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<div>The dissertation consists of two research topics.</div><div><br></div><div>The first research direction is to study stability of functional and geometric inequalities. Stability problem is to estimate the deficit of a functional or geometric inequality in terms of the distance from the class of optimizers or a functional that identifies the optimizers. In particular, we investigate the logarithmic Sobolev inequality, the Beckner-Hirschman inequality (the entropic uncertainty principle), and isoperimetric type inequalities for the expected lifetime of Brownian motion. </div><div><br></div><div>The second topic of the thesis is a stochastic representation of fractional integrals and nonlocal operators. We extend the Hardy-Littlewood-Sobolev inequality to symmetric Markov semigroups. To this end, we construct a stochastic representation of the fractional integral using the background radiation process. The inequality follows from a new inequality for the fractional Littlewood-Paley square function. We also prove the Hardy-Stein identity for non-symmetric pure jump Levy processes and the L^p boundedness of a certain class of Fourier multiplier operators arising from non-symmetric pure jump Levy processes. The proof is based on Ito's formula for general jump processes and the symmetrization of Levy processes. <br></div>

<p>This thesis studies infinite dimensional Gaussian measures on Banach spaces. Let $\mu_0$ be a centered Gaussian measure on Banach space $\mathcal B$, and $\mu^\ast$ is a measure equivalent to $\mu_0$. We are interested in approximating, in sense of relative entropy (or KL divergence) the quantity $\frac{d\mu^z}{d\mu^\ast}$ where $\mu^z$ is a mean shift measure of $\mu_0$ by an element $z$ in the so-called ``Cameron-Martin" space $\mathcal H_{\mu_0}$. That is, we want to find the information projection</p> <p><br></p> <p>$$\inf_{z\in \mathcal H_{\mu_0}} D_{KL}(\mu^z||\mu_0)=\inf_{z\in \mathcal H_{\mu_0}} E_{\mu^z} \left(\log \left(\frac{d\mu^z}{d\mu^\ast}\right)\right).$$</p> <p><br></p> <p>We relate this information projection to a mode computation, to an ``open loop" control problem, and to a variational formulation leading to an Euler-Lagrange equation. Furthermore, we use this relationship to establish a kind of Feynman-Kac theorem for systems of ordinary differential equations. We demonstrate that the solution to a system of second order linear ordinary differential equations is the mode of a diffusion, analogous to the result of Feynman-Kac for parabolic partial differential equations. </p>

published_or_final_version / Civil and Structural Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / Industrial and Manufacturing Systems Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

Uncertainty propagation in engineering mechanics and dynamics is a highly challenging problem that requires development of analytical/numerical techniques for determining the stochastic response of complex engineering systems. In this regard, although Monte Carlo simulation (MCS) has been the most versatile technique for addressing the above problem, it can become computationally daunting when faced with high-dimensional systems or with computing very low probability events. Thus, there is a demand for pursuing more computationally efficient methodologies. Recently, a Wiener path integral (WPI) technique, whose origins can be found in theoretical physics, has been developed in the field of engineering dynamics for determining the response transition probability density function (PDF) of nonlinear oscillators subject to non-white, non-Gaussian and non-stationary excitation processes. In the present work, the Wiener path integral technique is enhanced, extended and generalized with respect to three main aspects; namely, versatility, computational efficiency and accuracy. Specifically, the need for increasingly sophisticated modeling of excitations has led recently to the utilization of fractional calculus, which can be construed as a generalization of classical calculus. Motivated by the above developments, the WPI technique is extended herein to account for stochastic excitations modeled via fractional-order filters. To this aim, relying on a variational formulation and on the most probable path approximation yields a deterministic fractional boundary value problem to be solved numerically for obtaining the oscillator joint response PDF. Further, appropriate multi-dimensional bases are constructed for approximating, in a computationally efficient manner, the non-stationary joint response PDF. In this regard, two distinct approaches are pursued. The first employs expansions based on Kronecker products of bases (e.g., wavelets), while the second utilizes representations based on positive definite functions. Next, the localization capabilities of the WPI technique are exploited for determining PDF points in the joint space-time domain to be used for evaluating the expansion coefficients at a relatively low computational cost. Subsequently, compressive sampling procedures are employed in conjunction with group sparsity concepts and appropriate optimization algorithms for decreasing even further the associated computational cost. It is shown that the herein developed enhancement renders the technique capable of treating readily relatively high-dimensional stochastic systems. More importantly, it is shown that this enhancement in computational efficiency becomes more prevalent as the number of stochastic dimensions increases; thus, rendering the herein proposed sparse representation approach indispensable, especially for high-dimensional systems. Next, a quadratic approximation of the WPI is developed for enhancing the accuracy degree of the technique. Concisely, following a functional series expansion, higher-order terms are accounted for, which is equivalent to considering not only the most probable path but also fluctuations around it. These fluctuations are incorporated into a state-dependent factor by which the exponential part of each PDF value is multiplied. This localization of the state-dependent factor yields superior accuracy as compared to the standard most probable path WPI approximation where the factor is constant and state-invariant. An additional advantage relates to efficient structural reliability assessment, and in particular, to direct estimation of low probability events (e.g., failure probabilities), without possessing the complete transition PDF. Overall, the developments in this thesis render the WPI technique a potent tool for determining, in a reliable manner and with a minimal computational cost, the stochastic response of nonlinear oscillators subject to an extended range of excitation processes. Several numerical examples, pertaining to both nonlinear dynamical systems subject to external excitations and to a special class of engineering mechanics problems with stochastic media properties, are considered for demonstrating the reliability of the developed techniques. In all cases, the degree of accuracy and the computational efficiency exhibited are assessed by comparisons with pertinent MCS data.

Wai-tung Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 81-83). / Chapter CHAPTER 1 --- OVERVIEW OF CONSTRAINTED ESTIMATION OF STRUCTURAL EQUATION MODEL --- p.1 / Chapter CHAPTER 2 --- MULTISAMPLE ANALYSIS OF STRUCTURAL EQUATION MODELS WITH STOCHASTIC CONSTRAINTS --- p.4 / Chapter 2.1 --- The Basic Model --- p.4 / Chapter 2.2 --- Bayesian Approach to Nuisance Parameters --- p.5 / Chapter 2.3 --- Estimation and Algorithm --- p.8 / Chapter 2.4 --- Asymptotic Properties of the Bayesian Estimate --- p.11 / Chapter CHAPTER 3 --- MULTISAMPLE ANALYSIS OF STRUCTURAL EQUATION MODELS WITH EXACT AND STOCHASTIC CONSTRAINTS --- p.17 / Chapter 3.1 --- The Basic Model --- p.17 / Chapter 3.2 --- Bayesian Approach to Nuisance Parameters and Estimation Procedures --- p.18 / Chapter 3.3 --- Asymptotic Properties of the Bayesian Estimate --- p.20 / Chapter CHAPTER 4 --- SIMULATION STUDIES AND NUMERICAL EXAMPLE --- p.24 / Chapter 4.1 --- Simulation Study for Identified Models with Stochastic Constraints --- p.24 / Chapter 4.2 --- Simulation Study for Non-identified Models with Stochastic Constraints --- p.29 / Chapter 4.3 --- Numerical Example with Exact and Stochastic Constraints --- p.32 / Chapter CHAPTER 5 --- DISCUSSION AND CONCLUSION --- p.34 / APPENDICES --- p.36 / TABLES --- p.66 / REFERENCES --- p.81

Stochastic analysis on fractals is, as one might expect, a subfield of analysis on fractals. An intuitive starting point is to observe that on many fractals, one can define diffusion processes whose law is in some sense invariant with respect to the symmetries and self-similarities of the fractal. These can be interpreted as fractal-valued counterparts of standard Brownian motion on Rd. One can study these diffusions directly, for example by computing heat kernel and hitting time estimates. On the other hand, by associating the infinitesimal generator of the fractal-valued diffusion with the Laplacian on Rd, it is possible to pose stochastic partial differential equations on the fractal such as the stochastic heat equation and stochastic wave equation. In this thesis we investigate a variety of questions concerning the properties of diffusions on fractals and the parabolic and hyperbolic SPDEs associated with them. Key results include an extension of Kolmogorov's continuity theorem to stochastic processes indexed by fractals, and existence and uniqueness of solutions to parabolic SPDEs on fractals with Lipschitz data.

Chen, Jian. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 81-85). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Literature Review --- p.4 / Chapter 1.2.1 --- Vehicle Routing Problem with Stochastic Demands --- p.5 / Chapter 1.2.2 --- Vehicle Routing Problem with Stochastic Travel Times --- p.8 / Chapter 1.3 --- The Vehicle Routing Problem with Time Windows and Stochastic Travel Times --- p.10 / Chapter 2 --- Notations and Formulations --- p.12 / Chapter 2.1 --- Problem Definitions --- p.12 / Chapter 2.2 --- A Two-Index Stochastic Programming Model --- p.14 / Chapter 2.3 --- The Second Stage Problem --- p.17 / Chapter 3 --- The Scheduling Problem --- p.20 / Chapter 3.1 --- The Overtime Cost Problem --- p.22 / Chapter 3.2 --- The Waiting and Late Cost Problem --- p.27 / Chapter 3.3 --- The Algorithm --- p.37 / Chapter 4 --- The Integer L-Shaped Method --- p.40 / Chapter 4.1 --- Linearization of the Objective Function --- p.41 / Chapter 4.2 --- Handling the Constraints --- p.42 / Chapter 4.3 --- Branching --- p.44 / Chapter 4.4 --- The Algorithm --- p.44 / Chapter 5 --- Feasibility Cuts --- p.47 / Chapter 5.1 --- Connected Component Methods --- p.48 / Chapter 5.2 --- Shrinking Method --- p.49 / Chapter 6 --- Optimality Cuts --- p.52 / Chapter 6.1 --- Lower Bound I for the EOT Cost --- p.53 / Chapter 6.2 --- Lower Bounds II and III for the EOT Cost --- p.56 / Chapter 6.3 --- Lower Bound IV for the EWL Cost --- p.57 / Chapter 6.4 --- Lower Bound V for Partial Routes --- p.61 / Chapter 6.5 --- Adding Optimality Cuts --- p.66 / Chapter 7 --- Numerical Experiments --- p.70 / Chapter 7.1 --- Effectiveness in Separating the Rounded Capacity Inequalities --- p.71 / Chapter 7.2 --- Effectiveness of the Lower Bounds --- p.72 / Chapter 7.3 --- Performance of the L-shaped Method --- p.74 / Chapter 8 --- Conclusion and Future Research --- p.79 / Bibliography --- p.81 / Chapter A --- Generation of Test Instances --- p.86