to depth dose data but not beam profiles. Furthermore, the two algorithms show no significant difference and therefore it is recommended that beam be taken with the jaws defining the field.

This thesis is in the area of ultrasound tomography, which is a non-destructive imaging method that attempts to create quantitative images of the acoustical properties of an object of interest (OI). Specifically, three quantitative images per OI are created in this thesis, two of which correspond to the complex compressibility profile of the OI, and the other corresponds to its density profile. The focus of this thesis is on the development of an appropriate two-dimensional inverse scattering algorithm to create these quantitative images. The core of this algorithm is the Born iterative method that is used in conjunction with a fast and efficient method of moments forward solver, a Krylov subspace regularization technique, and a balancing method. This inversion algorithm is capable of simultaneous inversion of multiple-frequency data, and can handle a large imaging domain. This algorithm is finally tested against synthetic and measured data.

Thesis (M.A.) in Mathematics--University of Maine, 2008. / Includes vita. Includes bibliographical references (leaf 34).

Thesis (Master)--İzmir Institute of Technology, İzmir, 2004. / Includes bibliographical references (leaves. 82-84).