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<p>We are interested in backward-in-time solution techniques for evolutionary PDE problems arising in fluid mechanics. In addition to their intrinsic interest, such techniques have applications in recently proposed retrograde data assimilation. As our model system we consider the terminal value problem for the Kuramoto-Sivashinsky equation in a l D periodic domain. The Kuramoto-Sivashinsky equation, proposed as a model for interfacial and combustion phenomena, is often also adopted as a toy model for hydrodynamic turbulence because of its multiscale and chaotic dynamics. Such backward problems are typical examples of ill-posed problems, where any disturbances are amplified exponentially during the backward march. Hence, regularization is required to solve such problems efficiently in practice. We consider regularization approaches in which the original ill-posed problem is approximated with a less ill-posed problem, which is achieved by adding a regularization term to the original equation. While such techniques are relatively well-understood for linear problems, it is still unclear what effect these techniques may have in the nonlinear setting. In addition to considering regularization terms with fixed magnitudes, we also explore a novel approach in which these magnitudes are adapted dynamically using simple concepts from the Control Theory.</p> / Thesis / Master of Science (MSc)

The culture in which people work and dwell is instrumental in shaping their sense of self. The decision to migrate from the country of one's heritage culture may result in the modification of self-identity in order to accommodate new experiences within the host culture. For working professionals, such modifications may be manifested in a number of different domains, including attitudes, behaviors, values, and sense of culture. When considering America's diverse workforce and the pressures placed upon people to be competitive, educated, and reasonably assimilated, the process of acculturation must also be addressed. This process is best understood when heritage and mainstream cultures are viewed independently. Formally educated foreign-born African women were the focus of this research. The purpose was to increase understanding of the employment status of African women with respect to identity style and acculturation strategies. Two hundred thirty-eight (238) women in the Metropolitan Washington D.C. Area were surveyed with respect to acculturation, identity style and employment status. The Vancouver Index of Acculturation was used to measure the heritage and mainstream dimensions of acculturation. The Identity Style Inventory was used to measure aspects of individual identity. Differences were found for the acculturation dimension of mainstream acculturation, which was observed to be higher for employed subjects for three of the four analyses used for employment status. No statistically significant differences were found for any of the identity style measures due to employment status, with one exception. The underemployed group of women may have been characterized by an identity orientation based on family and friends. If these women appear to experience problems associated with acculturation and identity, they may require more time to learn about the U.S. culture. These women represent a heterogeneous group with an amazing diversity in terms of language, culture, religion, and national backgrounds. This research suggests that their goal of securing or maintaining a professional career in the United States while residing in a major metropolitan area does not require assimilating into the U.S. culture at the expense of their own culture. Although, given that the majority of these women plan to remain in the United States as permanent residents, learning as much as possible about their host culture could perhaps benefit them with respect to employment. / Ph. D.

Variational inverse problems integrate computational simulations of physical phenomena with physical measurements in an informational feedback control system. Control parameters of the computational model are optimized such that the simulation results fit the physical measurements.The solution procedure is computationally expensive since it involves running the simulation computer model (the emph{forward model}) and the associated emph {adjoint model} multiple times. In practice, our knowledge of the underlying physics is incomplete and hence the associated computer model is laden with emph {model errors}. Similarly, it is not possible to measure the physical quantities exactly and hence the measurements are associated with emph {data errors}. The errors in data and model adversely affect the inference solutions. This work develops methods to address the challenges posed by the computational costs and by the impact of data and model errors in solving variational inverse problems. Variational inverse problems of interest here are formulated as optimization problems constrained by partial differential equations (PDEs). The solution process requires multiple evaluations of the constraints, therefore multiple solutions of the associated PDE. To alleviate the computational costs we develop a parallel in time discretization algorithm based on a nonlinear optimization approach. Like in the emph{parareal} approach, the time interval is partitioned into subintervals, and local time integrations are carried out in parallel. Solution continuity equations across interval boundaries are added as constraints. All the computational steps - forward solutions, gradients, and Hessian-vector products - involve only ideally parallel computations and therefore are highly scalable. This work develops a systematic mathematical framework to compute the impact of data and model errors on the solution to the variational inverse problems. The computational algorithm makes use of first and second order adjoints and provides an a-posteriori error estimate for a quantity of interest defined on the inverse solution (i.e., an aspect of the inverse solution). We illustrate the estimation algorithm on a shallow water model and on the Weather Research and Forecast model. Presence of outliers in measurement data is common, and this negatively impacts the solution to variational inverse problems. The traditional approach, where the inverse problem is formulated as a minimization problem in $L_2$ norm, is especially sensitive to large data errors. To alleviate the impact of data outliers we propose to use robust norms such as the $L_1$ and Huber norm in data assimilation. This work develops a systematic mathematical framework to perform three and four dimensional variational data assimilation using $L_1$ and Huber norms. The power of this approach is demonstrated by solving data assimilation problems where measurements contain outliers. / Ph. D.