DSpace Collection:http://dspace.cityu.edu.hk:80/handle/2031/7742014-04-24T18:25:33Z2014-04-24T18:25:33ZSome nonlinear problems on manifolds arising from conformal geometryZhu, Huan (祝歡)http://dspace.cityu.edu.hk:80/handle/2031/72132014-04-22T02:55:13Z2012-01-01T00:00:00ZTitle: Some nonlinear problems on manifolds arising from conformal geometry
Authors: Zhu, Huan (祝歡)
Abstract: This thesis focuses on the the fully nonlinear Yamabe-type problem on manifolds with
boundary of admissible negative curvature, which is an important problem in Geometric Analysis and has been extensively studied.
This problem is essentially an elliptic problem with Neumann boundary condition. Firstly, we use Maximum Principle and perturbation method to derive C0 bound.
Then, we creatively utilize the tubular neighborhood coordinates to derive C1 and C2
boundary estimates. When the C2 estimate is established, this problem turns out to be
uniformly elliptic. So, by the theory of Lieberman and Trudinger’s and the concave
condition, the C2,α estimate can be obtained. Furthermore, using the standard Schauder estimate, we can get C4,α estimate. Finally, the existence result is gained through
method of continuity and the uniqueness of this problem is derived by Maximum Principle.
Notes: CityU Call Number: QA649 .Z45 2012; iv, 83 p. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [79]-83)2012-01-01T00:00:00ZSome existence and stability problems of the Boltzmann equationsWang, Ying (王穎)http://dspace.cityu.edu.hk:80/handle/2031/72042014-04-22T02:54:50Z2012-01-01T00:00:00ZTitle: Some existence and stability problems of the Boltzmann equations
Authors: Wang, Ying (王穎)
Abstract: This thesis is concerned with some existence and stability theories on the Boltzmann
equations under certain conditions. The research for the Boltzmann equations
has been one of the most important and challenging field in Partial Differential
Equations because of its rich physical background and practical applications.
Thus, it is very important to reveal the properties of the Boltzmann equations
mathematically.
Fluid passing through porous media (e.g. the underground water passing
through the earth) can be modeled by the Euler equations with frictional force
which have been extensively studied. Since the Boltzmann equations are closely
related to the equations of gas dynamics, we investigate in the first part of this
thesis, the Boltzmann equation with frictional force when the external force is
proportional to the macroscopic velocity. We discuss the Cauchy problem of
the Boltzmann equations with frictional force mainly for the hard sphere model.
We give not only the existence theory but also the optimal time convergence
rates of the solutions to the Boltzmann equations with frictional force towards
equilibrium.
In the second part, we consider the specular re
flective boundary problem for
the one-dimensional Boltzmann equations with soft potentials. It is shown that
the solution converges to a global Maxwellian under certain initial conditions.
Note that the result for hard potentials case has already been established, thus
our result here is a good supplement of this problem.
Notes: CityU Call Number: QC173.4.P67 W37 2012; v, 74 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [71]-74)2012-01-01T00:00:00ZBoundary integral equation methods for computational photonicsLu, Wangtao (魯汪濤)http://dspace.cityu.edu.hk:80/handle/2031/71912014-04-22T02:54:17Z2012-01-01T00:00:00ZTitle: Boundary integral equation methods for computational photonics
Authors: Lu, Wangtao (魯汪濤)
Abstract: Optical waveguides are structures that guide the propagation of light. They are
the fundamental components in communications systems and integrated optical
circuits. In recent years, many optical waveguides with complicated structures
have appeared. As a special class of optical waveguides, photonic crystal fibers
(PCFs) have been extensively studied because of their many unique properties
which are not available in traditional waveguides. The propagation of light in a
PCF is strongly controlled by the geometry of its cross section. Periodic structures,
such as diffraction gratings and photonic crystals (PhCs) are important
optical components that can be used to control and manipulate light. Accurate
and efficient numerical methods are essential in the analysis, design and optimization
of optical waveguides and periodic structures.
For a given optical waveguide or PCF, numerical methods that discretize
the cross section of the structure give rise to linear matrix eigenvalue problems.
The discretization can be obtained by using the finite difference method, the finite
element method (FEM), the multi-domain pseudospectral method, etc. However,
for PCFs with many holes and complicated geometries and for general optical
waveguides with high-index contrast, sharp corners and complex micro-structures,
the resulting matrices can be very large and the matrix eigenvalue problem can
only be solved by iterative methods and the accuracy may be limited. A better
approach is to formulate a nonlinear eigenvalue problem of which the resulting
matrix is much smaller. Numerical methods using the nonlinear approach include
the film mode matching method, the multipole method and the boundary integral
equation (BIE) method. The film mode matching method is quite successful, but it is only applicable to optical waveguides with vertical and horizontal refractive
index discontinuities. The multipole method is accurate for PCFs with wellseparated
and circular inclusions, but it cannot be easily extended to other optical
waveguides.
For diffraction gratings, existing numerical methods include general-purpose
methods such as the finite-difference time-domain (FDTD) method and the FEM,
and more special methods such as the analytic modal method, numerical modal
methods, the BIE methods, etc. Although FDTD and FEM are extremely versatile,
they are typically less efficient than the special methods. Analytic and
numerical modal methods require that the structure consists of uniform layers.
For gratings with high index-contrast and sharp corners in their profiles, all
modal methods converge slowly and may even fail to converge, due to the possible
field singularity at the corners. And for two dimensional (2D) PhCs with circular
cylinders, existing numerical methods such as the FDTD method, the FEM
method, the multipole method, the scattering matrix method and the Dirichletto-
Neumann map method are effective. However, if the cylinder in each unit
cell contains corners, the above methods still suffer from a considerable loss of
accuracy in the presence of the field singularity at corners.
In this thesis, high order boundary integral equation methods are developed
for analyzing optical waveguides including PCFs, diffraction gratings and photonic
crystals of arbitrary unit cells. The methods rely on a standard Nyström
method for discretizing integral operators and they do not require analytic properties
of the electromagnetic field (which are singular) at the corners. For PCFs
with smooth interfaces, we develop a new high order BIE mode solver. The
method solves two functions on the interfaces and is more efficient than existing
BIE methods. The key step is to use the kernel-splitting technique for discretizing
the hyper-singular boundary integral operators. For optical waveguides with high
index-contrast and sharp corners, a new full-vectorial waveguide mode solver is
developed based on a new formulation of boundary integral equations and the socalled
Neumann-to-Dirichlet (NtD) maps for sub-domains of constant refractive index. The method uses the normal derivatives of the two transverse magnetic
field components as the basic unknown functions, and it offers higher order of
accuracy where the order depends on a parameter used in a graded mesh for
handling the corners. For diffraction gratings, we present a high order BIE-NtD
method which is an improved-version of a BIE-NtD method in earlier works. The
improvements include a revised formulation that is more stable numerically, and
more accurate methods for computing tangential derivatives along material interfaces
and for matching boundary conditions with the homogeneous top and
bottom regions. For 2D PhCs of arbitrary unit cells, a new BIE-NtD method is
used to calculate the NtD map for each unit cell. We study two basic problems
encountered in the analysis of 2D PhCs. A projection technique is used for further
reducing the size of the reduced NtD map for each unit cell, and it makes
our method more effective.
Notes: CityU Call Number: TK8304 .L8 2012; vii, 109 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [94]-109)2012-01-01T00:00:00ZKernel-based algorithms in statistical learning theoryFeng, Yunlong (馮雲龍)http://dspace.cityu.edu.hk:80/handle/2031/71842014-04-22T02:54:00Z2012-01-01T00:00:00ZTitle: Kernel-based algorithms in statistical learning theory
Authors: Feng, Yunlong (馮雲龍)
Abstract: Statistical learning theory plays an important role in some subjects of various areas
such as science, engineering, and finance. As a research field, it provides theoretical
foundations for algorithms in machine learning. Generally speaking, statistical learning
aims at learning function features or data structures from observations. By making
use of kernel methods, data can be mapped into a high dimensional feature space, in
which various methods can be employed to find relations. In this thesis, we mainly focus
on several different kernel-based learning algorithms in the framework of statistical
learning theory.
Firstly, we study the q-norm regularized least-squares regression with dependent
samples. We conduct error analysis of the least-squares regularized regression algorithm
when the sampling sequence is weakly dependent satisfying an exponentially
decaying α-mixing condition and when the regularizer takes the q-penalty with
0 < q ≤ 2. We use a covering number argument and derive learning rates in terms
of the α-mixing decay, an approximation condition, and the capacity of balls of the
reproducing kernel Hilbert space.
Secondly, we concentrate on the coefficient-based regularized regression problem.
The lq-regularized least-squares regression problem with 1 ≤ q ≤ 2 and data dependent
hypothesis spaces is addressed. Algorithms in data dependent hypothesis spaces
perform well with the property of flexibility. We conduct a unified error analysis by a
stepping stone technique. An empirical covering number technique is also employed in our study to improve the sample error. Compared with existing results, we make a few
improvements: First, we obtain a significantly sharper learning rate of type O(m−θ)
with θ arbitrarily close to 1 under reasonable conditions, which is regarded as the best
learning rate in learning theory. Second, our results cover the case q = 1, which is
novel. Finally, our results hold under very general conditions.
Finally, we address the pairwise ranking problem via a kernel-based learning approach.
Various settings for the pair-wise ranking problem are compared. We adopt
a preference-based two-stage setting while the empirical data is generated in a different
manner. For the first learning stage, we learn a preference function by reducing
ranking to classification. Learning results concerning the learnability of the ranking
rule we learned are presented as in classification. For the second stage, we present an
optimization algorithm to produce a scoring function that might be used to yield an
ordering.
Notes: CityU Call Number: Q325.5 .F46 2012; vi, 70 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [60]-70)2012-01-01T00:00:00Z