For more than 100 years, the Navier-Stokes equations and various linearizations have been used as a model to study fluid dynamics. Recently, attention has been directed toward studying the nonnormality of linearized problems and developing convergent numerical schemes for simulation of these sytems. Numerical schemes for optimal control problems often require additional properties that may not be necessary for simulation; these properties can be critical when studying nonnormal problems. This research is concerned with approximating infinite dimensional optimal control problems with nonnormal system operators. We examine three different finite element methods for a specific convection-diffusion equation and prove convergence of the infinitesimal generators. Additionally, for two of these schemes, we prove convergence of the associated feedback gains. We apply these three schemes to control problems and compare the performance of all three methods. / Ph. D.

Recent technological advances in fields like medicine and genomics have produced high-dimensional data sets and a challenge to correctly interpret experimental results. The Optimal Discovery Procedure (ODP) (Storey 2005) builds on the framework of Neyman-Pearson hypothesis testing to optimally test thousands of hypotheses simultaneously. The method relies on the assumption of normally distributed data; however, many applications of this method will violate this assumption. This thesis investigates the sensitivity of this method to detection of significant but nonnormal data. Overall, estimation of the ODP with the method described in this thesis is satisfactory, except when the nonnormal alternative distribution has high variance and expectation only one standard deviation away from the null distribution.

Regression analysis is very popular among researchers in various fields but almost all the researchers use the classical methods which assume that X is nonstochastic and the error is normally distributed. However, in real life problems, X is generally stochastic and error can be nonnormal. Maximum likelihood (ML) estimation technique which is known to have optimal features, is very problematic in situations when the distribution of X (marginal part) or error (conditional part) is nonnormal. Modified maximum likelihood (MML) technique which is asymptotically giving the estimators equivalent to the ML estimators, gives us the opportunity to conduct the estimation and the hypothesis testing procedures under nonnormal marginal and conditional distributions. In this study we show that MML estimators are highly efficient and robust. Moreover, the test statistics based on the MML estimators are much more powerful and robust compared to the test statistics based on least squares (LS) estimators which are mostly used in literature. Theoretically, MML estimators are asymptotically minimum variance bound (MVB) estimators but simulation results show that they are highly efficient even for small sample sizes. In this thesis, Weibull and Generalized Logistic distributions are used for illustration and the results given are based on these distributions. As a future study, MML technique can be utilized for other types of distributions and the procedures based on bivariate data can be extended to multivariate data.

Generalized linear stability theory is used to calculate the optimal initial conditions that result in transient amplification of the thermohaline circulation (THC) in a zonally-averaged single basin ocean model. The eigenmodes of the tangent linear model verify that the system is asymptotically stable but the nonnormality of the system permits the growth of perturbations for a finite period through the interference of nonorthogonal eigenmodes. It is found that the maximum amplification of the THC anomalies occurs after 6 years with both the thermally driven and salinity driven components playing major roles in the amplification process. The transient amplification of THC anomalies is due to the constructive and destructive interference of a large number of eigenmodes and the evolution over time is determined by how the interference pattern evolves. It is found that five of the most highly nonnormal eigenmodes are critical to the initial cancellation of the salinity and temperature contributions to the THC while 11 oscillating modes with decay timescales ranging from 2 to 6 years are the major contributors at the time of maximum amplification. This analysis demonstrates that the different dynamics of salinity and temperature anomalies allows the dramatic growth of perturbations to the THC on relatively short (interannual to decadal) timescales. In addition the ideas of generalized stability theory are used to calculate the stochastic optimals which are the spatial patterns of stochastic forcing that are most efficient at generating variance growth in the THC. It is found that the optimal stochastic forcing occurs at high latitudes and induces low-frequency THC variability by exciting the salinity-dominated modes of the THC. The first stochastic optimal is found to have its largest projection on the same five highly nonnormal eigenmodes found to be critical to the structure of the optimal initial conditions. The model’s response to stochastic forcing is not controlled by the least damped eigenmodes of the tangent linear model but rather by the linear interference of these highly nonnormal eigenmodes. The process of pseudoresonance suggests that the nonnormal eigenmodes are excited and sustained by stochastically induced perturbations which in turn lead to maximum THC variance. Finally, it was shown that the addition of wind stress did not have a large impact on the nonnormal dynamics of the linearised system. Adding wind allowed the value of the vertical diffusivity to be reduced to achieve the same maximum linearised THC amplitude as was used in the case with no wind stress.

Generalized linear stability theory is used to calculate the optimal initial conditions that result in transient amplification of the thermohaline circulation (THC) in a zonally-averaged single basin ocean model. The eigenmodes of the tangent linear model verify that the system is asymptotically stable but the nonnormality of the system permits the growth of perturbations for a finite period through the interference of nonorthogonal eigenmodes. It is found that the maximum amplification of the THC anomalies occurs after 6 years with both the thermally driven and salinity driven components playing major roles in the amplification process. The transient amplification of THC anomalies is due to the constructive and destructive interference of a large number of eigenmodes and the evolution over time is determined by how the interference pattern evolves. It is found that five of the most highly nonnormal eigenmodes are critical to the initial cancellation of the salinity and temperature contributions to the THC while 11 oscillating modes with decay timescales ranging from 2 to 6 years are the major contributors at the time of maximum amplification. This analysis demonstrates that the different dynamics of salinity and temperature anomalies allows the dramatic growth of perturbations to the THC on relatively short (interannual to decadal) timescales. In addition the ideas of generalized stability theory are used to calculate the stochastic optimals which are the spatial patterns of stochastic forcing that are most efficient at generating variance growth in the THC. It is found that the optimal stochastic forcing occurs at high latitudes and induces low-frequency THC variability by exciting the salinity-dominated modes of the THC. The first stochastic optimal is found to have its largest projection on the same five highly nonnormal eigenmodes found to be critical to the structure of the optimal initial conditions. The model’s response to stochastic forcing is not controlled by the least damped eigenmodes of the tangent linear model but rather by the linear interference of these highly nonnormal eigenmodes. The process of pseudoresonance suggests that the nonnormal eigenmodes are excited and sustained by stochastically induced perturbations which in turn lead to maximum THC variance. Finally, it was shown that the addition of wind stress did not have a large impact on the nonnormal dynamics of the linearised system. Adding wind allowed the value of the vertical diffusivity to be reduced to achieve the same maximum linearised THC amplitude as was used in the case with no wind stress.