Ng, Chit Yu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 89-90). / Abstracts in English and Chinese.

Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) play an important role in many areas of engineering and applied sciences such as atmospheric sciences, mechanical and aerospace engineering, geosciences, and finance. Equilibrium statistics and long-time solutions of these equations are pertinent to many applications. Typically, these models contain several uncertain parameters which need to be propagated in order to facilitate uncertainty quantification and prediction. Correspondingly, in this thesis, we propose a generalization of the Polynomial Chaos (PC) framework for long-time solutions of SDEs and SPDEs driven by Brownian motion forcing. Polynomial chaos expansions (PCEs) allow us to propagate uncertainties in the coefficients of these equations to the statistics of their solutions. Their main advantages are: (i) they replace stochastic equations by systems of deterministic equations; and (ii) they provide fast convergence. Their main challenge is that the computational cost becomes prohibitive when the dimension of the parameters modeling the stochasticity is even moderately large. In particular, for equations with Brownian motion forcing, the long-time simulation by PC-based methods is notoriously difficult as the dimension of stochastic variables increases with time. With the goal in mind to deliver computationally efficient numerical algorithms for stochastic equations in the long time, our main strategy is to leverage the intrinsic sparsity in the dynamics by identifying the influential random parameters and construct spectral approximations to the solutions in terms of those relevant variables. Once this strategy is employed dynamically in time, using online constructions, approximations can retain their sparsity and accuracy; even for long times. To this end, exploiting Markov property of Brownian motion, we present a restart procedure that allows PCEs to expand the solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future Brownian motion. In case of SPDEs, the Karhunen-Loeve expansion (KLE) is applied at each restart to select the influential variables and keep the dimensionality minimal. Using frequent restarts and low degree polynomials, the algorithms are able to capture long-time solutions accurately. We will also introduce, using the same principles, a similar algorithm based on a stochastic collocation method for the solutions of SDEs. We apply the methods to the numerical simulation of linear and nonlinear SDEs, and stochastic Burgers and Navier-Stokes equations with white noise forcing. Our methods also allow us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations, and show that the algorithms compare favorably with standard Monte Carlo methods in terms of accuracy and computational times. To demonstrate the efficiency of the algorithms for long-time simulations, we compute invariant measures of the solutions when they exist.

No description available.

In the study of differential equations there are two fundamental questions: is there a solution? and what is it? One of the most elegant ways to prove that an equation has a solution is to pose it as a fixed point problem, that is, to find a function f such that x is a solution if and only if f (x) = x. Results from fixed point theory can then be employed to show that f has a fixed point. However, the results of fixed point theory are often nonconstructive: they guarantee that a fixed point exists but do not help in finding the fixed point. Thus these methods tend to answer the first question, but not the second. One such result is Schauder’s fixed point theorem. This theorem is broadly applicable in proving the existence of solutions to differential equations, including the Navier-Stokes equations under certain conditions. Recently a semi-constructive proof of Schauder’s theorem was developed in Rizzolo and Su (2007). In this thesis we go through the construction in detail and show how it can be used to search for multiple solutions. We then apply the method to a selection of differential equations.

Thesis (M.S.)--University of Nevada, Reno, 2005. / "May 2005." Includes bibliographical references (leaves 27-28). Online version available on the World Wide Web.

Thesis (Ph. D.)--Oregon State University, 1970. / Typescript (photocopy). Includes bibliographical references (leaves 90-91). Also available on the World Wide Web.

Thesis (Ph.D.)--Baylor University, 2005. / Includes bibliographical references (p. 54-58).

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (February 28, 2007) Vita. Includes bibliographical references.

In this thesis, it is proposed to examine the difference equation:(z-h) ∆-hW(z) = A(z)W(z)(1) where W(z) is a vector with two components,∆-hW(h) = W(z) – W(z-h)/h(2)Here, A(z) is a 2x2 matrix, whose elements admit factorial series representations:A (z) = R + Σ∞s=0 As+1S!/z(z+h) ••• (z+sh)(3)R and As+l are square matrices of order two and independent of z. We also assume that eigen values of R do not differ by an integer. We hope to show that if (3) converges in some half plane, then (1) will have a fundamental matris solution of the form: W(z) = S(z)ZR where S(z) is a 2x2 matrix, whose elements have convergent factorial representation in some half plane.

University of Macau / Faculty of Science and Technology / Department of Mathematics