Thesis (M.S.)--University of Texas at Dallas, 2008. / Includes vita. Includes bibliographical references (leaf 54)

Thesis (Ph.D.)--Drexel University, 2007. / Includes abstract and vita. Includes bibliographical references (leaves 97-108).

Thesis (M.E.E.)--University of Delaware, 2007. / Principal faculty advisor: Kenneth E. Barner, Dept. of Electrical and Computer Engineering. Includes bibliographical references.

We study the convergence of a nonlinear approximation method introduced in the engineering literature for the numerical solution of a high-dimensional Fokker--Planck equation featuring in Navier--Stokes--Fokker--Planck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le~Bris, Leli\`evre and Maday (Const. Approx. 30: 621--651, 2009) in the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, where they exploited the connection of the approximation method with the greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173--187, 1996). We extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le~Bris, Leli\`evre and Maday to the technically more complicated situation of the elliptic Fokker--Planck equation, where the role of the Laplace operator is played out by a high-dimensional Ornstein--Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker--Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space $\mathsf{D} = D_1 \times \dotsm \times D_N$ contained in $\mathbb{R}^{N d}$, where each set $D_i$, $i=1, \dotsc, N$, is a bounded open ball in $\mathbb{R}^d$, $d = 2, 3$. We exploit detailed information on the spectral properties and elliptic regularity of the Ornstein--Uhlenbeck operator to give conditions on the true solution of the Fokker--Planck equation which guarantee certain rates of convergence of the greedy algorithms. We extend the analysis to discretized versions of the greedy algorithms.

This thesis addresses orchestration of servers amongst multiple co-hosted internet services such as e-Banking, e-Auction and e-Retail in hosting centres. The hosting paradigm entails levying fees for hosting third party internet services on servers at guaranteed levels of service performance. The orchestration of server ensemble in hosting centres is considered in the context of maximising the hosting centre's revenue over a lengthy time horizon. The inspiration for the server orchestration approach proposed in this thesis is drawn from nature and generally classed as swarm intelligence, specifically, sophisticated collective behaviour of social insects borne out of primitive interactions amongst members of the group to solve problems beyond the capability of individual members. Consequently, the approach is self-organising, adaptive and robust. A new scheme for server ensemble orchestration is introduced in this thesis. This scheme exploits the many similarities between server orchestration in an internet hosting centre and forager allocation in a honeybee (Apis mellifera) colony. The scheme mimics the way a honeybee colony distributes foragers amongst flower patches to maximise nectar influx, to orchestrate servers amongst hosted internet services to maximise revenue. The scheme is extended by further exploiting inherent feedback loops within the colony to introduce self-tuning and energy-aware server ensemble orchestration. In order to evaluate the new server ensemble orchestration scheme, a collection of server ensemble orchestration methods is developed, including a classical technique that relies on past history to make time varying orchestration decisions and two theoretical techniques that omnisciently make optimal time varying orchestration decisions or an optimal static orchestration decision based on complete knowledge of the future. The efficacy of the new biomimetic scheme is assessed in terms of adaptiveness and versatility. The performance study uses representative classes of internet traffic stream behaviour, service user's behaviour, demand intensity, multiple services co-hosting as well as differentiated hosting fee schedule. The biomimetic orchestration scheme is compared with the classical and the theoretical optimal orchestration techniques in terms of revenue stream. This study reveals that the new server ensemble orchestration approach is adaptive in a widely varying external internet environments. The study also highlights the versatility of the biomimetic approach over the classical technique. The self-tuning scheme improves on the original performance. The energy-aware scheme is able to conserve significant energy with minimal revenue performance degradation. The simulation results also indicate that the new scheme is competitive or better than classical and static methods.