First, we investigate the Bouguer-Lambert-Beer (BLB) law as applied to the transmission of ultrashort pulses through water in the linear absorption regime. We present a linear theory for propagation of ultrashort laser pulses, and related experimental results are in excellent agreement with this theory. Thus we conclude that recent claims of the BLB law violations are inconsistent with the experimental data obtained by our group. Second, we study the dynamics of ultrashort pulses in a Lorentz medium and in water via the saddle point method. It shows that the saddle point method is a more efficient and faster method than the direct integration method to study one-dimensional pulse propagation over macroscopic distances (that is, distance comparable to the wavelength) in a general dielectric medium. Comments are also made about the exponential attenuation of the generalized Sommerfeld and Brillouin precursors. By applying the saddle point method, we also determined that the pulse duration estimated by the group velocity dispersion (GVD) approximation is within 2% of the value computed with the actual refractive index for a propagation distance of 6 m in water.

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Functional materials with controllable phase transitions have been widely used in devices for information storage (e.g. hard-disk, CD-ROM, memory) and energy storage (e.g. battery, shape memory alloy). One of the important issues to design such materials is to realize the desirable phase transition processes, in which atomistic simulation can be used for the prediction of materials properties. The accuracy of the prediction is largely dependent on searching the true value of the transition rate, which is determined by the minimum energy barrier between stable states, i.e. the saddle point on a potential energy surface (PES). Although a number of methods that search for saddle points on a PES have been developed, they intend to locate only one saddle point with the maximum energy along the transition path at a time. In addition, they do not consider the input uncertainty associated with the calculation of potential energy. To overcome the limitations, in this dissertation, new saddle point search methods are developed to provide a global view of energy landscape with improved efficiency and robustness. First, a concurrent search algorithm for multiple phase transition pathways is developed. The algorithm is able to search multiple local minima and saddle points simultaneously without prior knowledge of initial and final stable configurations. A new representation of transition paths based on parametric Bézier curves is introduced. A curve subdivision scheme is developed to dynamically locate all the intermediate local minima and saddle points along the transition path. Second, a curve swarm search algorithm is developed to exhaustively locate the local minima and saddle points within a region concurrently. The algorithm is based on the flocking of multiple groups of curves. A collective potential model is built to simulate the communication activities among curves. Third, a hybrid saddle-point search method using stochastic kriging models is developed to improve the efficiency of the search algorithm as well as to incorporate model-form uncertainty and numerical errors associated with density functional theory calculation. These algorithms are demonstrated by predicting the hydrogen diffusion process in FeTiH and body-centered iron Fe8H systems.

This dissertation considers the composite saddle-point (CSP) problem which is motivated by real-world applications in the areas of machine learning and image processing. Two new accelerated algorithms for solving composite saddle-point problems are introduced. Due to the two-block structure of the CSP problem, it can be solved by any algorithm belonging to the block-decomposition hybrid proximal extragradient (BD-HPE) framework. The framework consists of a family of inexact proximal point methods for solving a general two-block structured monotone inclusion problem which, at every iteration, solves two prox sub-inclusions according to a certain relative error criterion. By exploiting the fact that the two prox sub-inclusions in the context of the CSP problem are equivalent to two composite convex programs, the first part of this dissertation proposes a new instance of the BD-HPE framework that approximately solves them using an accelerated gradient method. It is shown that this new instance has better iteration-complexity than the previous ones. The second part of this dissertation introduces a new algorithm for solving a special class of CSP problems. The new algorithm is a special instance of the hybrid proximal extragradient (HPE) framework in which a Nesterov's accelerated variant is used to approximately solve the prox subproblems. One of the advantages of the this method is that it works for any constant choice of proximal stepsize. Moreover, a suitable choice of the latter stepsize yields a method with the best known (accelerated inner) iteration complexity for the aforementioned class of saddle-point problems. Experiment results on both synthetic CSP problems and real-world problems show that the two method significantly outperform several state-of-the-art algorithms.

In this thesis uniform asymptotic approximations of integrals are discussed. In order to derive these approximations, two well-known methods are used i.e., the saddle point method and the Bleistein method. To start with this, examples are given to demonstrate these two methods and a general idea of how to approach these techniques. The asymptotics of the hypergeometric functions with large parameters are discussed i.e., 2F1 (a + e1λ, b + e2λ c + e3λ ; z)where ej = 0,±1, j = 1, 2, 3 as |λ|→ ∞, which are valid in large regions of the complex z-plane, where a, b and c are fixed. The saddle point method is applied where the saddle point gives a dominant contributions to the integral representations of the hypergeometric functions and Bleistein’s method is adopted to obtain the uniform asymptotic approximations of some cases where the coalescence takes place between the critical points of the integrals. As a special case, the uniform asymptotic approximation of the hypergeometric function where the third parameter is large, is obtained. A new method to estimate the remainder term in the Bleistein method is proposed which is created to deal with new type of integrals in which the usual methods for the remainder estimates fail. Finally, using the asymptotic property of the hypergeometric function when the third parameter is large, the uniform asymptotic approximation of the monic Meixner Sobolev polynomials Sn(x) as n → ∞ , is obtained in terms of Airy functions. The asymptotic approximations for the location of the zeros of these polynomials are also discussed. As a limit case, a new asymptotic approximation for the large zeros of the classical Meixner polynomials is provided.

<p>We investigate mathematical tools, Edgeworth series expansion and the saddle point method, which are approximation techniques that help us to estimate the distribution function for the standardized mean of independent identical distributed random variables where we will take into consideration the lattice case. Later on we will describe one important application for these mathematical tools where game developing companies can use them to reduce the amount of time needed to satisfy their standard requests before they approve any game</p>

We investigate mathematical tools, Edgeworth series expansion and the saddle point method, which are approximation techniques that help us to estimate the distribution function for the standardized mean of independent identical distributed random variables where we will take into consideration the lattice case. Later on we will describe one important application for these mathematical tools where game developing companies can use them to reduce the amount of time needed to satisfy their standard requests before they approve any game

The aggregate interference distribution in cognitive radio networks is studied in a rigorous and analytical way using the popular Poisson point process model. While a number of results are available for this model for non-cognitive radio networks, cognitive radio networks present extra levels of difficulties for the analysis, mainly due to the exclusion region around the primary receiver, which are typically addressed via various ad-hoc approximations (e.g., based on the interference cumulants) or via the large-deviation analysis. Unlike the previous studies, we do not use here ad-hoc approximations but rather obtain the asymptotic interference distribution in a systematic and rigorous way, which also has a guaranteed level of accuracy at the distribution tail. This is in contrast to the large deviation analysis, which provides only the (exponential) order of scaling but not the outage probability itself. Unlike the cumulant-based analysis, our approach provides a guaranteed level of accuracy at the distribution tail. Additionally, our analysis provides a number of novel insights. In particular, we demonstrate that there is a critical transition point below which the outage probability decays only polynomially but above which it decays super-exponentially. This provides a solid analytical foundation to the earlier empirical observations in the literature and also reveals what are the typical ways outage events occur in different regimes. The analysis is further extended to include interference cancelation and fading (from a broad class of distributions). The outage probability is shown to scale down exponentially in the number of canceled nearest interferers in the below-critical region and does not change significantly in the above-critical one. The proposed asymptotic expressions are shown to be accurate in the non-asymptotic regimes as well.

In this thesis, we address the statistics of mutual information of amplify-and-forward (AF) multiple-input multiple-output (MIMO) two-hop relay channels, where the source terminal (ST), relay terminal (RT) and destination terminal (DT) are equipped with a number of correlated antennas and there is a line-of-sight (LOS) component (Rician fading) of each link. To the best of our knowledge, deriving analytical expressions for the statistics of mutual information of the relay channel is difficult and still unsolvable. To circumvent the mathematical difficulties, we consider this problem under the large-system regimen in which the numbers of antennas at the transmitter and receiver go to infinity with a fixed ratio. In the large-system regimen, this thesis has made the following contributions: 1) We get the mean and the variance of the mutual information of the concerned relay channel. 2) We show that the mutual information distribution converges to the Gaussian distribution. The analytical results are derived by mean of two powerful tools developed in the context of theoretical physics: emph{saddle-point approximation} and emph{superanalysis}. The derived analytical results are very general and can degenerate to several previously results as special cases. From a degenerated case, we realize that the previous result by Wagner {em et al.} cite{Wag-08} is wrong and thus we provide the corrected result. Finally, Simulation results demonstrate that even for a moderate number of antennas at each end, the proposed analytical results provide undistinguishable results as those obtained by Monte-Carlo results.

We study computational theory and methods for finding multiple unstable solutions (corresponding to saddle points) to three types of nonlinear variational elliptic systems: cooperative, noncooperative, and Hamiltonian. We first propose a new Lorthogonal selection in a product Hilbert space so that a solution manifold can be defined. Then, we establish, respectively, a local characterization for saddle points of finite Morse index and of infinite Morse index. Based on these characterizations, two methods, called the local min-orthogonal method and the local min-max-orthogonal method, are developed and applied to solve those three types of elliptic systems for multiple solutions. Under suitable assumptions, a subsequence convergence result is established for each method. Numerical experiments for different types of model problems are carried out, showing that both methods are very reliable and efficient in computing coexisting saddle points or saddle points of infinite Morse index. We also analyze the instability of saddle points in both single and product Hilbert spaces. In particular, we establish several estimates of the Morse index of both coexisting and non-coexisting saddle points via the local min-orthogonal method developed and propose a local instability index to measure the local instability of both degenerate and nondegenerate saddle points. Finally, we suggest two extensions of an L-orthogonal selection for future research so that multiple solutions to more general elliptic systems such as nonvariational elliptic systems may also be found in a stable way.