Thesis (M. Phil.)--University of Hong Kong, 1992.

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2009. / Includes bibliographical references (p. 83-86).

Thesis (M.S)--Washington State University, December 2006. / Includes bibliographical references (p. 49-53).

Thesis (Ph. D.)--Washington State University, August 2006. / Includes bibliographical references (p. 67-68).

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2005. / Includes bibliographical references (leaves 52-55). Also available in electronic version.

Following a stochastic approach, this thesis presents a numerical framework for elastostatics of random media. Firstly, after a mathematically rigorous investigation of the popular white noise model in an engineering context, the smooth spatial stochastic dependence between material properties is identified as a fundamental feature of practical random media. Based on the recognition of the probabilistic essence of practical random media and driven by engineering simulation requirements, a comprehensive random medium model, namely elementary random media (ERM), is consequently defined and its macro-scale properties including stationarity, smoothness and principles for material measurements are systematically explored. Moreover, an explicit representation scheme, namely the Fourier-Karhunen-Loeve (F-K-L) representation, is developed for the general elastic tensor of ERM by combining the spectral representation theory of wide-sense stationary stochastic fields and the standard dimensionality reduction technology of principal component analysis. Then, based on the concept of ERM and the F-K-L representation for its random elastic tensor, the stochastic partial differential equations regarding elastostatics of random media are formulated and further discretized, in a similar fashion as for the standard finite element method, to obtain a stochastic system of linear algebraic equations. For the solution of the resulting stochastic linear algebraic system, two different numerical techniques, i.e. the joint diagonalization solution strategy and the directed Monte Carlo simulation strategy, are developed. Original contributions include the theoretical analysis of practical random medium modelling, establishment of the ERM model and its F-K-L representation, and development of the numerical solvers for the stochastic linear algebraic system. In particular, for computational challenges arising from the proposed framework, two novel numerical algorithms are developed: (a) a quadrature algorithm for multidimensional oscillatory functions, which reduces the computational cost of the F-K-L representation by up to several orders of magnitude; and (b) a Jacobi-like joint diagonalization solution method for relatively small mesh structures, which can effectively solve the associated stochastic linear algebraic system with a large number of random variables.

This study gives a phenomenologically based stochastic model of space -time rainfall. Specifically, two random variables on the spatial rainfall, e.g. the cumulative rainfall within a season and the maximum cumulative rainfall per rainfall event within a season are considered. An approach is given to determine the cumulative distribution function (c.d.f.) of the cumulative rainfall per event, based on a particular random structure of space -time rainfall. Then the first two moments of the cumulative seasonal rainfall are derived based on a stochastic dependence between the cumulative rainfall per event and the number of rainfall events within a season. This stochastic dependence is important in the context of the spatial rainfall process. A theorem is then proved on the rate of convergence of the exact c.d.f. of the seasonal cumulative rainfall up to the ith year, i > 1, to its limiting c.d.f. Use of the limiting c.d.f. of the maximum cumulative rainfall per rainfall event up to the ith year within a season is given in the context of determination of the 'design rainfall'. Such information is useful in the design of hydraulic structures. Special mathematical applications of the general theory are developed from a combination of empirical and phenomenological based assumptions. A numerical application of this approach is demonstrated on the Atterbury watershed in the Southwestern United States.

Models governing stochasticity in various systems are typically calibrated from data, therefore are subject to statistical errors/uncertainties which can lead to inferior decision making. This thesis develops statistically and computationally efficient data-driven methods for problems in stochastic simulation and optimization to quantify and hedge impacts of these uncertainties. The first half of the thesis focuses on efficient methods for tackling input uncertainty which refers to the simulation output variability arising from the statistical noise in specifying the input models. Due to the convolution of the simulation noise and the input noise, existing bootstrap approaches consist of a two-layer sampling and typically require substantial simulation effort. Chapter 2 investigates a subsampling framework to reduce the required effort, by leveraging the form of the variance and its estimation error in terms of the data size and the sampling requirement in each layer. We show how the total required effort is reduced, and explicitly identify the procedural specifications in our framework that guarantee relative consistency in the estimation, and the corresponding optimal simulation budget allocations. In Chapter 3 we study an optimization-based approach to construct confidence intervals for simulation outputs under input uncertainty. This approach computes confidence bounds from simulation runs driven by probability weights defined on the data, which are obtained from solving optimization problems under suitably posited averaged divergence constraints. We illustrate how this approach offers benefits in computational efficiency and finite-sample performance compared to the bootstrap and the delta method. While resembling distributionally robust optimization, we explain the procedural design and develop tight statistical guarantees via a generalization of the empirical likelihood method. The second half develops uncertainty quantification techniques for certifying solution feasibility and optimality in data-driven optimization. Regarding optimality, Chapter 4 proposes a statistical method to estimate the optimality gap of a given solution for stochastic optimization as an assessment of the solution quality. Our approach is based on bootstrap aggregating, or bagging, resampled sample average approximation (SAA). We show how this approach leads to valid statistical confidence bounds for non-smooth optimization. We also demonstrate its statistical efficiency and stability that are especially desirable in limited-data situations. We present our theory that views SAA as a kernel in an infinite-order symmetric statistic. Regarding feasibility, Chapter 5 considers data-driven optimization under uncertain constraints, where solution feasibility is often ensured through a "safe" reformulation of the constraints, such that an obtained solution is guaranteed feasible for the oracle formulation with high confidence. Such approaches generally involve an implicit estimation of the whole feasible set that can scale rapidly with the problem dimension, in turn leading to over-conservative solutions. We investigate validation-based strategies to avoid set estimation by exploiting the intrinsic low dimensionality of the set of all possible solutions output from a given reformulation. We demonstrate how our obtained solutions satisfy statistical feasibility guarantees with light dimension dependence, and how they are asymptotically optimal and thus regarded as the least conservative with respect to the considered reformulation classes.

This thesis focuses primarily on generalizations and enhancements of the Wiener path integral (WPI) technique for stochastic response analysis and optimization of diverse nonlinear dynamic systems of engineering interest. Concisely, the WPI technique, which has proven to be a potent mathematical tool in theoretical physics, has been recently extended to address problems in stochastic engineering dynamics. Herein, the WPI technique has been significantly enhanced in terms of computational efficiency and versatility; these results are presented in Chapters 2-5. Specifically, in Chapter 2 a brief introduction to the standard WPI solution approach is outlined. In Chapter 3, a novel methodology is presented, which utilizes theoretical results from calculus of variations to extend the WPI for determining marginalized response PDFs of n-degree-of-freedom (n-DOF) nonlinear systems. The associated computational cost relates to the dimension of the PDF and is essentially independent from the dimension n of the system. In several commonly encountered cases, the aforementioned methodology improves the computational efficiency of the WPI by orders of magnitude, and exhibits a significant advantage over the commonly utilized Monte-Carlo-simulation (MCS). Moreover, in Chapter 4, an extension of the WPI technique is presented for addressing the challenge of determining the stochastic response of nonlinear dynamical systems under the presence of singularities in the diffusion matrix. The key idea behind this approach is to partition the original system into an underdetermined system of SDEs corresponding to a nonsingular diffusion matrix and an underdetermined system of homogeneous differential equations; the latter is treated as a dynamic constraint that allows for employing constrained variational/optimization solution methods. In Chapter 5, this approach is applied for the stochastic response analysis and optimization of electromechanical vibratory energy harvesters. Next, in Chapter 6, a technique from computational algebraic geometry has been developed, which is based on the concept of Gröbner basis and is capable of determining the entire solution set of systems of polynomial equations. This technique has been utilized to address diverse challenging problems in engineering mechanics. First, after formulating the WPI as a minimization problem, it is shown in Chapter 7 that the corresponding objective function is convex, and thus, convergence of numerical schemes to the global optimum is guaranteed. Second, in Chapter 8, the computational algebraic geometry technique has been applied to the challenging problem of determining nonlinear normal modes (NNMs) corresponding to multi-degree-of-freedom dynamical systems as defined in [1], and has been shown to yield improvements in accuracy compared to the standard treatment in the literature.

Ce travail de thèse est consacré à l'extension de la formule d'Itô au cas de chemins à variations bornées à valeurs dans l'espace des distributions tempérées composés par des processus réguliers au sens de Malliavin. On s'attache en particulier à faire des hypothèses de régularité minimales, ce qui donne accès à un certain nombre d'applications de notre principal résultat, en particulier à l'étude d'un problème variationnel. Le premier chapitre est consacré à des rappels de calcul de Malliavin. Le deuxième donne des résultats sur la topologie sur la classe de Schwartz et l'espace des distributions tempérées. Dans le troisième chapitre, on donne des conditions optimales sous lesquelles on peut définir la composition d'une distribution tempérée par une variable aléatoire et quelle est la régularité au sens de Malliavin de l'objet ainsi construit. Des techniques d'interpolation permettent d'obtenir des résultats pour des espaces fractionnaires. On donne également des résultats pour le cas où la distribution est elle-même stochastique. Ces résultats nous permettent d'écrire, au chapitre 4, une formule d'Ito faible s'appliquant sous des hypothèses beaucoup plus faibles que celles généralement proposées dans la littérature. On donne aussi une version anticipative et une formule de type Ito-Wentzell. On donne des résultats plus précis dans le cas où le processus auquel on applique notre formule est la solution d'une EDS simple et on applique ce résultat à l'étude de la régularité du temps local en dimension quelconque. Enfin le cinquième chapitre résout un problème variationnel simple en affaiblissant considérablement une hypothèse d'ellipticité faite par la plupart des auteurs. / This dissertation studies the extension of the Itô formula to the case of distibution-valued paths of bounded variation lifted by processes which are regular in the sense of Malliavin calculus. We make optimal hypotheses, which gives us access to many applications. The first chapter is a primer in Malliavin calculus. The second chapter provides useful results on the toplogy of the schwartz class and of the space of tempered distributions. in the third chapter, we give optimal conditions under which a tempered distribution may be composed by a random variable and we study the malliavin regularity of the object thus defined. Interpolation techniques give access to results in fractional spaces. We also give results for the case where the tempered distribution is itself stochastic. These results allow us to obtain, in chapter 4, a weak Itô formula under hypotheses which are much weaker than those usually made in the litterature. We also give an Itô-Wentzell and an anticipative version. In the case where the process to which the ito formula is applied is the solution to an SDE, we give a more precise result, which we use to study the reguarity of the multi-dimensional local time. Finally the fifth chapter solves a variational problem under hypotheses which are much weaker than the usual assumption of hypoellipticity